SOLSTICE: AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS. Volume V, Number 1. Summer, 1994. \hsize = 6.5 true in %THE FIRST 14 LINES ARE TYPESETTING CODE. %------------------------------------------------------------ %SOLSTlCE is typeset, using TeX, for the %reader to download including mathematical notation. %------------------------------------------------------------ %BACK ISSUES open anonymous FTP host um.cc.umich.edu %account IEVG (do not type the percent sign in any of the %following instructions). After you are in the system, type %cd IEVG and then type on the next line, ls. Then type %get filename (substitute a name from the directory). 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(MTS) equipment. \setpointsize{12}{9}{8} %same as previous comment line; set font for 12 point type. \parskip=3pt \baselineskip=14 pt \mathsurround=1pt \headline = {\ifnum\pageno=1 \hfil \else {\ifodd\pageno\righthead \else\lefthead\fi}\fi} \def\righthead{\sl\hfil SOLSTICE } \def\lefthead{\sl Summer, 1994 \hfil} \def\ref{\noindent\hang} \font\big = cmbx17 %this may cause problems in some installations--replace %if it does with a different font. \font\tn = cmr10 \font\nn = cmr9 \font\bn = cmbx9 \font\sn = cmsl9 \font\ee = cmr8 \font\be = cmbx8 \font\se = cmsl8 %The code has been kept simple to facilitate reading as e-mail %---------------------------------------------------------------- %---------------------------------------------------------------- \centerline{\big SOLSTICE:} \vskip.5cm \centerline{\bf AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS} \vskip5cm \centerline{\bf SUMMER, 1994} \vskip12cm \centerline{\bf Volume V, Number 1} \smallskip \centerline{\bf Institute of Mathematical Geography} \vskip.1cm \centerline{\bf Ann Arbor, Michigan} \vfill\eject %---------------------------------------------------------------- %---------------------------------------------------------------- \hrule \smallskip \centerline{\bf SOLSTICE} \line{Founding Editor--in--Chief: {\bf Sandra Lach Arlinghaus} \hfil} \line{Institute of Mathematical Geography and University of Michigan \hfil} \smallskip \centerline{\bf EDITORIAL BOARD} \smallskip \line{{\bf Geography} \hfil} \line{{\bf Michael Goodchild}, University of California, Santa Barbara. \hfil} \line{{\bf Daniel A. Griffith}, Syracuse University. \hfil} \line{{\bf Jonathan D. Mayer}, University of Washington; joint appointment in School of Medicine.\hfil} \line{{\bf John D. Nystuen}, University of Michigan.\hfil} \smallskip \line{{\bf Mathematics} \hfil} \line{{\bf William C. Arlinghaus}, Lawrence Technological University. \hfil} \line{{\bf Neal Brand}, University of North Texas. \hfil} \line{{\bf Kenneth H. Rosen}, A. T. \& T. Bell Laboratories. \hfil} \smallskip \line{{\bf Engineering Applications} \hfil} \line{{\bf William D. Drake}, University of Michigan, \hfil} \smallskip \line{{\bf Education} \hfil} \line{{\bf Frederick L. Goodman}, University of Michigan, \hfil} \smallskip \line{{\bf Business} \hfil} \line{{\bf Robert F. Austin, Ph.D.} \hfil} \line{President, Austin Communications Education Services \hfil} \smallskip \hrule \smallskip The purpose of {\sl Solstice\/} is to promote interaction between geography and mathematics. Articles in which elements of one discipline are used to shed light on the other are particularly sought. Also welcome, are original contributions that are purely geographical or purely mathematical. These may be prefaced (by editor or author) with commentary suggesting directions that might lead toward the desired interaction. Individuals wishing to submit articles, either short or full-- length, as well as contributions for regular features, should send them, in triplicate, directly to the Editor--in--Chief. Contributed articles will be refereed by geographers and/or mathematicians. Invited articles will be screened by suitable members of the editorial board. IMaGe is open to having authors suggest, and furnish material for, new regular features. The opinions expressed are those of the authors, alone, and the authors alone are responsible for the accuracy of the facts in the articles. \smallskip \noindent {\bf Send all correspondence to:} Sandra Arlinghaus, Institute of Mathematical Geography, 2790 Briarcliff, Ann Arbor MI 48105. Solstice@um.cc.umich.edu; SArhaus@umich.edu \smallskip Suggested form for citation. If standard referencing to the hardcopy in the IMaGe Monograph Series is not used (although we suggest that reference to that hardcopy be included along with reference to the e-mailed copy from which the hard copy is produced), then we suggest the following format for citation of the electronic copy. Article, author, publisher (IMaGe) -- all the usual--plus a notation as to the time marked electronically, by the process of transmission, at the top of the recipients copy. Note when it was sent from Ann Arbor (date and time to the second) and when you received it (date and time to the second) and the field characters covered by the article (for example FC=21345 to FC=37462). This document is produced using the typesetting program, {\TeX}, of Donald Knuth and the American Mathematical Society. Notation in the electronic file is in accordance with that of Knuth's {\sl The {\TeX}book}. The program is downloaded for hard copy for on The University of Michigan's Xerox 9700 laser-- printing Xerox machine, using IMaGe's commercial account with that University. Unless otherwise noted, all regular ``features" are written by the Editor--in--Chief. \smallskip {\nn Upon final acceptance, authors will work with IMaGe to get manuscripts into a format well--suited to the requirements of {\sl Solstice\/}. Typically, this would mean that authors would submit a clean ASCII file of the manuscript, as well as hard copy, figures, and so forth (in camera--ready form). Depending on the nature of the document and on the changing technology used to produce {\sl Solstice\/}, there may be other requirements as well. Currently, the text is typeset using {\TeX}; in that way, mathematical formul{\ae} can be transmitted as ASCII files and downloaded faithfully and printed out. The reader inexperienced in the use of {\TeX} should note that this is not a ``what--you--see--is--what--you--get" display; however, we hope that such readers find {\TeX} easier to learn after exposure to {\sl Solstice\/}'s e-files written using {\TeX}!} {\nn Copyright will be taken out in the name of the Institute of Mathematical Geography, and authors are required to transfer copyright to IMaGe as a condition of publication. There are no page charges; authors will be given permission to make reprints from the electronic file, or to have IMaGe make a single master reprint for a nominal fee dependent on manuscript length. Hard copy of {\sl Solstice\/} is available at a cost of \$15.95 per year (plus shipping and handling; hard copy is issued once yearly, in the Monograph series of the Institute of Mathematical Geography. Order directly from IMaGe. It is the desire of IMaGe to offer electronic copies to interested parties for free. Whether or not it will be feasible to continue distributing complimentary electronic files remains to be seen. Presently {\sl Solstice\/} is funded by IMaGe and by a generous donation of computer time from a member of the Editorial Board. Thank you for participating in this project focusing on environmentally-sensitive publishing.} \vskip.5cm \copyright Copyright, June, 1994 by the Institute of Mathematical Geography. All rights reserved. \vskip1cm {\bf ISBN: 1-877751-56-1} {\bf ISSN: 1059-5325} \vfill\eject %---------------------------------------------------------------- %---------------------------------------------------------------- \centerline{\bf TABLE OF CONTENT} \smallskip \noindent{\bf 1. WELCOME TO NEW READERS AND THANK YOU} \smallskip \noindent{\bf 2. PRESS CLIPPINGS---SUMMARY} \smallskip \noindent{\bf 3. REPRINTS} \smallskip \noindent{\bf Getting Infrastructure Built} \smallskip \noindent{\bf Virginia Ainslie and Jack Licate} \smallskip Transmitted as part 2 of 13. \smallskip Cleveland Infrastructure Team Shares Secrets of Success; What Difference Has the Partnership Approach Made? How Process Affects Products --- Moving Projects Faster Means Getting More Public Investment; How Can Local Communities Translate These Successes to Their Own Settings? \smallskip \noindent{\bf Center Here; Center There; Center, Center Everywhere} \smallskip \noindent {\bf Frank E. Barmore}. \smallskip Transmitted as parts 3 and 4 of 13. \smallskip Reprinted from {\sl The Wisconsin Geographer\/}, Vol. 9, pp. 8-21, 1993. Publication of the Wisconsin Geographical Society, reprinted here with permission of that Society. \smallskip Abstract; Introduction; Definition of Geographic Center; Geographic Center of a Curved Surface; Geographic Center of Wisconsin; Geographic Center of the Conterminous United States; Geographic Center of the United States; Summary and Recommendations; Appendix A: Calculation of Wisconsin's Geographic Center; Appendix B: Calculation of the Geographical Center of the Conterminous United States; References \smallskip \noindent{\bf 4. ARTICLES} \smallskip \noindent{\bf Equal-Area Venn Diagrams of Two Circles: Their Use with Real-World Data} \smallskip \noindent{\bf Barton R. Burkhalter} \smallskip Transmitted as parts 5 and 6 of 13. \smallskip General Problem; Definition of the Two-Circle Problem; Analytic Strategy; Derivation of $B\%$ and $AB\%$ as a Function of $r_B$ and $d_{AB}$. \smallskip \noindent{\bf Los Angeles, 1994 --- A Spatial Scientific Study} \smallskip Transmitted as parts 7, 8, 9, 10, 11, 12 of 13 (with body of text in part 7 and supporting tables and computer program in five subsequent parts). \smallskip \noindent{\bf Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary, John D. Nystuen} \smallskip Los Angeles, 1994; Policy Implications; References. Tables and Complicated Figures. \smallskip \noindent{\bf 5. DOWNLOADING OF SOLSTICE} \smallskip \noindent{\bf 6. INDEX to Volumes I (1990), II (1991), III (1992), and IV (1993) of {\sl Solstice}.} \smallskip \noindent{\bf 7. OTHER PUBLICATIONS OF IMaGe } \vfill\eject %---------------------------------------------------------------- %---------------------------------------------------------------- \centerline{\bf 1. WELCOME TO NEW READERS AND THANK YOU} Welcome to new subscribers! We hope you enjoy participating in this means of journal distribution. Instructions for downloading the typesetting have been repeated in this issue, near the end. They are specific to the {\TeX} installation at The University of Michigan, but apparently they have been helpful in suggesting to others the sorts of commands that might be used on their own particular mainframe installation of {\TeX}. New subscribers might wish to note that the electronic files are typeset files---the mathematical notation will print out as typeset notation. For example, $$ \Sigma_{i=1}^n $$ when properly downloaded, will print out a typeset summation as $i$ goes from one to $n$, as a centered display on the page. Complex notation is no barrier to this form of journal production. \vskip.5cm Thanks much to subscribers who have offered input. Helpful suggestions are important in trying to keep abreast, at least somewhat, of the constantly changing electronic world. Some suggestions from readers have already been implemented; others are being worked on. Indeed, it is particularly helpful when the reader making the suggestion becomes actively involved in carrying it out. We hope you continue to enjoy {\sl Solstice\/}. \smallskip %--------------------------------------------------------------- %--------------------------------------------------------------- \centerline{\bf 2. PRESS CLIPPINGS---SUMMARY} \noindent Volume 72, Number 4, October 1993 issue of {\sl Papers in Regional Science: The Journal of the Regional Science Association\/} carried an article by Gunther Maier and Andreas Wildberger entitled ``Wide Area Computer Networks and Scholarly Communication in Regional Science." Maier and Wildberger noted that ``Only one journal in this directory can be considered to be related to Regional Science, {\sl Solstice: An Electronic Journal of Geography and Mathematics\/}." Beyond that, brief write-ups about {\sl Solstice\/} have appeared in the following publications: \noindent 1. {\bf Science}, ``Online Journals" Briefings. [by Joseph Palca] 29 November 1991. Vol. 254. \smallskip \noindent 2. {\bf Science News}, ``Math for all seasons" by Ivars Peterson, January 25, 1992, Vol. 141, No. 4. \smallskip \noindent 3. {\bf Newsletter of the Association of American Geographers}, June, 1992. \smallskip \noindent 4. {\bf American Mathematical Monthly}, ``Telegraphic Reviews" --- mentioned as ``one of the World's first electronic journals using {\TeX}," September, 1992. \smallskip \noindent 5. {\bf Harvard Technology Window}, 1993. \smallskip \noindent 6. {\bf Graduating Engineering Magazine}, 1993. \noindent 7. {\bf Earth Surface Processes and Landforms}, 18(9), 1993, p. 874. \noindent 8. {\bf On Internet}, 1994. If you have read about {\sl Solstice\/} elsewhere, please let us know the correct citations (and add to those above). Thanks. We are happy to share information with all and are delighted when others share with us, as well. \vfill\eject Publications of the Institute of Mathematical Geography have, in addition, been reviewed or noted in \smallskip 1. {\sl The Professional Geographer\/} published by the Association of American Geographers; \smallskip 2. The {\sl Urban Specialty Group Newsletter\/} of the Association of American Geographers; \smallskip 3. {\sl Mathematical Reviews\/} published by the American Mathematical Society; \smallskip 4. {\sl The American Mathematical Monthly\/} published by the Mathematical Association of America; \smallskip 5. {\sl Zentralblatt\/} fur Mathematik, Springer-Verlag, Berlin \smallskip 6. {\sl Mathematics Magazine\/}, published by the Mathematical Association of America. \smallskip 7. {\sl Newsletter\/} of the Association of American Geographer. \smallskip 8. {\sl Journal of The Regional Science Association\/}. \smallskip 9. {\sl Journal of the American Statistical Association\/}. \smallskip \vfill\eject \centerline{\bf 3. REPRINTS} \smallskip \centerline{ \bf Getting Infrastructure Built} \smallskip \smallskip \centerline{Virginia Ainslie} \centerline{Technical Liaison to Congress} \smallskip \centerline{Jack Licate (Ph.D., Geography)} \centerline{Director, Build Up Greater Cleveland Program and} \centerline{Director of Federal Programs for the Greater Cleveland Growth Association.} \smallskip \noindent Reprinted with permission from Land Development/Spring-Summer 1994. \smallskip The profitability and success of a development project often hinge on the timely completion of improvements to adjacent highways, bridges, sewers, and transit services. In recent years, complex environmental and construction requirements have increased the lead time and costs required for many infrastructure improvements. At the same time, the public funding needed to finance road widening, interchange and bridge construction, sewer improvements, and transit development has come under severe budgetary constraints at all levels of government. In northeast Ohio, public and private sector leaders have entered into a successful partnership to solve infrastructure development problems. The lessons learned from Cleveland's partnership can be readily translated to other communities. \noindent{\bf Cleveland Infrastructure Team Shares the Secrets of Success} Founded in 1983 and commonly referred to as ``BUGC", Build Up Greater Cleveland is a unique partnership that consists of elected and appointed officials from local, state, and federal governments as well as dedicated private sector executive volunteers from engineering, banking, investment, manufacturing, utility, accounting, and law firms. The development community has actively participated in all aspects of BUGC's activities since the program was founded. A team effort that was born of crisis but matured over a decade of wrenching economic upheaval, BUGC has earned national recognition for its ability to attract public financing needed for the improvement, repair, and construction of roads, bridges, and sewer, water, and transit facilities. The first secret of BUGC's success is its systematic strategy for gaining the commitment of public funding through coordinated and simultaneous advocacy efforts at the local, state, and federal levels. The strategy calls for the aggressive and persistent pursuit of a fair and equitable share of state and federal infrastructure investment. Central to the strategy is the involvement of elected officials. Specifically, these officials enacted legislation that maximizes the return of tax dollars to northeast Ohio. At the federal level, the formula used to divide highway and bridge funding has been amended in favor of Ohio and certain other states in each piece of major surface transportation legislation since 1987. The Ohio Congressional delegation's leadership for this effort was based, in large part, on technical assistance and networking support from BUGC. For their part, private sector volunteers help develop the data that form the basis for BUGC's ``fair share" advocacy efforts. Corporate expertise has been particularly useful in quantifying the public benefits of infrastructure investment. Private sector executives also play an active role in various task forces charged with solving problems and coordinating road and bridge repair work with utilities. In addition, both public and private sector members of BUGC are involved in long-term efforts to educate the public on the importance of infrastructure to the community and its economy. The second secret of BUGC's success is its focus on improving the process by which projects move from identified need to construction. Between 1988 and 1993, the greater Cleveland area posted a 166 percent increase in the number of completed road and bridge projects. This surge resulted largely from the adoption of new procedures that reduced project completion time by 44 percent. BUGC's approach to achieving performance to achieving performance has helped reshape Cleveland's skyline and, at the same time, contributed to a major renaissance in regional economic development. \noindent{\bf What Difference Has the Partnership Approach Made?} Over the last decade, BUGC's advocacy program has yielded more than one billion dollars in unprogrammed funds that the greater Cleveland area would not have otherwise received. Much of the credit for the funds goes to northeast Ohio's congressional delegation. The delegation orchestrated a multistate/multiyear coalition effort that has increased the return of Ohio's share of the federal gas tax from 61 cents on the dollar in 1981 to 90 cents on the dollar in 1993. Changes in the formulas used to distribute federal highway and bridge dollars have also brought more than 1.5 billion dollars in new funds to Ohio and more than two million dollars in new funds to greater Cleveland. At the state level, BUGC played a pivotal role in establishing the Ohio Public Works Commission and its program of needs-based funding for local capital assets. This effort has resulted in an estimated annual increase of 14.5 million dollars for county roads, bridges, and sewers. BUGC worked locally for legislation that increased motor vehicle license tag fees, which now generate more than 13 million dollars per year in road and bridge repair funds. BUGC has lobbied successfully at the federal and state levels for project-specific funding for infrastructure improvements essential to Tower City Center, a new baseball stadium and arena project, the Rock and Roll Hall of Fame, and many other major development projects. BUGC has also been a key player in the cleanup of the Cuyahoga River, the overhaul and rehabilitation of Cleveland's transit system, and the completion of the Interstate highway network in northeast Ohio. BUGC's efforts have generated 396 million dollars for road and bridge improvements, 260.7 million dollars for transit development, 396 million dollars for water projects, and 341 million dollars for sewer needs. BUGC Chairman James M. Delaney of Deloitte and Touche points out, ``We now are witnessing a shift in investment from rehabilitation of existing infrastructure to increased expenditures for new facilities, which support high impact economic development projects." With the increase in funding for necessary infrastructure repairs and improvements, it became clear to city and county engineers, the Ohio Department of Transportation, the Greater Cleveland Regional Transit Authority, and the Northeast Ohio Regional Sewer District that projects were moving too slowly from design to construction. The various agencies shared many problems, particularly the burden of meeting new and complex environmental requirements while processing more and larger projects with a limited number of professional staff. To meet this challenge, the several agencies worked with private sector executives to develop, test, and implement performance improvement measures. The payoff has been a dramatic increase in the agencies' ability to complete road and bridge projects at a much faster pace. Whenever possible, environmental impact and engineering tasks are performed simultaneously rather than sequentially. Scoping meetings that include public and private sector participants are conducted early in the project planning process. The meetings sort out which agency or entity will assume primary responsibility for each task and institute cooperative mechanisms to ensure that projects remain on schedule and that problems are addressed quickly. The application of value engineering techniques helps make certain that the right project is initiated for the right reasons within a time frame that makes sense for all involved. BUGC's recommendations for fast tracking highway projects have worked so well the Governor George Voinovich has encouraged the Ohio Department of Transportation and the Ohio Environmental Protection Agency to implement similar action on a statewide basis. \noindent {\bf How Process Affects Products --- Moving Projects Faster Means Getting More Public Investment} The fast-track approach has reduced costs, improved completion times, and helped finance infrastructure projects. Given that a great deal of federal and state financing for highway, bridge, sewer, and transit project s is distributed on a ``first come, first served" basis, it is hustle --- and the ability to keep the bureaucratic pipelines full of ready-to-go projects --- that determines where public money is spent. Most of the federal funding for highways and bridges is disbursed with time constraints; that is, a jurisdiction must spend federal funds within a certain number of years or lose its funding allotment to other states. Therefore, a state department of transportation must, for example, meet all federal and state planning and programming requirements while spending its allotment of federal funds within the stated time limit. Accordingly, cities and counties with ready-to-go projects consistently receive funding. On the other hand, communities that fail to put together ready-to-go projects are unable to attract anything close to their fair share of public investment. \noindent {\bf How Can Local Communities Translate These Successes to Their Own Settings?} \item{1.} Develop a public/private infrastructure partnership team. BUS-C will be pleased to provide you with advice and written material on creating a partnership and, in return, asks only for feedback on what actions you take and what works in your community. In the meantime, even if you have a partnership in place, consider the actions outlined below. \item{2.} Visit your metropolitan planning organization (MPO) to find out about the availability and requirements for state and federal funding. If you do not live in an urban area, visit the nearest field office of your state department of transportation. Planners and engineers at this and other agencies can assist you in determining whether federal participation is appropriate for a given project. If participation is appropriate, staff will tell you what steps are necessary to ensure federal financing. \item{\phantom{2.}} It is important to recognize that projects can usually be completed more quickly in the absence of assistance. The federal government conditions the receipt of funds on compliance with federal standards for planning, environmental, public participation, and programming actions. For large, expensive public works improvements, however, federal financing is typically essential. \vfill\eject \item{\phantom{2.}} If your project can be successfully undertaken with only state and local financing, your MPO will advise you accordingly. Your MPO can also greatly assist in guiding you through the federal and state funding processes and introducing you to the key players. \item{3.} Identify the most appropriate public agency or government sponsor for your project and secure that party's agreement to fund your project. Early on, talk to the head of the appropriate agency about what you need, when you need it, and the rationale for the project. Ask for and follow the agency head's advice on how to ``feed the agency." Identify the staff members who will be assigned to your project and get to know those individuals. In other words, find out who is responsible for your project, what they need from you, and when you need to complete key steps to ensure adherence to a mutually workable and realistic project schedule. \item{4.} Be sure that elected officials are familiar with and support your project. Find out which public bodies must sign off on your project and what specific actions are necessary. Visit the appropriate elected officials and describe how your project will contribute to their vision of and priorities for the community. Is a consent ordinance needed from city council? Must the MPO board of directors include your project in its short- and long-term plans and project lists? Does the state need to file project-related documents with the U. S. Department of Transportation or U. S. Environmental Protection Agency? Does the state legislature assign funds for projects such as yours? Elected officials can be of immense assistance in spurring timely action by public agencies, especially if the officials are involved in the process early and consider themselves stakeholders in the project. \item{5.} Assume 100 percent proactive responsibility for keeping your project moving. If a delay occurs, identify the reason. To the greatest extent possible, help the person who must resolve the problem get whatever he or she needs to move the project forward. Keep all involved parties informed of project progress, and alert key public agency staff to changes as soon as possible. \item{6.} Say thank you often and keep your word; deliver on your promises. Most staff at public works agencies labor under hiring freezes and have not seen a significant pay increase in years. At the same time, they are responsible for large numbers of projects and must comply with new and confusing regulations. They are answerable to a diverse set of interests. Let them know that you appreciate their efforts. \item{\phantom{6.}} Encourage public agency staff to let you know if a problem arises or if you can do something to help keep your project on track. If staff members ask you for information, drawings, or legal or other information, tell them when you will submit the requested materials to them. Make sure that you deliver what you promised when you promised. Make certain that any material is delivered in a form that most readily serves agency purposes. Confirm that the right person has in fact received your information. \item{\phantom{6.}} If the various steps look like a lot of work --- and for some projects they represent a full-time job --- remember that the Cleveland experience has proven to be well worth the effort. The actions described here have led to considerable success, but the process can be expanded and improved. We look forward to hearing from you about your experiences in building public/private partnerships. \vfill\eject To make comments or to request further information write or phone: Jack Licate, Director; Build Up Greater Cleveland; 200 Tower City Center; 50 Public Square; Cleveland, Ohio 44114; 216/621-3300 \vfill\eject \centerline{ \bf Center Here; Center There; Center, Center Everywhere!} \smallskip \smallskip \centerline{The Geographic Center of Wisconsin and the U.S.A.:} \centerline{Concepts, Comments, and Misconceptions} \smallskip \centerline{Frank E. Barmore} \centerline{Department of Physics} \centerline{University of Wisconsin --- La Crosse} \smallskip \noindent Reprinted with permission from the Wisconsin Geographical Society: {\sl The Wisconsin Geographer\/}, Volume 9, 1993, pp. 8-21. \smallskip \smallskip \noindent{\bf Abstract} Published locations of geographic centers are found to be inaccurate, inconsistently determined and in serious need of revision. The definition of geographic center is clarified. Methods of computation of two-dimensional distributions on curved surfaces are given. An accurate location for the center of Wisconsin is determined to be at latitude of $44^{\circ}38'04''$ N., longitude of $89^{\circ}42'35''$ W. The uncertainty in the geographic center of the United States is discussed. Recommendations for future further work are given. \noindent{\bf Introduction} For more than two thirds of a century the U.S. Geological Survey has published information about the area and geographic center of the various states and the United States (Douglas, 1923, 1930; Van Zandt, 1966, 1976; and pamphlets of the U.S. Geological Survey, 1967, 1991). These publications are careful to point out the uncertainty and limitation of the data and results. For example, Douglas (1923, p. 221) states that ``That exact position of the center of each State can not be determined from the data available, $\ldots $" and Van Zandt (1966, p. 265) states that ``There being no generally accepted definition of `geographic center' and no completely satisfactory method for determining it, a State or country may have as many geographic centers as there are definitions of the term." and, ``Because many factors, such as the curvature of the earth, large bodies of water, and irregular surfaces, affect the determination of geographic centers, the locality of the centers should be considered as approximations only." Since first published, the information (with minor exceptions) has not been revised. Some things are getting better. Adequate data are now available. There are satisfactory definitions. Computers and powerful software are now widely available. There are analytical means of taking into account the Earth's surface curvature. Large bodies of water are just as much a part of the whole as is the land and should be included. As a result, it is now possible to determine ``geographic centers" to high accuracy. This paper will discuss these points and their impact on the determination of the geographic centers of Wisconsin and the United States. \noindent{\bf Definition of Geographic Center} The lack of agreement on a definition of geographic center (center of area) is unfortunately true. Opinions range from despair of any suitable solution existing, expressed by Adams (1932), to enthusiasm over the existence of an infinity of centers, all equally valid (if not equally popular), outlined by Neft (1966, p. 21). I suggest that a reading of the literature will show that an intermediate view is widely held and a single definition of ``center" is agreed on: {\bf The center of any distribution of things is the average location of those things}. It corresponds to the ``center of gravity" or ``balance point" of the distribution. In Euclidean spaces the average location is most easily calculated by taking the weighted vector sum of the location vectors (vectors whose magnitude and direction are the distance and direction of the various things in the distribution) and dividing by the total weight or total population of the things. Such a center has the additional property that the sum of the squares of the distances between the center and the location of the various things in the distribution is minimum. This definition is equally suitable for distributions in one-, two- three-, or higher-dimensional Euclidean spaces. Almost a century ago, Hayford (1902) convincingly argued that the average location was the most appropriate center. D. I. Mendeleev (1907 and before) used formul{\ae} which may be derived from the ``balance point" concept (derived by his son I. D. Mendeleev) for finding the geographic center and population center of Russia. Deetz (1918, p. 57) states that the ``~`Geographic center of the United States' is here considered as a point analogous to the center of gravity of a spherical surface equally weighted (per unit area) and of the outline of the country, and hence it may be found by means similar to those employed to find the center of gravity." All six Geological Survey publications, cited in the Introduction, appeal to the ``balance point" concept. For more than a century the U. S. Bureau of the Census has used the concept of a ``center of gravity" or ``balance point" as defining the U.S.A. population center (Barmore, 1991). In spite of this long tradition, there are still dissenters. Kumler and Goodchild (1992, p.~ 278) recommend that the point of minimum aggregate travel (M.A.T.) is the best measure of center of population. I find it hard to accept some of their reasons for this recommendation. First, they say that when calculating the mean or average location, ``the points, or people, are effectively weighted proportionally to their distance from the center --- more distant people have greater influence on the location of the mean center than people nearby." But, it is {\bf location} that is being averaged (weighted by population), {\bf not people} being averaged (weighted by distance). Each individual has exactly the same weight in finding the average location. Second, they believe the M.A.T. ``point does have one flaw --- it is insensitive to radial movement: If a person moves 1,000 kilometers directly toward or away from the mat [M.A.T.], the point will not move; if that same person, however moves only a few kilometers in any other direction the [M.A.T.] point will move accordingly." And this shortcoming ``is the least severe" shortcoming of the various measures of center of population they discuss. I disagree. Are we to have preferred or elite directions? Shouldn't the center of a distribution be equally sensitive to the motion of its component parts in any direction? I suggest that the term, center, should be reserved for the average (arithmetic mean) location. Other statistical concepts that are found to be useful should be labeled with names (other than center) that are descriptive of what they represent. For example, ``the point of minimum aggregate travel," is just that; it should not be called the center. To do otherwise is to invite a return to the confusion that existed earlier in this century when the point of minimum aggregate travel, the center (or average) location and the median latitude (and/or longitude) of an area were often and incorrectly thought to be the same (Eells, 1930). \noindent{\bf Geographic Center of a Curved Surface} As mentioned in the Introduction, one difficulty that must be dealt with is the curvature of the Earth's surface. If the Earth's surface were flat, or if there existed a flat map projection which left area, distance, and direction undistorted, the determination of geographic center of portions of the Earth's surface would be much simplified. However, distributions on the Earth's curved surface are spread over a two-dimensional non-Euclidean space. Traditionally there have been two different ways of responding to this problem. One response is to find a higher dimension space that is Euclidean in which to embed the non-Euclidean space. Then the necessary calculations can be carried out using the familiar Euclidean geometry. Thus, one can embed the two-dimensional Earth's surface in a three-dimensional Earth's surface in a three-dimensional Euclidean space and calculate the three-dimensional average location, balance point, or ``center of gravity". This three-dimensional approach is equivalent to the one sentence definition given by Deetz (1918) and results in the formul{\ae} given and used by Mendeleev (1907) for population and geographic centers on a spherical Earth. The method can easily be extended for distributions on the surface of an ellipsoid of revolution representing the Earth, though the formul{\ae} are more complex. The resulting centers are below the surface and I find this distasteful. The second response is to adapt and restrict the calculations to the two-dimensional non-Euclidean space. As I have previously described in some detail (Barmore, 1991, 1992) this second solution is preferable. The result is a method that restricts the computations of average location and the outcome to the surface of a sphere or an ellipsoid of revolution which very closely approximates the Earth's surface. \noindent{\bf Geographic Center of Wisconsin} There exists, several hundred feet south of the geometric center of the City of Pittsville, Wood Co., Wisconsin, a monument with the following text: \smallskip \smallskip \hrule \smallskip \centerline{\bf Center of the State of Wisconsin} \smallskip \smallskip \centerline{\sl In the early 1950's Governor Walter J. Kohler, Jr.} \centerline{\sl frequently visited the Pittsville area.} \centerline{\sl On one such trip he Proclaimed Pittsville to be} \centerline{\sl the exact center of the State by Official Proclamation} \centerline{\sl on the 27th of June, 1952.} \centerline{\sl Professional Land Surveyors established the corner} \centerline{\sl lying 250 feet North of where you are now standing.} \smallskip \smallskip \centerline{\nn This monument donated by the Central Chapter} \centerline{\nn of the Wisconsin Society of Land Surveyors} \centerline{\nn Erected July 1987} \smallskip \smallskip \centerline{\nn Wayside construction donated by Cedar Corporation, Marshfield.} \centerline{\nn Dale Decker Surveying; Esser Trucking, Arpin;} \centerline{\nn Mid State Associates; People's State Bank, Pittsville.} \smallskip \hrule \smallskip \smallskip The text of the proclamation (Kohler, 1952) gives no hint of how or when it was determined that the center of Wisconsin was at Pittsville. The Geological Survey places the Wisconsin geographic center at ``9 miles southeast of Marshfield." This point is 16 km from the Pittsville monument. The geographic centers of the various states were first published by the U.S. Geological Survey (Douglas, 1923, p. 221-222). Since then, and until as recently as 1991, the centers for most of the States and particularly for Wisconsin have remained unrevised. Thus, the most recently published center of Wisconsin reflects the boundaries and geographic data quality as of 1923 or earlier. Also, according to the very brief definition accompanying the list of centers and Adams' (1932) lament that no analytical process was available, the outcome is only approximate. Thus, the results are of low accuracy. Third, the Great Lakes and some islands were not included when determining the centers. Thus, significant portions of Wisconsin were not included. Clearly, these centers are ripe for revising. It is now possible to calculate the geographic center of Wisconsin to much higher accuracy. I have determined the geographic center of Wisconsin with an uncertainty of less than 0.1 km. The determination was done for the center of all land and water areas including those portions of the Great Lakes within Wisconsin. The center is in the east central portion of Sec. 19, R 7 E, T 25 N, in the Town of Eau Pleine, Portage Co. A second determination was done for the center of the land area and ``inland waters" for comparison with the previous determination given by the Geological Survey. This ``center" is near the center of Sec. 23, R 4 E, T 25 N, a little northeast of the northeast corner of the city of Auburndale in Wood Co. and is about 8 km from the point published by the Geological Survey. Based on these results, it would be reasonable to assume that one could expect similar errors in the existing published locations of the other state centers and they are also in need of revision. These and previous results for Wisconsin are given in Table 1 and displayed on a map in Figure 1. (The computational details and assumptions are given in Appendix A). \vskip.5in \vskip.5in \topinsert \hrule \vskip.5cm \noindent{Table 1. Wisconsin Geographic Center According to Various Sources} \vskip.5cm \hrule \smallskip \settabs\+\quad &Center of land and ``inland waters"\quad &USGS 1923\quad &$44.5728^{\circ}$\quad &$89.7098^{\circ}$\quad &\cr %sample line \+&{\bf Description of Computation}&{\bf Source} &{\bf N. lat.}&{\bf W. long.}& \cr \smallskip \+&Center of all of Wisconsin&this work &$44.6344^{\circ}$&$89.7098^{\circ}$ &\cr \+&Center of land and ``inland waters"&this work &$44.6351^{\circ}$&$89.9923^{\circ}$& \cr \+&9 miles southeast of Marshfield, WI&USGS 1923 &$44.5728^{\circ}$&$90.0441^{\circ}$& \cr \+&On the Pittsville, WI monument&Gov. 1952 &$44.4384^{\circ}$&$90.1301^{\circ}$& \cr \vskip.5cm \hrule \smallskip \hrule \smallskip\smallskip\smallskip \vskip.5cm \hrule \vskip.5cm \noindent{Table 2. Geographic Center of the Conterminous United States} \vskip.5cm \hrule \smallskip \settabs\+\quad &On a Lambert Azimuthal Equal Area map\quad &Deetz 1918\quad &$39.7872^{\circ}$\quad &$98.9830^{\circ}$\quad &\cr %sample line \+&{\bf Description of Computation}&{\bf Source} &{\bf N. lat.}&{\bf W. long.}& \cr \smallskip \+&On Clarke's (1866) ellipsoid surface&&&&\cr \+&\quad a) land \& inland waters only&this work &$39.7872^{\circ}$&$98.9830^{\circ}$ &\cr \+&\quad b) all land \& water areas&this work &$39.9074^{\circ}$&$98.6843^{\circ}$& \cr \+&In three dimensions&this work &$39.9020^{\circ}$&$98.6909^{\circ}$& \cr \+&On a Lambert Azimuthal Equal Area map&this work &$39.8785^{\circ}$&$98.6593^{\circ}$& \cr \+&On Albers Equal Area Conic projection&this work &$39.8352^{\circ}$&$98.6896^{\circ}$& \cr \+&Analogue: Balancing flat map (??)&Deetz 1918 &$39.8333^{\circ}$&$98.5833^{\circ}$& \cr \vskip.5cm \hrule \smallskip \hrule \endinsert \topinsert \vskip6.3in \noindent{\bf Figure 1.} Wisconsin Geographic Centers according to various sources. The point labeled ``CENTER OF WISCONSIN" is the center calculated for all the land and water area within the boundaries of Wisconsin. The location uncertainty of the point is not noticeable on a map of this scale. The point labeled ``CENTER, LAND ONLY" is the center calculated for all the land and ``inland waters" but excluding the portions of the Great Lakes lying within Wisconsin. The location uncertainty of this point is not noticeable on a map of this scale. The point labeled ``CENTER, USGS, 1923" is the center published by the U.S. Geological Survey since 1923. The ``error bars" indicate the probable uncertainty implied by the manner in which the various State center locations were stated. The point labeled ``CENTER, KOHLER, 1952" inside the boundaries of Pittsville is the result of Governor Kohler's 1952 Official Proclamation. The location uncertainty and method of determination of this point are unknown. \endinsert \noindent{\bf Geographic Center of the Conterminous United States} The geographic center of the ``Conterminous" United States (48 States and the District of Columbia) is widely published on maps, in atlases and in government documents, as being near Lebanon, Smith County, Kansas, at latitude of $39^{\circ} 50'$ N and at longitude of $98^{\circ} 35'$ W. All sources for this and similar statements that can be traced, ultimately refer to a one sentence statement with a brief footnote published by Deetz (1918, p. 57) that reads: {\narrower\smallskip\noindent ``The Geographic center ($^*$) of the United States is approximately in latitude $39^{\circ} 50'$ and longitude $98^{\circ} 35'$. \smallskip \smallskip $(^*)$ `Geographic center of the United States' is here considered as a point analogous to the center of gravity of a spherical surface equally weighted (per unit area) and of the outline of the country, and hence it may be found by means similar to those employed to find the center of gravity" \smallskip} There is a hint as to how this might have been determined in the melancholy paper by Adams (1932) which states: {\narrower\smallskip\noindent ``A method that was used in the Coast and Geodetic Survey a number of years ago was the following: An equal-area map of the United States was constructed on thin cardboard and then the outline map was cut out along the various boundaries. The center of gravity of this outline map was then determined." \smallskip} As this was done in an analogue way (on what must have been a map of modest scale) rather than calculated in a precise way, the result is probably of modest accuracy. Note that: (a) It is a flat (and therefore distorted) map not a spherical map whose center was found. (b) It is not stated which map projection was used to produce the map. (c) It is not stated what boundaries were used. In an attempt to reproduce Deetz's result, this geographic center was recomputed in a variety of ways. If only the areas and centers of the land and ``inland waters" of the various states were used the agreement was very poor. However, if the list of areas and centers used was expanded to include the portions of the Great Lakes within the United States and to include the various sounds, straits, bays and coastal waters that are not part of the ``inland waters" of the various states, then modest agreement could be achieved (see Appendix B for details of these calculations). The results are summarized in Table 2 and displayed on a map in Figure 2. Because of the low quality of the data used in the computation, these results should not be considered accurate. \topinsert \vskip6.3in \noindent{\bf Figure 2.} Geographic Center of the Conterminous United States determined by various computational methods. The point labeled ``CENTER, 48 STATES, ALL AREAS" is the average location of all the various areas that make up the conterminous United States calculated on the surface of Clarke's (1866) ellipsoid using the preferred method (Barmore, 1991, 1992). The point labeled ``CENTER, LAND \& INLAND WATERS ONLY" is the average location of the various areas (the Great Lakes and other ``non-inland waters" being excluded) that make up the conterminous United States calculated in the same manner. Because of the limited accuracy of the data used, neither of these locations nor the other center locations displayed in this figure should be considered as accurate. The point labeled ``3-D" is the three-dimensional average location, projected onto the surface, of all the areas that make up the conterminous United States. The point labeled ``DEETZ, USCGS, 1918" is the widely quoted result. The points labeled ``L" and ``A" are the centers determined by an analytical computation that is equivalent to finding the balance point of a Lambert Azimuthal Equal Area map and an Albers Conical Equal map, respectively, of all areas of the conterminous United States. \endinsert \noindent{\bf Geographic Center of the United States} Apparently, the geographic center of The United States (50 States and the District of Columbia) was determined by the U.S. Coast and Geodetic Survey (ca. 1959) in a manner described, if nowhere else, in several news releases. The accuracy of this result is questionable for reasons outlined below. The center of all 50 states was apparently determined, piecemeal, as follows: The 48 states were represented as being 3,022,400 square miles in area at the previously determined location given by Deetz (1918) at latitude $39^{\circ} 50'$ N., longitude $98^{\circ}35'$. Alaska's land and ``inland waters" area were represented as being 586,400 square miles at latitude $63^{\circ}50'$ N., longitude $152^{\circ} 00'$ W. The balance point of these two areas was found to be at latitude $44^{\circ} 59'$ N., longitude $103^{\circ} 38'$ W. on the (presumed great circle) arc between them. Then when Hawaii joined the Union, the process was repeated. The 49 states were represented as the sum of the previous two areas (3,608,800 square miles) located at their balance point. Hawaii's land and ``inland waters" area were represented as 6424 square miles at latitude $20^{\circ} 15'$ N., longitude $156^{\circ} 20'$ W. The balance point of these two areas was found to be at latitude $44^{\circ} 58'$ N., longitude $103^{\circ} 46'$ W. If the Earth's surface were flat, this procedure would be as accurate as the data used would allow. However, the surface is not flat, but curved. When the distances are as large as those between the various states of the United States, ignoring the curvature can result in a substantial error (Barmore, 1991, 1992). If the center is to be determined with distances measured on the curved surface of the Earth, it must be redone from the beginning with each addition. Another difficulty has to do with using data of mixed consistency. The 3,022,400 square mile figure for the 48 States is the land plus ``inland waters" only. The location used for this area is apparently the center of a different area --- the land, ``inland waters," {\bf and a substantial area of ``non-inland waters}." (These ``non-inland waters" have an area of about 74,364 square miles, 2.4\% of the 3,022,400 square mile figure [U.S. Bureau of the Census, 1940].) In order to illustrate the differences that can result, the geographic center for the entire United States was calculated various ways. The results are summarized in Table 3 and displayed on a map in Figure 3. the same methods and data were used that were used in the preceding example with the exception that the total of all land and all water areas for Alaska and Hawaii were those given most recently (U.S. Bureau of the Census, 1992, table 340). Because the centers and areas used have not been revised (with the exception of the Alaskan and Hawaiian areas) the results should not be taken as accurate. \vskip.5in \vskip.5in \topinsert \hrule \vskip.5cm \noindent{Table 3. Geographic Center of the United States (All 50 States)} \vskip.5cm \hrule \smallskip \settabs\+\quad &\quad ``inland waters" only. For comparison. \quad &news release\quad &$45.4344^{\circ}$\quad &$104.3524^{\circ}$\quad &\cr %sample line \+&{\bf Description of Computation}&{\bf Source} &{\bf N. lat.}&{\bf W. long.}& \cr \smallskip \+&On Clarke's (1866) ellipsoid surface&this work &$45.4344^{\circ}$&$104.3524^{\circ}$& \cr \+&In three dimensions&this work &$45.2517^{\circ}$&$104.1776^{\circ}$& \cr \+&On ellipsoid surface. Land and& && & \cr \+&\quad ``inland waters" only. For comparison.&this work &$44.9482^{\circ}$&$104.1189^{\circ}$& \cr \+&U.S. Coast and Geodetic Survey&news release &$44.9667^{\circ}$&$103.7667^{\circ}$& \cr \vskip.5cm \hrule \smallskip \hrule \smallskip\smallskip\smallskip \vskip.5cm \hrule \vskip.5cm \noindent{Table 4. Wisconsin Geographic Center Calculated Two Ways} \vskip.5cm \hrule \smallskip \settabs\+\quad &On a two-dimensional non-Euclidean surface\quad &$44.6343739^{\circ}$\quad &$89.7097544^{\circ}$\quad &2.4 km\quad &\cr %sample line \+&{\bf Description of Computation} &{\bf N. lat.}&{\bf W. long.}&{\bf depth}& \cr \smallskip \+&In a three-dimensional Euclidean volume &$44.6343739^{\circ}$&$89.7097544^{\circ}$&2.4 km & \cr \+&On a two-dimensional non-Euclidean surface &$44.6343818^{\circ}$&$89.7097566^{\circ}$&0.0 km & \cr \vskip.5cm \hrule \smallskip \hrule \endinsert \topinsert \vskip6.3in \noindent{\bf Figure 3.} Geographic Center of the United States (all land and water areas of all 50 States and the District of Columbia) determined various ways. The point labeled ``CENTER, ALL AREAS OF USA" is the average location of all the various land and water areas that make up the United States calculated on the surface of Clarke's (1866) ellipsoid using the preferred method (Barmore, 1991, 1992). Because of the limited accuracy and limited internal consistency of the data used, neither this location nor the other center locations displayed in this figure should be considered as accurate. The point labeled ``3-D" is the three-dimensional average location, projected onto the surface, of the same areas, that make up the United States. The point labeled ``USCGS, 1959" is the widely quoted result. The point labeled ``CENTER, LAND \& INLAND WATERS ONLY" is the center of all land area combined with only the ``inland waters" area, the Great Lakes and ``non-inland waters" being excluded. This center, calculated on the Earth's curved surface, corresponds most closely to the U.S. Coast and Geodetic Survey procedure for determining the geographic center. It is presented here for comparison. \endinsert \noindent{\bf Summary and Recommendations} The geographic centers and areas of the various States and The United States are in serious need of revision for several reasons. In the seventy years that have passed since the centers were determined, much has happened. Mapping of the United States is much improved. Computational capability is now widely available ---it should no longer be necessary to make compromises for computational reasons. Data on land and water area are much improved. It should now be possible to compute the location of the various centers to an accuracy of ca. $10$ m. The following recommendations are made for this revision and any similar sort of statistical analysis. \item{I.} The term {\bf center} of spatial distributions should be reserved for the average (arithmetic mean) location. Other statistics of spatial distributions that are found to be useful should be given other names to avoid confusion. \item{II.} If the distribution covers enough of a curved surface for the curvature of the surface to be noticeable, then special care must be taken. Unless appropriate compensation is made for the Earth's surface curvature, these calculations may not be properly done using any flat map projection. There is no flat map of the Earth's curved surface that leaves area, distance, and direction undistorted. For distributions on the surface of the earth, the computations of average location should be carried out on the surface and the results restricted to the surface. The method of doing this is outlined in some detail elsewhere (Barmore, 1991, 1992). Alternately, the computations can be carried out in three dimensions using more familiar procedures, but the computation of two-dimensional distribution statistics in two dimensions is preferable. \item{III.} If geographic centers of hierarchical sets of areas are presented, they should be done in a consistent way so that comparisons are easy within a level and between levels. It should be possible at any level to find the average of the larger group by averaging over its component parts. In particular, if centers at one level for separate land and water areas are given, the centers for the subdivisions should be separated in the same manner. If ``non-inland waters" are excluded at one level they should be excluded at all levels. \item{IV.} What is included (or excluded) should be clearly stated. The absence of any discussion of what is meant by the term ``North America" makes meaningless the statements concerning the center of North America published by the U.S. Geological Survey (Douglas, 1930; and pamphlets by the U. S. Geological Survey, 1967 and 1991). Is Greenland included? Are ``non-inland waters" included? Are off-shore islands included? \vfill\eject \noindent{\bf Appendix A: Calculation of Wisconsin's Geographic Center} All areas and centers were determined assuming they lay on the surface of Clarke's (1866) ellipsoid ($a=6378.2064$ km and $e=0.08227185$). The State's surface and adjacent areas were divided into $30 \times 60$ minute quadrangles. For $30 \times 60$ minute quadrangles that lay completely inside the State boundaries (or had more than half their area within the boundaries) the areas and centers were calculated using the ellipsoid geometry found in Bomford (1977). These results are very accurate. Wherever the boundary cut a quadrangle, the areas and centers were determined from the 1:100000, $30 \times 60$ minute quadrangle maps published by the Geological Survey. If less than half the quadrangle's area was within Wisconsin, only the portion within the State was considered. If more than half the quadrangle's area was within the State, the area and center of the portion to be excluded were determined and subtracted from the previously calculated values for the entire quadrangle. This process minimizes the areas that had to be measured rather than calculated. The areas and centers that had to be measured were done as follows: a) If the areas were composed of quadrilaterals or triangles, the areas and centers were calculated from measurements taken directly from the map. b) If the areas were irregular, they were carefully traced onto a uniform sheet whose areal density had been previously determined with the aid of an electronic ``balance," cut out, reweighed to determine their area and suspended from several points to determine their centers. c) The latitude and longitude of the centers were then determined directly from the geographic grid on the map. d) The areas were then corrected for scale changes. The scale changes have two causes: First, very small variations in scale resulting from the Universal Transverse Mercator projection (Snyder, 1987, p. 58-64). Second, scale changes due to expansion or shrinkage of the map paper caused by humidity changes (determined from measurements of the 10000 m grid on the map). This process created a collection of 111 area elements representing the State. Over 87\% of the area (represented by the 37 full $30 \times 60$ minute quadrangles) in the calculations of center have calculated areas and centers for which the accuracy is very high. For the remaining 13\% of the area (represented by 74 fractional areas averaging 325 sq. km) the accuracy of the areas is probably limited by how well the areas were corrected for scale changes caused by humidity changes. As a check, the total area of land and ``inland waters" was found to be 145435.166 sq. km = 56152.8 sq. miles. This compares favorably with the 56153 sq. miles listed as the area of Wisconsin in the 1980 Census (U.S. Bureau of the Census, 1983). Also, the total area of Wisconsin (including the portion of the Great Lakes falling within Wisconsin) was found to be 169609.8 sq. km. The Bureau of the Census (1992) reports the total area of Wisconsin to be 169653 sq. km. The difference of 43 sq. km may be due to disagreement about the boundaries of the State in Lake Michigan. I have used the boundaries shown on the 1:100000 scale, $30 \times 60$ minute series maps published by the Geological Survey. These boundaries, in turn, are in agreement with those given in Van Zandt (1976) and further clarified in the 1948 Compact between Michigan, Wisconsin, and Minnesota which finally settled the boundary (U.S. Statutes at Large, 1948). Other sources show a different boundary --- {\sl The National Atlas\/} (U.S. Geological Survey, 1970, p. 17, 19, 313) or the Geological Survey map, {\sl State of Wisconsin\/}, 1:500,000 scale, 1966 comp., 1968 ed., for example. In the worst possible case an error of this magnitude would shift the center of Wisconsin two or three seconds of arc or about 50 m on the surface. The State center was then calculated by finding the average location of the 111 area elements. This calculation was done two ways: first as a three-dimensional volume distribution and second as a two-dimensional surface distribution (Barmore, 1991, 1992). For areas the size of Wisconsin, there is little difference between the two results except for depth. For example, see Table 4. The difference is only a few hundredths of a second of arc, and corresponds to a distance of one or two meters on the surface. In order to provide a comparison for the Center of Wisconsin given by Douglas (1923), that included the land and ``inland waters" only, this center was also redetermined. Therefore, the process, outlined above, was repeated for a somewhat different collection of 111 area elements (30 full $30 \times 60$ minute quadrangles and 81 fractional areas averaging 197 sq. km; representing 82\% and 18\%, respectively, of the areas used in the calculation). These area elements represent the area of the land and ``inland waters", but not the Great Lakes, within the boundaries of Wisconsin. \noindent{\bf Appendix B: Calculation of the Geographical Center of the Conterminous United States} The geographic center of the conterminous United States was calculated using methods previously described. The centers calculated on the curved surface in two dimensions or when treating the areas as a three-dimensional volume distribution assumed Clarke's (1866) ellipsoid (though the data quality hardly justifies such accuracy). The centers calculated by distributing the areas on the surface of various flat maps used equations for the projections given by Snyder (1987, p. 100-101, 185-187) for a spherical earth. The Lambert Azimuthal Equal Area map was centered at $38^{\circ}$ N. latitude, $95^{\circ}$ W. longitude, following Deetz (1918, p. 57) and the Albers Equal Area map used two standard parallels at $29^{\circ} 30'$ and $45^{\circ} 30'$ N. latitude as suggested by Deetz and Adams (1945, p. 94). The data used consisted of two parts. The first part was the areas of land and ``inland waters" and centers as given by Douglas (1923, p. 219, 222) for the 48 States and the District of Columbia. If the example of Wisconsin is typical, the accuracy of this data is not high. More recent and probably better data were not used because the 1923 data for area are nearly identical to that given by Gannet (1906, p. 7, 8) and thus more characteristic of the data available to Deetz than more modern material. The second part of the data was for the ``non-inland waters". The areas included are those delineated earlier (U.S. Bureau of the Census, 1942, Map I, and Table IV). The approximate centers for these ``non-inland water" areas were determined from maps in {\sl The National Atlas\/} (U.S. Geological Survey, 1970). Because of the uncertainty in the areas and centers of the area elements whose locations were averaged to get these results, they should not be considered accurate. \vfill\eject \noindent{\bf References} \smallskip \ref Adams, Oscar S. 1932. ``Geographical Centers." {\sl The Military Engineer\/}, Vol. XXIV, No. 138, pp. 586-7. \ref Barmore, Frank E. 1991. ``Where Are We? Comments on the Concept of the `Center of Population' " {\sl The Wisconsin Geographer\/}, Vol. 7, 40-50. (Reprinted (with the example data set used and with several corrections) in {\sl Solstice: An Electronic Journal of Geography and Mathematics\/}, Vol. III, No. 2, pp. 22-38. Winter 1992. (Inst. of Mathematical Geography, Ann Arbor, MI)). \ref Barmore, Frank E. 1992. ``The Earth Isn't Flat. And It Isn't Round Either! Some Significant and Little Known Effects of the Earth's Ellipsoidal Shape." {\sl The Wisconsin Geographer\/}, Vol. 8, 1-9. (Reprinted in {\sl Solstice: An Electronic Journal of Geography and Mathematics\/}, Vol. IV, No. 1, pp. 26-38. Summer 1993. (Inst. of Mathematical Geography, Ann Arbor, MI)). \ref Bomford, G. 1977. {\sl Geodesy\/}. (Oxford UK, Clarendon Press) a reprinting (with corrections) of the 1971 3rd Ed. \ref Deetz, Charles H. 1918. {\sl The Lambert Conformal Conic Projection with Two Standard Parallels, etc.\/} (Special Publication No. 47). Washington DC, U.S. Coast and Geodetic Survey. \ref Deetz, Charles H. and Oscar Adams, 1945. {\sl Elements of Map Projection with Applications to Map and Chart Construction\/} (U.S. Coast and Geodetic Survey Special Publication 68, 5th Ed., 1944 revision). Washington DC, U.S. Government Printing Office. \ref Douglas, Edward M. 1923. {\sl Boundaries, Areas, Geographic Centers and Altitudes of The United States and the Several States, etc.\/} (Bulletin 689). Washington DC, U.S. Geological Survey. \ref Douglas, Edward M. 1930. {\sl Boundaries, Areas, Geographic Centers and Altitudes of the United States and the Several States, etc.\/}, 2nd Edition (Bulletin 817). Washington DC, U.S. Geological Survey. \ref Eells, W.C. 1930. ``A mistaken conception of the center of population." {\sl Journal of the American Statistical Association\/}, New Series No. 169, Vol. 25, pp. 33-40. \ref Gannett, Henry 1906. {\sl The Areas of the United States, The States, and the Territories\/} (Geological Survey Bulletin 302). Washington DC, U.S. Geological Survey. \ref Hayford, John F. 1902. ``What is the center of an area, or the center of a population?" {\sl Journal of the American Statistical Association\/}, New Series No. 58, Vol. 8, pp. 47-58. \ref Kohler, Walter J. Jr. 1952. No trace of the Proclamation could be found in the papers of Gov. Kohler or in the State Archives housed in the State Historical Society of Wisconsin, Madison. A copy of the Proclamation is on display in the City Council Chamber, Pittsville, Wisconsin. \ref Kumler, Mark P. and Michael F. Goodchild. 1992. ``The Population Center of Canada -- Just North of Toronto?!?" in Donald G. Janelle (Editor) {\sl Geographical Snapshots of North America; etc.\/} New York, NY, The Guilford Press. pp. 275-279. \ref Mendeleev, D.I. 1907. {\sl K Poznaniyu Rossii\/}, 5th ed. (St. Petersburg, A. S. Suvorina) p. 139. \ref Neft, David S. 1966. {\sl Statistical Analysis for Areal Distributions\/} (Monograph Series Number Two), Philadelphia, PA, Regional Science Research Institute. \ref Snyder, J.P. 1987. {\sl Map Projections --- A Working Manual\/}, (USGS Professional Paper, 1395). Washington DC, U.S. Government Printing Office. \ref U. S. Bureau of the Census. 1942. {\sl Sixteenth Census of the United States: 1940. Areas of the United States: 1940\/}. Washington DC, U.S. Government Printing Office. \ref U. S. Bureau of the Census. 1983. {\sl 1980 Census of Population, Vol. 1, Chapter A, Part 1\/} [PC80-1-A1]. Washington DC, U.S. Department of Commerce, Bureau of the Census. Table 11, pp. 1-47. \ref U. S. Bureau of the Census. 1992. {\sl Statistical Abstract of the United States: 1992\/}. (112th Ed.). Washington DC, U.S. Government Printing Office. \ref U.S. Coast and Geodetic Survey. ca. 1959. News releases titled: ``New Geographic Center of the United States," Aug.(?) 1958; ``Geographic Center of U.S. Moved Again with Admission of 50th State," March 1959; and, ``Geographic Center of the United States," n.d. Copies supplied the Earth Sciences Information Center (ESIC), U.S. Geological Survey, Reston, VA. \ref U.S. Geological Survey. 1967. {\sl Geographic Centers of the United States\/} (pamphlet). Washington DC, U.S. Geological Survey. \ref U.S. Geological Survey. 1970. {\sl The National Atlas of the United States of America\/}. Washington DC; U.S. Department of the Interior, Geological Survey. \ref U.S. Geological Survey. 1991. {\sl Elevations and Distances in the United States\/} (pamphlet). Washington DC, U.S. Geological Survey. \ref U.S. Statutes at Large. 1948. Vol. 62, p. 1152, Chapter 757. S.J. Res. 206. June 30, 1948. Public Law 844. \ref Van Zandt, Franklin K. 1966. {\sl Boundaries of the United States and the Several States\/} (Geological Survey Bulletin 1212). Washington DC, U.S. Geological Survey. \ref Van Zandt, Franklin K. 1976. {\sl Boundaries of the United States and the Several States\/} (Geological Survey Professional Paper 909). Washington DC, U.S. Geological Survey. \vfill\eject \centerline{\bf 4. ARTICLES} \smallskip \centerline{\bf Equal-Area Venn Diagrams of Two Circles: Their Use with Real-World Data.} \vskip.5cm \centerline{Barton R. Burkhalter} \centerline{Senior Program Officer} \centerline{Academy for Educational Development} \centerline{Washington, D.C.} \vskip.2cm \centerline{First draft 12/21/90; revised 2/18/91.} \centerline{Communicated by John D. Nystuen to Solstice, 5/3/93.} \vskip.5cm \noindent{\bf General Problem} We are concerned with populations whose members have two discrete characteristics; that is, characteristics which are either present or absent in each member. In populations having two characteristics, the populations can be described by the proportion of the population that has both characteristics present, one or the other (but not both) characteristics present, or both characteristics absent. One well-known way to present this data is with a two circle Venn Diagram in which each circle represents one of the characteristics, the intersection of the circles represents the members with both characteristics, and the region outside both circles but within the universe of discourse (depicted as a bounded figure surrounding the circles --- often a rectangle) represents the members with neither characteristic. Appropriate regions might then be labelled with suitable percentages, whether or not the geometric intersection pattern is suggestive of the numeric partition of the sample. At this point, it may be useful to the reader to draw a two-circle Venn diagram. For example, consider a population of countries with national child vaccination programs. Some of the countries in the population use a campaign strategy, some a clinic-based strategy, some use both strategies, and a few use neither strategy. Construct a Venn diagram to represent this grouping of mixed strategies. Draw Circle $\alpha $ on the left and draw an intersecting Circle $\beta $ on the right. Draw a rectangle that is large enough to easily contain all of the intersecting circle configuration. In this Venn diagram, Circle $\alpha $ could represent countries using a campaign strategy and Circle $\beta $ could represent countries with a clinic-based strategy. Partition each of these circles according to their intersection pattern using the following notation. \vskip.5cm \hrule \vskip.5cm \noindent {\sl Notation} \noindent Let the symbol $A$ denote the area of Circle $\alpha $ that does NOT also lie within the Circle $\beta $. \noindent Let the symbol $B$ denote the area of Circle $\beta $ that does NOT also lie within the Circle $\alpha $. \noindent Let the symbol $AB$ denote the area of the intersection of Circles $\alpha $ and $\beta $. \vskip.5cm \hrule \vskip.5cm In the vaccination program interpretation of the two circle Venn diagram, Area $A$ is proportional to the countries using only a campaign strategy, Area $B$ to the countries using only a clinic-based strategy, and Area $AB$ to the countries using both a campaign and a clinic-based strategy. Countries using neither strategy are not represented; indeed, only participant populations will be considered for the remainder of this analysis, although we do note the existence of the logical category of the ``neither" class. Data such as this is sometimes illustrated with bar or pie charts. Such illustrations are inadequate because they do not allow easy portrayal on a single diagram of both the intersecting areas and the total percentage of each characteristic. The advantage of using a diagram of intersecting circles is that it portrays all the data clearly in a single diagram. The objective here is to draw the intersecting circles so that the different areas are exactly proportional to the data, in much the way that an equal area map is drawn so that different areas are exactly proportional to the size of the landmass. Equal area graphic displays, be they maps or diagrams, are critical in making accurate visual comparisons of mapped or plotted data. The remainder of this paper is devoted to displaying the detail of the calculations required to construct equal-area Venn diagrams from real-world data. Fundamentally, these calculations rest on the problem of finding the radii of the circles and the location of their centers, given the various common (intersecting) and non-common areas. \vskip.5cm \noindent{\bf Definition of the two-circle problem} Given two intersecting circles, $\alpha $ and $\beta $, and given their common and non-common areas, $AB$, $A$ and $B$, respectively. Find the radii of both circles, $r_A$ and $r_B$, and the distance between the centers of the two circles $d_{AB}$ such that the centers can be located and the circles drawn. It is clear that if the center of Circle $\alpha $ is at the origin, the Cartesian coordinates of the center of Circle $\beta $ are $(d_{AB}, 0)$. We can transform the three areas into percentages of the total area covered by the two intersecting circles by noting that the total area covered is $A+B+AB$. $$ A\% = 100 \times (A/(A+B+AB)) \leqno (1) $$ gives the $A$-only area percent; $$ B\% = 100 \times (B/(A+B+AB)) \leqno (2) $$ gives the $B$-only area percent; $$ AB\% = 100 \times (AB/(A+B+AB)) \leqno (3) $$ gives the $AB$ area percent; \noindent No generality is lost by requiring Circle $\alpha $ to be the larger circle and by standardizing the size of Circle $\alpha $ by setting its radius equal to 1: $r_A=1$. (Naturally, this assumes that $\alpha >\beta$ and that the bigger real-world characteristic is assigned to Circle $\alpha $.) As a result, the area $A+AB$ of Circle $\alpha $ is $\pi $. The problem can now be restated, more simply, as follows. Given: $A\%$, $B\%$, and $AB\%$, where $A\%+B\%+AB\% = 100$. Find: $r_B$, the radius of Circle $\beta $, and $d_{AB}$, the distance between the centers. \vskip.5cm \noindent{\bf Analytic Strategy} In order to solve the two-circle problem, define the {\sl chord} of the intersection to be the straight line joining the two points where the perimeters of the two circles intersect --assuming here, and throughout the remainder of the text, that one of the two circles is not fully contained within the other. There are two situations that arise: one in which the chord lies between the two centers and a second in which the chord lies to one side of both centers. To visualize this relationship, draw one pair of circles with a relatively small area of intersection; in this case the chord lies between the centers. In what follows, this configuration will be referred to as one of type Case I. Alternatively, draw two circles with a relatively large area of overlap; in this case the chord lies on one side of both centers. In what follows, this configuration will be referred to as one of type Case II. Starting with $A\%$, $B\%$, and $AB\%$, it is straightforward to derive $r_B$, but not to derive $d_{AB}$. Therefore, we reverse the situation and seek the function that yields $A\%$, $B\%$, and $AB\%$ given $r_A$, $r_B$, and $d_{AB}$. Several derivations are possible. The simplest one (not using integral calculus) is presented below to maximize accessibility of content. Functions for $A\%$, $B\%$, and $AB\%$ in terms of $r_A$, $r_B$, and $d_{AB}$ were obtained for Case I and for Case II. These functions are sufficiently complex to obstruct the derivation of an inverse function that would yield $d_{AB}$ in terms of $A\%$, $B\%$, and $AB\%$. Consequently, a numerical approach was used in which $d_{AB}$ was calculated for a grid of values of $B\%$ and $AB\%$. The results are presented in Tables 1 and 2. The value $r_A$ is assumed equal to one and $r_B$ is readily calculated from $A\%$, $B\%$, and $AB\%$. With these results, several options are available to estimate $d_{AB}$ from $A\%$, $B\%$, and $AB\%$. The preferred option depends on the accuracy desired. Option 1 entails interpolating from the data in Tables 1 and 2. Option 2 entails using a polynomial in $B\%$ and $AB\%$ (obtained via regression) to estimate $d_{AB}$. Options 1 and 2 are the least accurate, both giving answers within one percent accuracy relative to the radius of the largest circle (Circle $\alpha $). Option 3, which will yield $d_{AB}$ to any desired accuracy, entails searching by trial and error using the functions $B\% = f_1(d_{AB}, r_B)$ and $AB\% = f_2(d_{AB}, r_B)$. The trial and error search is greatly simplified by the fact that $r_B$ can be calculated directly from $B\%$ and $AB\%$. \vskip.5cm \noindent{\bf Derivation of $B\%$ and $AB\%$ as a function of $r_B$ and $d_{AB}$} \vskip.2cm \noindent{\sl General formul{\ae}} Heron's formula for the area of a triangle is based on the lengths, $a$, $b$, and $c$, of its sides. Let $S=(1/2)\times (a+b+c)$. Then the area of the triangle is: $$ (S \times (S-a) \times (S-b) \times (S-c))^{1/2} \leqno (4) $$ A sector of a circle is the pie-shaped wedge cut from the center of the circle out to the edge. The region of overlap of two intersecting circles is called a ``lune." A sector can be decomposed into a triangle and a lune split longitudinally. We refer to the triangular portion as the ``triangle" of a sector and to the lunar portion as the ``segment" of a sector. The formula for the area of a sector of a circle with central angle $Q$ (measured in radians) and radius $r$ is: $$ (1/2) \times (Q/\pi )\times(\pi \times r^2) =(1/2)\times (Q\times r^2). \leqno (5) $$ The formula for the area of the corresponding triangle of a sector is: $$ (1/2)\times r^2\times \hbox{sin}\,Q. \leqno(6) $$ The formula for the area of the corresponding segment of a sector is: $$ (1/2)\times (Q\times r^2) - (1/2) \times r^2\times \hbox{sin}\,Q = (1/2)\times r^2 \times (Q - \hbox{sin}\,Q). \leqno(7) $$ \vskip.5cm \noindent{\sl Case I: Chord lies between the two centers} In Case I, the chord of the lune separates the centers of circles $\alpha $ and $\beta $. The distance $d_{AB}$ is the distance between the two centers, measured along the line of centers. Form a triangle using the line of centers as one side of length $d_{AB}$. The second side is formed by joining the center of circle $\alpha $ to the top intersection point of the lune; the acute angle enclosed by the line of centers and this side has measure $Q_A$ which is $1/2$ of the central angle subtending the chord of the lune from the center of circle $\alpha $. In a similar fashion, join the center of circle $\beta $ to the same third vertex to complete the triangle. The acute angle enclosed between the line of centers and this side has measure $Q_B$ which is $1/2$ of the central angle subtending the chord of the lune from the center of circle $\beta $. Let $h$ denote the altitude of this triangle from the vertex of the lune to the line of centers. Let $X$ denote the horizontal distance from the center of Circle $\alpha $ to the intersection with the chord. Let $Z$ denote the area of the triangle with sides of lengths $r_A=1$, $r_B$, and $d_{AB}$. Let $K_A$ and $K_B$ denote areas of the sectors in circles $\alpha $ and $\beta $ subtended by the chord. Let $L_A$ and $L_B$ denote the areas of the triangles of these two sectors. Finally, let $M_A$ and $M_B$ denote the areas of the segments of the two sectors. Then given $r_A$, $r_B$, and $d_{AB}$, find $h$, $B\%$, $AB\%$, and $A\%$ as follows. From equation (4), $$ S=(1+r_B+d_{AB})/2. \leqno(8) $$ From equations (4) and (8), we get the area of the triangle as $$ Z=(S\times (S-1)\times (S-r_B)\times (S-d_{AB}))^{1/2}. \leqno (9) $$ From equation (9), $$ h=2\times Z/d_{AB}, \leqno(10) $$ because $Z=(1/2)\times h \times d_{AB}$. From equation (10), $$ Q_A=\hbox{Arcsin}\,(h/r_A), \leqno(11) $$ because $\hbox{Sin}\,Q_A = h/r_A$; also from equation (10), $$ Q_B=\hbox{Arcsin}\,(h/r_B), \leqno(12) $$ because $\hbox{Sin}\,Q_B = h/r_B$. From equations (5) and (11), the sector $A$ area is $$ K_A = {r_A}^2\times (2\times Q_A)/2 = Q_A, \leqno(13) $$ and from equations (5) and (12), the sector $B$ area is $$ K_B = {r_B}^2\times (2\times Q_B)/2 = Q_B. \leqno(14) $$ From equation (9), the sum of the areas of the two sectors is $$ L_A + L_B = 2\times Z. \leqno(15) $$ To find the area $AB$ of the intersecting area (lune), view it as the sum of the two segments of the two sectors. From equation (7): $ AB = M_A + M_B =(K_A - L_A) + (K_B - L_B) =(K_A + K_B) - (L_A + L_B) $ so that from equations (9), (13), (14), (15), $ AB = Q_A + Q_B \times {r_B}^2 - 2 \times Z. $ Using equations (11) and (12), it follows that $ AB = \hbox{Arcsin}(h/r_A) + \hbox{Arcsin}(h/rB) \times {r_B}^2 -2\times Z $ and finally, noting that $r_A$=1, that $$ AB = \hbox{Arcsin}(h) +\hbox{Arcsin}(h/r_B) \times {r_B}^2 - 2\times Z. \leqno(16) $$ The $B$-only area is found by subtracting the area of the lune from the area of the whole circle as $$ B=(\pi \times {r_B}^2)-(AB). \leqno(17) $$ Subtracting out the extra intersection, the total area covered by the circles, denoted TOTAL, is (from equation (16)) $$ \hbox{TOTAL} = (\pi) +(\pi \times {r_B}^2) - AB. \leqno(18) $$ [Some may recognize the formula in (18) as one form of the Principle of Inclusion and Exclusion--ed.] From equations (16) and (18) it follows that $$ AB\% = 100 \times AB/\hbox{TOTAL}; \leqno(19) $$ from equations (17) and (18) it follows that $$ B\% = 100 \times B/ \hbox{TOTAL}; \leqno(20) $$ and from equations (19) and (20) it follows that $$ A\% = 100 - AB\% - B\%. \leqno(21) $$ These results hold for $d_{AB}$ greater than or equal to $X$, the distance from the origin to the chord, but not greater than $r_A +r_B$, that is: $$ X \leq d_{AB} \leq r_A+r_B, \leqno(22) $$ where, from equation (11), $X = \hbox{Cos}\,(Q_A/2)$. \vfill\eject \noindent{\sl Case II: Chord to one side of both centers} The same definitions apply as in the previous section, except in relation to the following situation. Draw two intersecting circles and associated lines, labelling them as follows. Draw the larger of the two circles on the left. Insert the center of the large circle as a distinguished dot. Draw a smaller circle intersecting the larger one in such a way that the center of the large circle is contained within the smaller circle. Much of the small circle is therefore necessarily contained within the large circle. Note the center of the small circle as a dot. Draw the chord joining the two intersection points of the small and large circles; half of it has length $h$. Draw the line segment joining the two circle centers, of length $d_{AB}$ and extend the segment to intersect the chord. The small circle now contains a right triangle which in turn contains a triangle with an obtuse angle. Label the radius of the larger circle as $r_A$; label the radius of the smaller circle as $r_B$. Label the constructed central angle in the larger circle as $Q_A$ and the constructed central angle in the smaller circle as $Q_B$. The area of the obtuse triangle is $Z_1$ and the area of the difference between the right triangle and the obtuse triangle is $Z_2$. The following formul{\ae} can then be readily deduced. From equation (4), $$ S_1 = (r_A + r_B +d_{AB})/2 = (1 + r_B +d_{AB})/2; \leqno(23) $$ from equations (5) and (23), $$ Z_1 = (S_1 \times (S_1-1)\times (S_1-r_B)\times (S_1-d_{AB}))^{1/2}; \leqno(24) $$ from equation (24), $$ h = 2\times Z_1/d_{AB} \leqno(25) $$ because $Z_1=(1/2)\times d_{AB} \times h$; from equation (25) $$ Q_A = \hbox{Arcsin}\,(h), \leqno(26) $$ because $\hbox{Sin}\,(Q_A) =h/r_A = h$; from equation (26) $$ Q_B = \hbox{Arcsin}\,(h/r_B), \leqno(27) $$ because $\hbox{Sin}\,(Q_B) = h/r_B$; from equations (25) and (27) $$ Z_2 = (1/2)\times h\times r_B\times \hbox{Cos}\,(Q_B)); \leqno(28) $$ from equations (5) and (26) $$ K_A = \hbox{Sector $A$ area} = (1/2)\times (2\times Q_A) \times {r_A}^2 = Q_A; \leqno(29) $$ from equations (24) and (28) $$ L_A=\hbox{Triangle Area of Sector $A$} =2\times (Z_1 + Z_2); \leqno(30) $$ from equations (7), (29), (30) $$ M_A=\hbox{Segment Area of Sector $A$} =K_A-L_A; \leqno(31) $$ from equations (5) and (27), $$ K_B=\hbox{Sector $B$ Area} =(1/2)\times (2\times Q_B)\times {r_B}^2 =Q_B\times {r_B}^2; \leqno(32) $$ from equation (28) $$ L_B=\hbox{Triangle Area of Sector $B$} =2\times Z_2; \leqno(33) $$ from equations (32) and (33) $$ M_B=\hbox{Segment Area of Sector $B$} =K_B-L_B; \leqno(34) $$ from equations (31) and (34) $$ \hbox{Area}\,W = M_B-M_A; \leqno(35) $$ from equation (35) $$ AB = \hbox{Area of Circle $B$}\, - \,\hbox{Area $W$} =\pi \times {r_B}^2-W; \leqno(36) $$ thus, $$ B=W; \leqno(37) $$ from equation (35) $$ \hbox{TOTAL, the total area covered by the circles}\, $$ $$ = \hbox{Area of Circle $A$} + \hbox{Area $W$} = \pi + W; \leqno(38) $$ from equations (36) and (38) $$ AB\%=100\times AB/\hbox{TOTAL}; \leqno(39) $$ from equations (37) and (38) $$ B\%=100\times B/ \hbox{TOTAL}; \leqno(40) $$ and from equations (39) and (40) $$ A\%=100-AB\%-B\%. \leqno(41) $$ These results hold for $d_{AB}$ greater than or equal to zero, but not greater than $X$, the distance from the origin to the chord, that is: $$ 0 \leq d_{AB} \leq X, \leqno(42) $$ where from equation (26) $X=\hbox{cos}\,(Q_A)$. \vfill\eject \noindent{\bf Methods for Computing $r_B$ and $d_{AB}$.} The computation of $r_B$ given $A\%$, $B\%$, and $AB\%$ is straightforward. However, this is not the case for $d_{AB}$ in light of the fact that we did not obtain a function for $d_{AB}$ in terms of $A\%$, $B\%$, and $AB\%$. We present three numerical methods for estimating $d_{AB}$, each with a different level of accuracy. First, however, we derive $AB\%$ and $B\%$ as a function of $A$, $B$, and $AB$, and $r_B$ as a function of $AB\%$ and $B\%$. These derivations of $AB\%$, $B\%$ and $r_B$ are the same for all three methods of estimating $d_{AB}$. \noindent{\sl $r_B$ as a Function of $B\%$ and $AB\%$} Let $A$, $B$, and $AB$ be the $B$-circle only area, the $A$-circle only area, and the area of intersection, respectively, and let $TA=A+B+AB$; then, $$ A\%=100\times A/TA,\quad A=A\%\times TA/100; \leqno(43) $$ $$ B\%=100\times B/TA,\quad B=B\%\times TA/100; \leqno(44) $$ $$ AB\%=100\times AB/TA,\quad AB=AB\%\times TA/100; \leqno(45) $$ $$ \hbox{Circle $A$ area} = A+AB=\pi; \leqno(46) $$ $$ \hbox{Circle $B$ area} = B+AB=\pi \times {r_B}^2. \leqno(47) $$ Substituting equation (46) in equation (47): $B+AB = \pi \times {r_B}^2 = (A + AB)\times {r_B}^2$, $$ {r_B}^2 = (B+AB)/(A+AB). \leqno(48) $$ Substituting equations (43), (44), and (45) in equation (48): $$ {r_B}^2={{(B\%\times TA)/100+(AB\%\times TA)/100} \over {(A\%\times TA)/100+(AB\%\times TA)/100}} ={{B\% + AB\%} \over {A\% + AB\%}}. \leqno(49) $$ $$ A\% =100-B\%-AB\%, \leqno(50) $$ because $A\% +B\% +AB\% =100$. \noindent Substituting equation (50) into equation (49): $$ {r_B}^2={{B\%+AB\%} \over {(100-B\%-AB\%)+AB\%}} ={{B\%+AB\%} \over {100 -B\%}}. $$ Thus, $$ r_B = ((B\%+AB\%)/(100-B\%))^{1/2}. \leqno(51) $$ \vfill\eject \noindent{\sl Look-up Table Method for Estimating $d_{AB}$} Table 1 (at end of article) gives the value of $d_{AB}$ to 6 decimal places for all values of $AB\%$ from 0 to 100 and for $B\%$ from 0 to 50 in 5 percentage point increments for values of $d_{AB}$ from 0 to $r_A + r_B$. Note from Table 1 that some of the values are calculated using the procedure for Case I and some using the Case II procedure. The procedure used to obtain the values in Table 1 is summarized here. For each combination of $B\%$ and $AB\%$ in Table 1, calculate $d_{AB}$ as follows. First calculate $r_B$ using equation (51). Then guess a value for $d_{AB}$ that is approximately correct, and guess whether Case I or Case II applies. (In most areas of the table this is obvious.) Then calculate the values of $B\%$ and $AB\%$ using the guessed value of $d_{AB}$, the calculated value of $r_B$ and either equations (8) through (20) in Case I, or equations (23) through (40) in Case II. Then adjust the guessed value of $d_{AB}$ up or down and recalculate until the resulting values of $B\%$ and $AB\%$ approximate the desired values as closely as desired (six decimal points) in Table 1. Check the final value of $d_{AB}$ to be sure the correct calculation procedure was used (Case I or II) with the inequalities (22) or (42). Table 2 contains values of $d_{AB}$ for values of $AB\%$ in the 0 to 10 range. Between 0 and 5, $AB\%$ is in increments of 1. This table was produced because of the large and non-linear increments in $d_{AB}$ in this range of $AB\%$. If the given values of $B\%$ and $AB\%$ are one of the combinations found in Table 1 or Table 2, then the value of $d_{AB}$ can be obtained directly from the tables to six decimal point accuracy. If the exact values of $B\%$ and $AB\%$ are not in either table, then an interpolation procedure can be used. In Table 1, the procedure would be as follows. (a) Assume the given values of $B\%$ and $AB\%$ are not along the lower diagonal of the table, so that they are bounded by table values of $B\%$ and $AB\%$ at four corners forming a rectangle within the table. Let $d(i,j)$ be the value of $d_{AB}$ for any values of $AB\%$ (or $i$) and $B\%$ (or $j$), within the defined range. Then $d(AB\%,B\%)$ is the value of $d_{AB}$ at the given values of $B\%$ and $AB\%$. If $e$ is the value of $AB\%$ just less than the given $AB\%$, $f$ is the value of $AB\%$ just greater than the given $AB\%$, $g$ is the value of $B\%$ just less than the given $B\%$, and $h$ is the value of $B\%$ just greater than the given $B\%$, then, $$ d_K = \hbox{estimated value of $d$ at point $K$} $$ $$ = d(e,g)+(d(f,g)-d(e,g)) \times (AB\% -e)/(f-e), \leqno(52) $$ where $K$ is the intersection point of a horizontal through $d(i,j)$ with the vertical line through $g$. $$ d_L = \hbox{estimated value of $d$ at point $L$} $$ $$ = d(e,h)+(d(f,h)-d(e,h)) \times (AB\% -e)/(f-e), \leqno(53) $$ where $L$ is the intersection point of a horizontal through $d(i,j)$ with the vertical line through $h$. $$ \hbox{Estimate of}\,\,\, d(AB\%,B\%)=d_K+(d_L-d_K)\times (B\%-g)/(h-g). \leqno(54) $$ (b) Assume the given values of $AB\%$ and $B\%$ are near the lower diagonal of the defined range such that the location is bounded by only three table values (rather than by four table values, as in case (a) above). Use the same labelling as in case (a) above for the bounding table entries; note, however, that d(f,h) is not defined in this case (because of the nearness of the table entry to the lower diagonal). At the boundaries where $B\%=0$ or $AB\%=0$, equation (54) holds. Otherwise, we have: $$ d_K = \hbox{estimated value of $d$ at point $K$} $$ $$ = d(e,g)+(d(f,g)-d(e,g)) \times (AB\% -e)/(f-e), \leqno(55) $$ where $K$ is the intersection point of a horizontal through $d(i,j)$ with the vertical line through $g$. $$ d_L = \hbox{estimated value of $d$ at point $L$} $$ $$ = d(e,h)+(d(e,h)-d(e,g)) \times (B\% -g)/(h-g), \leqno(56) $$ where $L$ is the intersection point of a vertical through $d(i,j)$ with the horizontal line through $e$. $$ d(AB\%,B\%)=d(e,g)+(d_K-d(e,g))+(k_L-d(e,g)) =d_K+d_L-d(e,g) $$ $$ = d(e,g)+(d(f,g)-d(e,g))\times (AB\%-e)/(f-e) +(d(e,h)-d(e,g))\times (B\%-g)/(h-g). \leqno(57) $$ The use of formulas (54) and (57) in conjunction with Table 1 will generally produce answers for $d_{AB}$ within 0.01 of the correct figures, with the exception of the range for $AB\%$ from 0 to 5. In some areas of this range, particularly for $B\%$ greater than 45, the error can be over 0.05. For example, this method produces an estimated value for $d(2.5, 47.5) = 1.7394$, compared to the correct value of 1.7927, an error of 0.053. (This error is 5.3\% of the radius of circle $A$, which is 1, and is $100*0.053/1.7927=3\%$ of the correct value of $d_{AB}$.) If Table 2 is used for values of $AB\%$ between 0 and 5, the error can be reduced to less than 0.02 in the worst cases. For example, the use of Table 2 produces an estimated value for $d(0.5,49.5)=1.93016$, compared to the correct value of 1.91299, an error of 0.01717. (This is 1.7\% of the radius of circle $A$ and 0.9\% of the correct value of $d_{AB}$.) \noindent{\sl Polynomial Estimation of $d_{AB}$} The regression formul{\ae} were used to obtain polynomials in $AB\%$ and $B\%$ that estimated $d_{AB}$, in effect interpolating for values of $AB\%$ and $B\%$ between the grid points in Table 1. The range of $AB\%$ and $B\%$ was separated into three subranges and a polynomial was obtained for each subrange. The three subranges are specified below and also denoted graphically, using a variety of typefaces, in Table 3. $$ \hbox{Subrange 1:}\, AB\% > 5 \,\hbox{and}\, B\%\geq 5. \leqno(58) $$ $$ \hbox{Subrange 2:}\, 0 \leq AB\% \leq 5 \, \hbox{and}\, 0 \leq B\% \leq 50. \leqno(59) $$ $$ \hbox{Subrange 3:}\, 0 \leq AB\% \leq 100 \, \hbox{and}\, 0 \leq B\% < 5. \leqno(60) $$ The polynomials obtained for each subrange are given below. \noindent Polynomial 1 for subrange 1: $$ \hbox{est}\, {d_{AB}}^1=c_0+c_1X_1+c_2X_2+c_3X_3+c_4X_4 +c_5X_5+c_6X_6+c_7X_7+c_8X_8, \leqno(61) $$ where \qquad $c_0=\phantom{-}0.994388189$ \qquad $c_1=\phantom{-}0.003790799$, $X_1=AB\%$ \qquad $c_2= -0.001818030$, $X_2=B\%$ \qquad $c_3= -0.129148003$, $X_3=(AB\%)^{1/2}$ \qquad $c_4=\phantom{-}0.130891455$, $X_4=(B\%)^{1/2}$ \qquad $c_5= -0.000147200$, $X_5=(AB\%)\times (B\%)$ \qquad $c_6= -0.000017449$, $X_6=(AB\%)^2$ \qquad $c_7=\phantom{-}0.000081024$, $X_7=(B\%)^2$ \qquad $c_8= -0.004913375$, $X_8=(AB\%\times B\%)^{1/2}$. \noindent Polynomial 2 for subrange 2: $$ \hbox{est}\, {d_{AB}}^2=c_0+c_1X_1+c_2X_2+c_3X_3+c_4X_4 +c_5X_5+c_6X_6+c_7X_7, \leqno(62) $$ where \qquad $c_0=\phantom{-}1.003584849$ \qquad $c_1= -0.009650203$, $X_1=AB\%$ \qquad $c_2=\phantom{-}0.002712922$, $X_2=B\%$ \qquad $c_3= -0.089520075$, $X_3=(AB\%)^{1/2}$ \qquad $c_4=\phantom{-}0.093223275$, $X_4=(B\%)^{1/2}$ \qquad $c_5= -0.000366121$, $X_5=(AB\%)\times (B\%)$ \qquad $c_6= -0.000521608$, $X_6=(AB\%)^2$ \qquad $c_7=\phantom{-}0.000075950$, $X_7=(B\%)^2$. \noindent Polynomial 3 for subrange 3: $$ \hbox{est}\, {d_{AB}}^3=c_0+c_1X_1+c_2X_2+c_3X_3+c_4X_4 +c_5X_5+c_6X_6+c_7X_7+c_8X_8, \leqno(63) $$ where \qquad $c_0=\phantom{-}1.009984781$ \qquad $c_1=\phantom{-}0.001043059$, $X_1=AB\%$ \qquad $c_2=\phantom{-}0.009094475$, $X_2=B\%$ \qquad $c_3= -0.106641306$, $X_3=(AB\%)^{1/2}$ \qquad $c_4=\phantom{-}0.088497298$, $X_4=(B\%)^{1/2}$ \qquad $c_5= -0.000104701$, $X_5=(AB\%)\times (B\%)$ \qquad $c_6= -0.000052845$, $X_6=(AB\%)^2$ \qquad $c_7= -0.000084727$, $X_7=(B\%)^2$ \qquad $c_8= -0.007209240$, $X_8=(AB\%\times B\%)^{1/2}$. \noindent{\sl Numerical Search on the Inverse Function} The value of $d_{AB}$ can be obtained to any desired accuracy for any combination of $AB\%$ and $B\%$ in the defined range using the same procedure as was used to derive Table 1. We summarize the procedure below. \item{(1)} Given $A$, $B$, and $AB$. \item{(2)} Calculate $B\%$ using equation (44). \item{(3)} Calculate $AB\%$ using equation (45). \item{(4)} Calculate $r_B$ using equation (51). \item{(5)} Estimate an approximate value for $d_{AB}$ using Table 1. \item{(6)} Estimate whether Case I or Case II applies for the calculated values of $B\%$ and $AB\%$ using Tables 1 and 2. \item{(7)} Calculate estimated values of $AB\%$ and $B\%$ using the calculated value of $r_B$ from step 4, the estimated value of $d_{AB}$ from step 5 and equations (19) and (20) for Case I or equations (39) and (40) for Case II. \item{(8)} If the estimated value of $B\%$ obtained in step 7 is too small, increase the estimated value of $d_{AB}$ and recalculate $AB\%$ and $B\%$ by recycling through step 7; if $B\%$ is too big, reduce the estimated value of $d_{AB}$ and recycle through step 7. The size of the adjustment depends on the approximate slope in the region of concern. For example, if we were in a region of Table 1 where $d_{AB}$ increased 0.025 while $B\%$ was increasing by 5 (as is approximately the case for $B\%$ between 10 and 15 and $AB\%$=60), then adjust $d_{AB}$ by 0.005 for each error increment of 1 in $B\%$. In this way, the error in $B\%$ can be made as small as desired by continued recycling. The value of $AB\%$ converges to the desired value along with $B\%$. \item{(9)} Check to be sure that the proper case (I or II) was used by applying the inequalities (22) or (42). \vfill\eject \smallskip {\bf Editor's Note:} In the original submission the author also considered Venn diagrams of three circles, noting that the three-circle Venn diagram contains insufficient degrees of freedom to provide a general solution to a three characteristic situation. The reader interested in generalizations of the two-circle case might wish to examine the literature of Boolean algebra, particularly Karnaugh maps used in the minimization of switching circuits. More detail is presented in this presentation than would be in traditional publications, suggesting yet another avenue to explore in the dissemination of information across disciplinary boundaries and one way to offer detail that might be required by engineers in the field to implement abstract ideas presented in journals. The increase in cost, to present extra detail that may not be necessary to all, is minuscule in an electronic format. \smallskip \smallskip {\bf Author's Note:} The author wishes to thank anonymous referees for suggesting the viewpoint of ``equal area" Venn diagrams, and for substantial help in making the context of the problem reflect this viewpoint. \vfill\eject \vskip.5cm \centerline{TABLE 1} \centerline{TWO INTERSECTING CIRCLES PROBLEM} \centerline{Distance between centers (d), given $AB\%$ and $B$-ONLY $\%$} \smallskip \smallskip \hrule \vskip.5cm {\ee \settabs\+ &{\ee $AB\%$} &{\be 0.776393}\quad&{\be 0.868866}\quad&{\be 0.657898}\quad&{\be 0.590844} &{\ee 0.395288} &{\ee 1.307136} &{\ee 1.374962} &{\ee 1.289705} &{\ee 1.350925} &{\ee 1.413929} &{\ee 50}&\cr %sample line \+&&&&&&{\ee $B$}-Only {\ee $\%$}&& &&&&&&&& \cr \+&$AB\%$&00&05&10&15&20&25&30&35&40&45&50 \cr \+&00 &1 &1.229399 &1.33333 &1.42008 &1.5 &1.57735 &1.65464 &1.7337 &1.8163 &1.90453 &2 \cr \+&05 &{\be 0.776393}&0.982242 &1.082793 &1.164457 &1.23779 &1.307136 &1.374962 &1.443 &1.5127 &1.58547 & \cr \+&10 &{\be 0.683773}&{\be 0.868866}&0.961981 &1.037539 &1.10493 &1.168068 &1.229156 &1.289705 &1.350925 &1.413929& \cr \+&15 &{\be 0.612702}&{\be 0.782479}&0.86877 &0.938611 &1.0005 &1.05796 &1.112964 &1.166795 &1.220433&& \cr \+&20 &{\be 0.552787}&{\be 0.710053}&{\be 0.79011} &0.8546 &0.9113 &0.96341 &1.012665 &1.060145 &1.106585 && \cr \+&25 &{\be 0.5} &{\be 0.646459}&{\be 0.72074} &0.78016 &0.831894 &0.878849 &0.922537 &0.963822 &&& \cr \+&30 &{\be 0.452278}&{\be 0.589054}&{\be 0.657898}&0.712452 &0.759365 &0.801269 &0.839453 &0.874543 &&& \cr \+&35 &{\be 0.408393}&{\be 0.536264}&{\be 0.599914}&0.649738 &0.691905 &0.728788 &0.761409 &&&& \cr \+&40 &{\be 0.367545}&{\be 0.487058}&{\be 0.545676} &{\be 0.590844} &0.628273 &0.660049 &0.686956 &&&& \cr \+&45 &{\be 0.32918} &{\be 0.440711}&{\be 0.494394} &{\be 0.534916} &0.567539 &0.594046 &&&&& \cr \+&50 &{\be 0.292894}&{\be 0.396685}&{\be 0.445464} &{\be 0.48128} &0.508937 &0.529864 &&&&& \cr \+&55 &{\be 0.258381}&{\be 0.354559}&{\be 0.398393}&0.429363 &0.451767 &&&&&& \cr \+&60 &{\be 0.225404}&{\be 0.313984}&{\be 0.352752}&0.378617 &0.395288 &&&&&& \cr \+&65 &{\be 0.193775}&{\be 0.27465} &{\be 0.308123}&0.328451 &&&&&&& \cr \+&70 &{\be 0.16334} &{\be 0.236231}&{\be 0.264046}&0.278098 &&&&&&& \cr \+&75 &{\be 0.133975}&{\be 0.198489}&{\be 0.219922} &&&&&&&& \cr \+&80 &{\be 0.105573}&{\be 0.160916}&0.174756 &&&&&&&& \cr \+&85 &{\be 0.078049}&{\be 0.122853} &&&&&&&&& \cr \+&90 &{\be 0.051317}&0.082698 &&&&&&&&& \cr \+&95 &{\be 0.025321} &&&&&&&&&& \cr \+&100&{\be 0} &&&&&&&&&& \cr } \vskip.5cm \hrule \vskip.5cm \centerline{Case I: Chord joining intersection points lies between the two centers} \centerline{\bf Case II: Chord lies to one side of both centers.} \vfill\eject \vskip.5cm \centerline{TABLE 2} \centerline{DATA TABLE FOR TWO-CIRCLE PROBLEM} \centerline{Distance between centers (d), given $AB\%$ and $B$-ONLY $\%$} \centerline{for $AB\% = 00 - 05$} \smallskip \smallskip \hrule \vskip.5cm \settabs\+\quad &$AB\%$\quad &{\bf 0.776397}$\,$&{\bf 0.856395}$\,$&{\bf 0.897278}$\,$&{\bf 0.929644}$\,$ &0.957428$\,$ &0.982242$\,$ &\cr %sample line \+&&&$B$-Only $\%$&&&& & \cr \+&$AB\%$&00&01&02&03&04&05& \cr \+&00 &1 &1.100503 &1.142857 &1.175863 &1.204124 &1.229399 & \cr \+&01 &{\bf 0.9} &0.996625 &1.041666 &1.076413 &1.105877 &1.132029 & \cr \+&02 &{\bf 0.858578}&{\bf 0.949095}&0.993166 &1.027487 &1.056694 &1.082652 & \cr \+&03 &{\bf 0.826794}&{\bf 0.91298} &{\bf 0.95593} &0.989618 &1.018366 &1.044 & \cr \+&04 &{\bf 0.8} &{\bf 0.882799}&{\bf 0.924676}&{\bf 0.957696} &0.985978 &1.011199 & \cr \+&05 &{\bf 0.776397}&{\bf 0.856395}&{\bf 0.897278}&{\bf 0.929644} &0.957428 &0.982242 & \cr \vskip.5cm \hrule \vskip.5cm \smallskip \smallskip \hrule \vskip.5cm \settabs\+\quad &$AB\%$\quad &1.082793$\,$ &1.164457$\,$ &1.2270744$\,$ &1.307136$\,$ &1.374962$\,$ &1.480464$\,$ &1.552075$\,$ &\cr %sample line \+&&&&$B$-Only $\%$&&&&& \cr \+&$AB\%$&10&15&20&25&30&35&40 \cr \+&00 &1.33333 &1.42008 &1.5 &1.57735 &1.65464 &1.7337 &1.8163 & \cr \+&01 &1.237653 &1.324053 &1.402561 &1.477746 &1.552221 &1.627878 &1.706373 & \cr \+&02 &1.187417 &1.272749 &1.349919 &1.423492 &1.496065 &1.569499 &1.645393 & \cr \+&03 &1.147486 &1.231629 &1.307493 &1.379587 &1.450471 &1.521965 &1.595619 & \cr \+&04 &1.113248 &1.196158 &1.270744 &1.341437 &1.410751 &1.480464 &1.552075 & \cr \+&05 &1.082793 &1.164457 &1.23779 &1.307136 &1.374962 &1.443 &1.5127 & \cr \vskip.5cm \hrule \vskip.5cm \smallskip \smallskip \hrule \vskip.5cm \settabs\+\quad &$AB\%$\quad &1.627057$\,$ &1.642596$\,$ &1.658348$\,$ &1.674328$\,$ &1.980196$\,$ &2$\,$&\cr %sample line \+&&&$B$-Only $\%$&&&&& \cr \+&$AB\%$&45&46&47&48&49&50 \cr \+&00 &1.90453 &1.922958 &1.941696 &1.960768 &1.980196 &2 &\cr \+&01 &1.789387 &1.806694 &1.824274 &1.842146 &1.860327 & &\cr \+&02 &1.725354 &1.741988 &1.758872 &1.776022 &1.793457 & &\cr \+&03 &1.67297 &1.689029 &1.705319 &1.721855 & & &\cr \+&04 &1.627057 &1.642596 &1.658348 &1.674328 & & &\cr \+&05 &1.58547 &1.600524 &1.615776 & & & &\cr \vskip.5cm \hrule \vskip.5cm \centerline{Case I: Chord joining intersection points lies between the two centers} \centerline{\bf Case II: Chord lies to one side of both centers.} \vfill\eject \vskip.5cm \centerline{TABLE 3} \centerline{TWO INTERSECTING CIRCLES PROBLEM} \centerline{Error in the Estimated Distance between the two centers: (dest - $d$)} \smallskip \smallskip \hrule \vskip.5cm {\ee \settabs\+ &{\ee $AB\%$} &{\be \phantom{-}0.003585}\quad&{\be -0.001890} &{\be -0.000220} &{\be \phantom{-}0.002339} &{\be \phantom{-}0.005130}&{\be \phantom{-}0.007643} &{\be \phantom{-}0.009292}&{\be \phantom{-}0.009392} &{\be \phantom{-}0.006917}&{\be \phantom{-}0.000295} &{\be -0.01170}&\cr %sample line \+&&&&&&{\ee $B$}-Only {\ee $\%$}&& &&&&&&&& \cr \+&$AB\%$&00&05&10&15&20&25&30&35&40&45&50 \cr \+&00 &{\be \phantom{-}0.003585 } &{\be -0.00189} &{\be -0.00022} &{\be \phantom{-}0.002339 } &{\be \phantom{-}0.005130} &{\be \phantom{-}0.007643 } &{\be \phantom{-}0.009292 } &{\be \phantom{-}0.009392 } &{\be \phantom{-}0.006917} &{\be \phantom{-}0.000295 } &{\be -0.01170 } \cr \+&05 &{\be -0.00796} &{\be \phantom{-}0.000948 } &{\be -0.00314} &{\be -0.00465 } &{\be -0.00442} &{\be -0.00306 } &{\be -0.00110} &{\be \phantom{-}0.000865 } &{\be \phantom{-}0.002138} &{\be \phantom{-}0.001824 } & \cr \+&10 &{\se -0.00111} &-0.00320 &\phantom{-}0.000151 &\phantom{-}0.000254 &-0.00029 &-0.00075 &-0.00096 &-0.00100 &-0.00117 &-0.00193 & \cr \+&15 &{\se -0.00128}&-0.00331 &-0.00005 &-0.00039 &-0.00121 &-0.00152 &-0.00099 &\phantom{-}0.000469 &\phantom{-}0.002778 && \cr \+&20 &{\se -0.00096}&-0.00263 &\phantom{-}0.000458 &-0.00030 &-0.00137 &-0.00157 &-0.00037 &\phantom{-}0.002464 &\phantom{-}0.007088 && \cr \+&25 &{\se -0.00044} &-0.00166 &\phantom{-}0.001121&-0.00009 &-0.00143 &-0.00152 &\phantom{-}0.000319 &\phantom{-}0.004538 &&& \cr \+&30 &{\se \phantom{-}0.000144}&-0.00065 &\phantom{-}0.001711\phantom{-}\phantom{-}&0.000014 &-0.00158 &-0.00154 &\phantom{-}0.001003 &\phantom{-}0.006707 &&& \cr \+&35 &{\se \phantom{-}0.000727}&\phantom{-}0.000235 &\phantom{-}0.002122 &-0.00005 &-0.00188 &-0.00163 &\phantom{-}0.001762 &&&& \cr \+&40 &{\se \phantom{-}0.001248}&\phantom{-}0.00930 &\phantom{-}0.002310&-0.00031 &-0.00229 &-0.00168 &\phantom{-}0.002798 &&&& \cr \+&45 &{\se \phantom{-}0.001670} &\phantom{-}0.001372 &\phantom{-}0.002261 &-0.00074 &-0.00272 &-0.00151 &&&&& \cr \+&50 &{\se \phantom{-}0.001965}&\phantom{-}0.001534 &\phantom{-}0.001995&-0.00124 &-0.00300 &-0.00084 &&&&& \cr \+&55 &{\se \phantom{-}0.002114}&\phantom{-}0.001407 &\phantom{-}0.001563&-0.00172 &-0.00292 &&&&&& \cr \+&60 &{\se \phantom{-}0.002102}&\phantom{-}0.001011 &\phantom{-}0.001052&-0.00195 &-0.00207 &&&&&& \cr \+&65 &{\se \phantom{-}0.001914}&\phantom{-}0.000389 &\phantom{-}0.000606&-0.00165 &&&&&&& \cr \+&70 &{\se \phantom{-}0.001542} &-0.00037 &\phantom{-}0.000606&-0.00165 &&&&&&& \cr \+&75 &{\se \phantom{-}0.000976}&-0.00113 &\phantom{-}0.001043 &&&&&&&& \cr \+&80 &{\se \phantom{-}0.000210}&-0.00160 &\phantom{-}0.003186 &&&&&&&& \cr \+&85 &{\se -0.00076}&-0.00124 &&&&&&&&& \cr \+&90 &{\se -0.00194}&\phantom{-}0.001460 &&&&&&&&& \cr \+&95 &{\se -0.00334} &&&&&&&&&& \cr \+&100&{\se -0.00496} &&&&&&&&&& \cr } \vskip.5cm \hrule \vskip.5cm \centerline{Actual Distance from Table 1; Estimated Distance from Polynomials 1, 2, and 3.} \centerline{Polynomial 1} \centerline{\bf Polynomial 2} \centerline{\sl Polynomial 3} \vfill\eject \centerline{\bf Los Angeles, 1994--A Spatial Scientific View} \smallskip \smallskip \centerline{Sandra L. Arlinghaus,} \centerline{Institute of Mathematical Geography and University of Michigan} \smallskip \centerline{William C. Arlinghaus,} \centerline{Lawrence Technological University} \smallskip \centerline{Frank Harary,} \centerline{New Mexico State University} \smallskip \centerline{John D. Nystuen,} \centerline{University of Michigan} \smallskip An algorithm discussed by Maria Hasse (Hasse, 1961; Harary, Norman, and Cartwright, 1965) offers a method for finding the shortest distance between any two nodes in a network of $n$ nodes when given only distances between adjacent nodes. The algorithm is one that focuses on structure alone, and it is therefore spatial. The procedure is similar in form to that used to multiply matrices, given two $n \times n$ matrices $A$ and $B$. To find the entries in their Hasse-sum, matrix $C$, take the minimum of the row-by-column sums; thus, the entry $$ c(21) = \hbox{min} \{a(21)+b(12),a(22)+b(22),a(23)+b(32),...,a(2n)+b(n2)\}. $$ The results below show the outcome of applying this tool from theoretical spatial science to the real-world: to one change in the Los Angeles freeway pattern following the recent devastating earthquake (January 17, 1994). \noindent{\bf Los Angeles, 1994.} When a recent earthquake caused a disastrous collapse of a span of the Santa Monica freeway, according to television reports the world's busiest freeway (carrying an estimated 300,000 vehicles per weekday), municipal authorities managed to keep the city moving. They employed a well-balanced combination of alternate routing using intelligent vehicle highway systems (IVHS) in which traffic lights along surface routes paralleling freeways were coordinated in response to user demand, together with media messages urging people to stay off roadways and the effective dispersal of information concerning alternate routes. Outside forecasters of doom predicted massive gridlock that did not occur in regions where alternate routing was available. In the analysis below, we test Hasse's algorithm against a changed adjacency configuration and interpret the results. Indeed, what would a forecaster using the Hasse algorithm have predicted in this situation? The map in Figure 1 shows a portion of the Los Angeles freeway system, and nearby major surface arterials, linking Los Angeles International Airport (LAX) to the Central City (CC). We tightened our focus to consider what sort of impact the partial closing of the Santa Monica freeway might have on travel times to and from the airport and the downtown region. The routing in Figure 1 is along freeways, only, that form a square envelope around the direct diagonal route (that does not exist in the real world) linking LAX to the CC. Any rupture along this circuit will completely destroy one of the two possible routes, sending all the traffic along one path only. Thus, when the Santa Monica freeway was ruptured (Figure 2)--cross-hatched area on Figure 1 -- all the traffic would have been forced due east and then north, if only freeway linkages were employed. \midinsert \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad\qquad&LAX\quad &4 &----- &{\bf X} &----- &6 &----- &8 &----- &10 &----- &12 &----- &$\bullet $ &\quad CC\cr %sample line \+& &4& & & &6& &8& &10& &12& &16&CC&\cr \+& &$\bullet $&-----&{\bf X}&-----&$\bullet $&-----&$\bullet $ &-----&$\bullet $&-----&$\bullet $&-----&$\bullet $&&\cr \+& &$\vert $ & & & & & & & & & & & &$\vert $&&\cr \+& &$\vert $ & & & & & & & & & & & &$\vert $&&\cr \+& &$\vert $ & & & & & & & & & & & &$\vert $&&\cr \+&${\phantom{LA}}$3 &$\bullet$&&&&&&&&&&&&$\bullet $&15&\cr \+& &$\vert $ & & & & & & & & & & & &$\vert $&&\cr \+& &$\vert $ & & & & & & & & & & & &$\vert $&&\cr \+& &$\vert $ & & & & & & & & & & & &$\vert $&&\cr \+&${\phantom{LA}}$2&$\bullet $&&&&&&&&&&&&$\bullet $&14&\cr \+& &$\vert $ & & & & & & & & & & & &$\vert $&&\cr \+& &$\vert $ & & & & & & & & & & & &$\vert $&&\cr \+& &$\vert $ & & & & & & & & & & & &$\vert $&&\cr \+& &$\bullet $&-----&-----&-----&$\bullet $&-----&$\bullet $ &-----&$\bullet $&-----&$\bullet $&-----&$\bullet $&&\cr \+&LAX &1& & & &5& &7& &9& &11& &13&&\cr \vskip.5cm \noindent{\bf Figure 1.} LAX denotes Los Angeles International Airport. CC denotes the Central City. Routes are along major expressways. The X indicates the rupture in the Santa Monica freeway caused by the January 17, 1994 earthquake. Consider that all lines, whether dashed or solid, represent continuous graphical linkage between adjacent nodes. The only break in the freeway is at the X. \vskip.5cm \hrule \endinsert \midinsert \vskip.5cm \hrule \vskip5in \noindent{\bf Figure 2.} Drawing based on a photo, showing damage to the Los Angeles freeways, from the {\sl New York Times\/}, Tuesday, January 18, 1994. \vskip.5cm \hrule \endinsert \topinsert \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad\qquad&LAX\quad &4 &----- &{\bf X} &----- &6 &----- &8 &----- &10 &----- &12 &----- &$\bullet $ &\quad CC\cr %sample line \+& &4& & & &6& &8& &10& &12& &16&CC&\cr \+& &$\bullet $&-----&{\bf X}&-----&$\bullet $&-----&$\bullet $ &-----&$\bullet $&-----&$\bullet $&-----&$\bullet $&&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\vert $ & & & &18& &20& &22& &24& &$\vert $&&\cr \+&${\phantom{LA}}$3 &$\bullet $&-----&-----&-----&$\bullet $ &-----&$\bullet $&-----&$\bullet $&----- &$\bullet $&-----&$\bullet $&15&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\vert $ & & & &17& &19& &21& &23& &$\vert $&&\cr \+&${\phantom{LA}}$2 &$\bullet $&-----&-----&-----&$\bullet $ &-----&$\bullet $&-----&$\bullet $&----- &$\bullet $&-----&$\bullet $&14&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\bullet $&-----&-----&-----&$\bullet $&-----&$\bullet $ &-----&$\bullet $&-----&$\bullet $&-----&$\bullet $&&\cr \+&LAX &1& & & &5& &7& &9& &11& &13&&\cr \vskip.5cm \noindent{\bf Figure 3.} Same basic pattern as Figure 1, with surface routes added, and intersections of surface routes added as nodes in the graph. LAX denotes Los Angeles International Airport. CC denotes the Central City. Routes are along major expressways. The X indicates the rupture in the Santa Monica freeway caused by the January 17, 1994 earthquake. Consider that all lines, whether dashed or solid, represent continuous graphical linkage between adjacent nodes. The only break in the freeway is at the X. \vskip.5cm \hrule \endinsert To overcome this apparently disastrous traffic situation, it is natural to introduce alternate routes along roads that are already present. Indeed, in earlier mathematical references there is consideration of this sort of rerouting problem after some edges of a network have been deleted (Menger, 1927; Ford and Fulkerson, 1962). One set of major surface routes is added to the map in Figure 1 to offer a number of different routes (Figure 3). The matrix $A$ (Figure 4) displays time-distances in tabular form across the network shown in Figure 3. The entry of 12 in the first row, second column indicates that it takes 12 quarter-minutes to travel from the node labelled 1 to the node labelled 2. A zero in this matrix indicates that there are zero quarter-minutes required as travel time--thus, zeroes appear in this matrix only along the main diagonal. Nodes are treated as points within which no travel is possible. An asterisk indicates that there is no direct linkage between corresponding entries-- an asterisk in the (1,3) position indicates that there is no single edge of the graph linking node 1 to node 3. All numerical entries are expressed in quarter-minutes; the Pascal program (Figure 5), was written to display integral results. (Use of a spreadsheet is possible but is far more time-consuming.) Travel times were calculated from distances in the 1993 Rand McNally Road Atlas, assuming (from field experience) an average speed of 40 mph. \topinsert \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad\qquad&LAX\quad &4 &----- &{\bf X} &----- &6 &----- &8 &----- &10 &----- &12 &----- &$\bullet $ &\quad CC\cr %sample line \+& &4& & & &6& &8& &10& &12& &16&CC&\cr \+& &$\bullet $&-----&{\bf X}&-----&$\bullet $&-----&$\bullet $ &-----&$\bullet $&-----&$\bullet $&-----&$\bullet $&&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+&${\phantom{LA}}$3 &$\bullet $&-----&-----&-----&----- &-----&-----&-----&-----&----- &-----&-----&$\bullet $&15&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+&${\phantom{LA}}$2 &$\bullet $&-----&-----&-----&----- &-----&-----&-----&-----&----- &-----&-----&$\bullet $&14&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\vert $ & & & &$\vert $ & &$\vert$ & &$\vert $ & &$\vert $ & &$\vert $&&\cr \+& &$\bullet $&-----&-----&-----&$\bullet $&-----&$\bullet $ &-----&$\bullet $&-----&$\bullet $&-----&$\bullet $&&\cr \+&LAX &1& & & &5& &7& &9& &11& &13&&\cr \vskip.5cm \noindent{\bf Figure 6.} Same basic pattern as Figure 1, with surface routes added; unlike Figure 2, in this case intersections of surface routes are NOT added as nodes in the graph. The surface routes have limited access. LAX denotes Los Angeles International Airport. CC denotes the Central City. Routes are along major expressways. The X indicates the rupture in the Santa Monica freeway caused by the January 17, 1994 earthquake. Consider that all lines, whether dashed or solid, represent continuous graphical linkage between adjacent nodes. The only break in the freeway is at the X. \vskip.5cm \hrule \endinsert Higher powers of the matrix $A$ count numbers of paths of longer length. The matrix $A^2$ counts paths of 2 edges as well as those of one edge. Thus, in $A^2$ one would expect to find an entry indicating the total time to travel from node 1 to node 3, as well as entries representing travel times across single graphical edges from node 1 to node 2 and from node 2 to node 3; indeed, as would be hoped the travel time of 30 quarter-minutes from nodes 1 to 3 is the sum of the travel times from 1 to 2 (12 quarter-minutes) and from 2 to 3 (18 quarter-minutes). The Hasse operator (erroneously referred to as the Hedetniemi operator in some earlier work, corrected by F. Harary who also notes that this procedure may also be present in literature earlier than Hasse's 1961 article) always selects the shortest path if more than one is available. Other algorithms execute similar calculations; however, Floyd's algorithm provides easy display of lengths only (and not the components that compose them), while Dijkstra's algorithm is not designed for easy display of results but does permit the determination of the actual position of the shortest path. Both of these algorithms require the same number of steps independent of the actual data; Hasse's does not -- it stops shorter than would Floyd's or Dijkstra's in many situations. Further detail has been published elsewhere (Harary, Norman, and Cartwright, 1960; Arlinghaus, Arlinghaus, and Nystuen, 1990). The matrices $A$ through $A^8$ show travel times across paths of varying length for the freeway system prior to the earthquake (Figure 4a). The algorithm stops when $A^{n+1}=A^n$; in this case, therefore, the last matrix with new entries is $A^8$--the matrix $A^9$ is calculated to know when to stop the iteration. A different initial matrix is required to capture the linkage pattern between LAX and CC following the 1994 earthquake (Figure 4b)--the Santa Monica freeway was shattered between nodes 4 and 6 on the graph in Figure 3. The matrix $B$ in Figure 6 indicates a new adjacency pattern; in $A$, the 4th row, 6th column contained a value of 30 to represent the direct linkage between nodes 4 and 6. The corresponding entry in matrix $B$ is an asterisk -- that is, there is no path, of a single graphical edge, available between nodes 4 and 6. When Hasse's algorithm is run using $B$ (Figure 4b), instead of $A$ (Figure 4a), as the initial matrix, the iteration requires the same number of stages; however, some of the entries are larger in $B$ than in $A$, reflecting the need for longer paths to provide alternate routes around the earthquake-altered freeway. In the eighth iteration, the $B$-iterate contains entries in the (4,6), (4,8), (4,10), (4,12), and (4,16) positions that are about 30 quarter-minutes larger than are the entries in the corresponding eighth A-iterate. This increase in the structural model comes purely from spatial pattern--it does not address the natural increase in congestion that one would also expect. The surface route pattern that was introduced permitted all turns at each of the surface route intersections; this sort of strategy appears desirable, but because turns (especially U.S. left-hand turns onto two way streets) generally force additional slowing of the traffic one might consider further alteration of the structural model. Figure 6 shows a modified form of the map in Figure 3; in it, the nodes 17 through 24 have been omitted. This graphical omission corresponds to the real-world notion of preventing intersecting traffic flows within the interchanges. One way to reduce congestion is to prohibit all turns. Another is to use traffic lights in a manner that responds to the traffic itself, rather that to estimates of traffic. The structural model in Figure 6 represents this sort of approach; the north-south route from node 5 to node 6 does not intersect any of the east-west surface vehicular flows. Figure 7a shows the initial matrix $C$ representing this particular structural model that permits restricted pre-earthquake travel across surface routes. Figure 7b shows the initial matrix $D$ representing the model with the rupture in the Santa Monica freeway. When Hasse's algorithm is run, there are clearly once again a number of locations that stand out: the (4,6) entry, for example, goes from 30 quarter-minutes to 114 quarter-minutes in this case. Indeed, there is not even any path available of length less than 5 for this entry: there is an asterisk in this position in $D^4$ --the only asterisk for this entry in the $C$ iteration sequence, with the bridge in, is in the first matrix. The last entries to come into the $D$ sequence iteration are (4,8) and (4,9)--this situation tallies with the relationships shown on the map in Figure 3. Traffic engineers might choose this latter model during times of the day when volumes are not high at the nodes showing large increases, or some other strategy that responds to traffic history. The path structure from node 1 (LAX) to node 16 (CC) is not altered; the Santa Monica freeway was not the shortest route from LAX to CC although its closure no doubt adds to the congestion along shorter routes. Most of the entries in the fourth row of $D^7$ to the right of, and including, the sixth column show increases in time -- some only slight and some substantial. Only the fourth row and the fourth column show altered time patterns, pre- and post-earthquake -- Hasse's algorithm shows that the underlying spatial structure of the road network is sufficient to provide alternate routing to and from LAX to CC and between many of the intervening locations. This finding matches what has apparently happened in the actual post-earthquake environment. \noindent{\bf Policy Implications} In order to turn the elegant theoretical tool of Hasse into one a traffic engineer might actually employ, there are a number of policy implications to consider -- policy changes can put real-world teeth into theory. \item{1.} No turns except onto expressways means maximum flow; however, this strategy is awkward for those living in the area. Indeed, even if it is assumed that people can turn off onto minor streets but cannot turn at major intersections, these local turns cause a lower average speed. \item{2.} Permit right hand turns only --not too disruptive of flow so speed is maintained. The algorithm still holds, even with an asymmetric matrix. \item{3.} Permit all turns -- there are a number of engineering strategies that might have corresponding structural components in a graphical model. Left hand turns slow the system. Insert different average speeds or times on the edges of the structural model. \item{4.} Use one-way streets--this strategy equalizes left and right turns; it, too, produces asymmetric adjacency matrices. \vfill\eject \noindent{\bf References} \ref Arlinghaus, S., Arlinghaus, W., and Nystuen, J. 1990. The Hedetniemi matrix sum: an algorithm for shortest path and shortest distance. {\sl Geographical Analysis\/}, Vol. 22, No. 4, 351-360. \ref Ford, L. R., Jr., and Fulkerson, D. R. 1962. {\sl Flows in networks\/}. Princeton, N.J.: Princeton University Press. \ref Harary, F., Norman R., and Cartwright, D. 1965. {\sl Structural Models: An Introduction to the Theory of Directed Graphs\/}. New York: Wiley. \ref Hasse, Maria. 1961. Uber die Behandlung graphentheorischer Probleme unter Verwendung der Matrizenrechnung, {\sl Wiss. Z. Techn. Univer. Dresden\/}, {\bf 10}, 1313-1316. \ref Menger, K. Zur allgemeinen Kurventheorie. 1927. {\sl Fund. Math.\/}, {\bf 10}, 96-115. \vfill\eject \centerline{\bf Figures containing tables} \smallskip \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&* &* &18&* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* \cr \+&12&00&18&* &* &* &* &* &* &* &* &* &* &* &* &* &6 &* &* &* &* &* &* &* \cr \+&* &18&00&24&* &* &* &* &* &* &* &* &* &* &* &* &* &18&* &* &* &* &* &* \cr \+&* &* &24&00&* &30&* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* \cr \+&18&* &* &* &00&* &12&* & *&* &* &* &* &* &* &* &12&* &* &* &* &* &* &* \cr \+&* &* &* &30&* &00&* &12&* &* &* &* &* &* &* &* &* &21&* &* &* &* &* &* \cr \+&* &* &* &* &12&* &00&* &6 &* &* &* &* &* &* &* &* &* &15&* &* &* &* &* \cr \+&* &* &* &* &* &12&* &00&* &9 &* &* &* &* &* &* &* &* &* &21&* &* &* &* \cr \+&* &* &* &* &* &* &6 &* &00&* &6 &* &* &* &* &* &* &* &* &* &15&* &* &* \cr \+&* &* &* &* &* &* &* &9 &* &00&* &6 &* &* &* &* &* &* &* &* &* &21&* &* \cr \+&* &* &* &* &* &* &* &* &6 &* &00&* &6 &* &* &* &* &* &* &* &* &* &15&* \cr \+&* &* &* &* &* &* &* &* &* &6 &* &00&* &* &* &6 &* &* &* &* &* &* &* &21\cr \+&* &* &* &* &* &* &* &* &* &* &6 &* &00&12&* &* &* &* &* &* &* &* &* &* \cr \+&* &* &* &* &* &* &* &* &* &* &* &* &12&00&12&* &* &* &* &* &* &* &3 &* \cr \+&* &* &* &* &* &* &* &* &* &* &* &* &* &12&00&21&* &* &* &* &* &* &* &3 \cr \+&* &* &* &* &* &* &* &* &* &* &* &6 &* &* &21&00&* &* &* &* &* &* &* &* \cr \+&* &6 &* &* &12&* &* &* &* &* &* &* &* &* &* &* &00&12&9 &* &* &* &* &* \cr \+&* &* &18&* &* &21&* &* &* &* &* &* &* &* &* &* &12&00&* &9 &* &* &* &* \cr \+&* &* &* &* &* &* &15&* &* &* &* &* &* &* &* &* &9 &* &00&12&9 &* &* &* \cr \+&* &* &* &* &* &* &* &21&* &* &* &* &* &* &* &* &* &9 &12&00&* &9 &* &* \cr \+&* &* &* &* &* &* &* &* &15&* &* &* &* &* &* &* &* &* &9 &* &00&12&6 &* \cr \+&* &* &* &* &* &* &* &* &* &21&* &* &* &* &* &* &* &* &* &9 &12&00&* &12\cr \+&* &* &* &* &* &* &* &* &* &* &15&* &* &3 &* &* &* &* &* &* &6 &* &00&12\cr \+&* &* &* &* &* &* &* &* &* &* &* &21&* &* &3 &* &* &* &* &* &* &12&12&00\cr \vskip.5cm \noindent{\bf Figure 4a.i This is the initial matrix, $A$}. Figure 4a contains a set of nine tables (i to ix) illustrating the use of Hasse's algorithm on part of the LA freeway/surface route system, shown in Figure 3, prior to the earthquake of January 17, 1994. Travel times are in one-quarter minutes. An asterisk indicates that the travel time between locations is too large to enter the matrix. A double-zero indicates an entry of 0. \vskip.5cm \hrule \vfill\eject \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&* &18&* &30&* &* &* &* &* &* &* &* &* &18&* &* &* &* &* &* &* \cr \+&12&00&18&42&18&* &* &* &* &* &* &* &* &* &* &* &6 &18&15&* &* &* &* &* \cr \+&30&18&00&24&* &39&* &* &* &* &* &* &* &* &* &* &24&18&* &27&* &* &* &* \cr \+&* &42&24&00&* &30&* &42&* &* &* &* &* &* &* &* &* &42&* &* &* &* &* &* \cr \+&18&18&* &* &00&* &12&* &18&* &* &* &* &* &* &* &12&24&21&* &* &* &* &* \cr \+&* &* &39&30&* &00&* &12&* &21&* &* &* &* &* &* &33&21&* &30&* &* &* &* \cr \+&30&* &* &* &12&* &00&* &6 &* &12&* &* &* &* &* &24&* &15&27&21&* &* &* \cr \+&* &* &* &42&* &12&* &00&* &9 &* &15&* &* &* &* &* &30&33&21&* &30&* &* \cr \+&* &* &* &* &18&* &6 &* &00&* &6 &* &12&* &* &* &* &* &21&* &15&27&21&* \cr \+&* &* &* &* &* &21&* &9 &* &00&* &6 &* &* &* &12&* &* &* &30&33&21&* &27\cr \+&* &* &* &* &* &* &12&* &6 &* &00&* &6 &18&* &* &* &* &* &* &21&* &15&27\cr \+&* &* &* &* &* &* &* &15&* &6 &* &00&* &* &24&6 &* &* &* &* &* &27&33&21\cr \+&* &* &* &* &* &* &* &* &12&* &6 &* &00&12&24&* &* &* &* &* &* &* &15&* \cr \+&* &* &* &* &* &* &* &* &* &* &18&* &12&00&12&33&* &* &* &* &9 &* &3 &15\cr \+&* &* &* &* &* &* &* &* &* &* &* &24&24&12&00&21&* &* &* &* &* &15&15&3 \cr \+&* &* &* &* &* &* &* &* &* &12&* &6 &* &33&21&00&* &* &* &* &* &* &* &24\cr \+&18&6 &24&* &12&33&24&* &* &* &* &* &* &* &* &* &00&12&9 &21&18&* &* &* \cr \+&* &18&18&42&24&21&* &30&* &* &* &* &* &* &* &* &12&00&21&9 &* &18&* &* \cr \+&* &15&* &* &21&* &15&33&21&* &* &* &* &* &* &* &9 &21&00&12&9 &21&15&* \cr \+&* &* &27&* &* &30&27&21&* &30&* &* &* &* &* &* &21&9 &12&00&21&9 &* &21\cr \+&* &* &* &* &* &* &21&* &15&33&21&* &* &9 &* &* &18&* &9 &21&00&12&6 &18\cr \+&* &* &* &* &* &* &* &30&27&21&* &27&* &* &15&* &* &18&21&9 &12&00&18&12\cr \+&* &* &* &* &* &* &* &* &21&* &15&33&15&3 &15&* &* &* &15&* &6 &18&00&12\cr \+&* &* &* &* &* &* &* &* &* &27&27&21&* &15&3 &24&* &* &* &21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4a.ii. This is the power 2 matrix, $A^2$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&* &30&* &36&* &* &* &* &* &* &* &18&30&27&* &* &* &* &* \cr \+&12&00&18&42&18&39&30&* &* &* &* &* &* &* &* &* &6 &18&15&27&24&* &* &* \cr \+&30&18&00&24&36&39&* &48&* &* &* &* &* &* &* &* &24&18&33&27&* &36&* &* \cr \+&54&42&24&00&* &30&* &42&* &51&* &* &* &* &* &* &48&42&* &51&* &* &* &* \cr \+&18&18&36&* &00&45&12&* &18&* &24&* &* &* &* &* &12&24&21&33&30&* &* &* \cr \+&* &39&39&30&45&00&* &12&* &21&* &27&* &* &* &* &33&21&42&30&* &39&* &* \cr \+&30&30&* &* &12&* &00&48&6 &* &12&* &18&* &* &* &24&36&15&27&21&33&27&* \cr \+&* &* &48&42&* &12&48&00&* &9 &* &15&* &* &* &21&42&30&33&21&42&30&* &36\cr \+&36&* &* &* &18&* &6 &* &00&48&6 &* &12&24&* &* &30&* &21&33&15&27&21&33\cr \+&* &* &* &51&* &21&* &9 &48&00&* &6 &* &* &30&12&* &39&42&30&33&21&39&27\cr \+&* &* &* &* &24&* &12&* &6 &* &00&48&6 &18&30&* &* &* &27&* &21&33&15&27\cr \+&* &* &* &* &* &27&* &15&* &6 &48&00&* &36&24&6 &* &* &* &36&39&27&33&21\cr \+&* &* &* &* &* &* &18&* &12&* &6 &* &00&12&24&45&* &* &* &* &21&* &15&27\cr \+&* &* &* &* &* &* &* &* &24&* &18&36&12&00&12&33&* &* &18&* &9 &21&3 &15\cr \+&* &* &* &* &* &* &* &* &* &30&30&24&24&12&00&21&* &* &* &24&21&15&15&3 \cr \+&* &* &* &* &* &* &* &21&* &12&* &6 &45&33&21&00&* &* &* &* &* &33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&* &* &* &* &* &* &* &00&12&9 &21&18&30&24&* \cr \+&30&18&18&42&24&21&36&30&* &39&* &* &* &* &* &* &12&00&21&9 &30&18&* &30\cr \+&27&15&33&* &21&42&15&33&21&42&27&* &* &18&* &* &9 &21&00&12&9 &21&15&27\cr \+&* &27&27&51&33&30&27&21&33&30&* &36&* &* &24&* &21&9 &12&00&21&9 &27&21\cr \+&* &24&* &* &30&* &21&42&15&33&21&39&21&9 &21&* &18&30&9 &21&00&12&6 &18\cr \+&* &* &36&* &* &39&33&30&27&21&* &27&* &21&15&33&30&18&21&9 &12&00&18&12\cr \+&* &* &* &* &* &* &27&* &21&39&15&33&15&3 &15&36&24&* &15&27&6 &18&00&12\cr \+&* &* &* &* &* &* &* &36&33&27&27&21&27&15&3 &24&* &30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4a.iii This is the power 3 matrix, $A^3$}. \vskip.5cm \hrule \vfill\eject \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&51&30&* &36&* &42&* &* &* &* &* &18&30&27&39&36&* &* &* \cr \+&12&00&18&42&18&39&30&48&36&* &* &* &* &* &* &* &6 &18&15&27&24&36&30&* \cr \+&30&18&00&24&36&39&48&48&* &57&* &* &* &* &* &* &24&18&33&27&42&36&* &48\cr \+&54&42&24&00&60&30&* &42&* &51&* &57&* &* &* &* &48&42&57&51&* &60&* &* \cr \+&18&18&36&60&00&45&12&54&18&* &24&* &30&* &* &* &12&24&21&33&30&42&36&* \cr \+&51&39&39&30&45&00&57&12&* &21&* &27&* &* &* &33&33&21&42&30&51&39&* &48\cr \+&30&30&48&* &12&57&00&48&6 &54&12&* &18&30&* &* &24&36&15&27&21&33&27&39\cr \+&* &48&48&42&54&12&48&00&54&9 &* &15&* &* &39&21&42&30&33&21&42&30&48&36\cr \+&36&36&* &* &18&* &6 &54&00&48&6 &54&12&24&36&* &30&42&21&33&15&27&21&33\cr \+&* &* &57&51&* &21&54&9 &48&00&54&6 &* &42&30&12&51&39&42&30&33&21&39&27\cr \+&42&* &* &* &24&* &12&* &6 &54&00&48&6 &18&30&51&36&* &27&39&21&33&15&27\cr \+&* &* &* &57&* &27&* &15&54&6 &48&00&48&36&24&6 &* &45&48&36&39&27&33&21\cr \+&* &* &* &* &30&* &18&* &12&* &6 &48&00&12&24&45&* &* &30&* &21&33&15&27\cr \+&* &* &* &* &* &* &30&* &24&42&18&36&12&00&12&33&27&* &18&30&9 &21&3 &15\cr \+&* &* &* &* &* &* &* &39&36&30&30&24&24&12&00&21&* &33&30&24&21&15&15&3 \cr \+&* &* &* &* &* &33&* &21&* &12&51&6 &45&33&21&00&* &* &* &42&42&33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&51&36&* &* &27&* &* &00&12&9 &21&18&30&24&36\cr \+&30&18&18&42&24&21&36&30&42&39&* &45&* &* &33&* &12&00&21&9 &30&18&36&30\cr \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&* &9 &21&00&12&9 &21&15&27\cr \+&39&27&27&51&33&30&27&21&33&30&39&36&* &30&24&42&21&9 &12&00&21&9 &27&21\cr \+&36&24&42&* &30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 &18\cr \+&* &36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 &12&00&18&12\cr \+&* &30&* &* &36&* &27&48&21&39&15&33&15&3 &15&36&24&* &15&27&6 &18&00&12\cr \+&* &* &48&* &* &48&39&36&33&27&27&21&27&15&3 &24&* &30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4a.iv This is the power 4 matrix, $A^4$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&51&30&60&36&* &42&* &48&* &* &* &18&30&27&39&36&48&42&* \cr \+&12&00&18&42&18&39&30&48&36&57&42&* &* &33&* &* &6 &18&15&27&24&36&30&42\cr \+&30&18&00&24&36&39&48&48&54&57&* &63&* &* &51&* &24&18&33&27&42&36&48&48\cr \+&54&42&24&00&60&30&72&42&* &51&* &57&* &* &* &63&48&42&57&51&66&60&* &72\cr \+&18&18&36&60&00&45&12&54&18&63&24&* &30&39&* &* &12&24&21&33&30&42&36&48\cr \+&51&39&39&30&45&00&57&12&63&21&* &27&* &* &51&33&33&21&42&30&51&39&57&48\cr \+&30&30&48&72&12&57&00&48&6 &54&12&60&18&30&42&* &24&36&15&27&21&33&27&39\cr \+&60&48&48&42&54&12&48&00&54&9 &60&15&* &51&39&21&42&30&33&21&42&30&48&36\cr \+&36&36&54&* &18&63&6 &54&00&48&6 &54&12&24&36&57&30&42&21&33&15&27&21&33\cr \+&* &57&57&51&63&21&54&9 &48&00&54&6 &54&42&30&12&51&39&42&30&33&21&39&27\cr \+&42&42&* &* &24&* &12&60&6 &54&00&48&6 &18&30&51&36&48&27&39&21&33&15&27\cr \+&* &* &63&57&* &27&60&15&54&6 &48&00&48&36&24&6 &57&45&48&36&39&27&33&21\cr \+&48&* &* &* &30&* &18&* &12&54&6 &48&00&12&24&45&39&* &30&* &21&33&15&27\cr \+&* &33&* &* &39&* &30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 &15\cr \+&* &* &51&* &* &51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr \+&* &* &* &63&* &33&* &21&57&12&51&6 &45&33&21&00&* &51&51&42&42&33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&* &00&12&9 &21&18&30&24&36\cr \+&30&18&18&42&24&21&36&30&42&39&48&45&* &39&33&51&12&00&21&9 &30&18&36&30\cr \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 &21&15&27\cr \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 &27&21\cr \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 &18\cr \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 &12&00&18&12\cr \+&42&30&48&* &36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 &18&00&12\cr \+&* &42&48&72&48&48&39&36&33&27&27&21&27&15&3 &24&36&30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4a.v This is the power 5 matrix, $A^5$}. \vskip.5cm \hrule \vfill\eject \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&51&30&60&36&69&42&* &48&45&* &* &18&30&27&39&36&48&42&54\cr \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&* &6 &18&15&27&24&36&30&42\cr \+&30&18&00&24&36&39&48&48&54&57&60&63&* &51&51&69&24&18&33&27&42&36&48&48\cr \+&54&42&24&00&60&30&72&42&78&51&* &57&* &* &75&63&48&42&57&51&66&60&72&72\cr \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&* &12&24&21&33&30&42&36&48\cr \+&51&39&39&30&45&00&57&12&63&21&69&27&* &60&51&33&33&21&42&30&51&39&57&48\cr \+&30&30&48&72&12&57&00&48&6 &54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr \+&60&48&48&42&54&12&48&00&54&9 &60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr \+&36&36&54&78&18&63&6 &54&00&48&6 &54&12&24&36&57&30&42&21&33&15&27&21&33\cr \+&69&57&57&51&63&21&54&9 &48&00&54&6 &54&42&30&12&51&39&42&30&33&21&39&27\cr \+&42&42&60&* &24&69&12&60&6 &54&00&48&6 &18&30&51&36&48&27&39&21&33&15&27\cr \+&* &63&63&57&69&27&60&15&54&6 &48&00&48&36&24&6 &57&45&48&36&39&27&33&21\cr \+&48&45&* &* &30&* &18&63&12&54&6 &48&00&12&24&45&39&51&30&42&21&33&15&27\cr \+&45&33&51&* &39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 &15\cr \+&* &45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr \+&* &* &69&63&* &33&63&21&57&12&51&6 &45&33&21&00&60&51&51&42&42&33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 &21&18&30&24&36\cr \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 &30&18&36&30\cr \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 &21&15&27\cr \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 &27&21\cr \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 &18\cr \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 &12&00&18&12\cr \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 &18&00&12\cr \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 &24&36&30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4a.vi This is the power 6 matrix, $A^6$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&51&30&60&36&69&42&75&48&45&57&* &18&30&27&39&36&48&42&54\cr \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&66&6 &18&15&27&24&36&30&42\cr \+&30&18&00&24&36&39&48&48&54&57&60&63&63&51&51&69&24&18&33&27&42&36&48&48\cr \+&54&42&24&00&60&30&72&42&78&51&84&57&* &75&75&63&48&42&57&51&66&60&72&72\cr \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&72&12&24&21&33&30&42&36&48\cr \+&51&39&39&30&45&00&57&12&63&21&69&27&72&60&51&33&33&21&42&30&51&39&57&48\cr \+&30&30&48&72&12&57&00&48&6 &54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr \+&60&48&48&42&54&12&48&00&54&9 &60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr \+&36&36&54&78&18&63&6 &54&00&48&6 &54&12&24&36&57&30&42&21&33&15&27&21&33\cr \+&69&57&57&51&63&21&54&9 &48&00&54&6 &54&42&30&12&51&39&42&30&33&21&39&27\cr \+&42&42&60&84&24&69&12&60&6 &54&00&48&6 &18&30&51&36&48&27&39&21&33&15&27\cr \+&75&63&63&57&69&27&60&15&54&6 &48&00&48&36&24&6 &57&45&48&36&39&27&33&21\cr \+&48&45&63&* &30&72&18&63&12&54&6 &48&00&12&24&45&39&51&30&42&21&33&15&27\cr \+&45&33&51&75&39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 &15\cr \+&57&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr \+&* &66&69&63&72&33&63&21&57&12&51&6 &45&33&21&00&60&51&51&42&42&33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 &21&18&30&24&36\cr \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 &30&18&36&30\cr \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 &21&15&27\cr \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 &27&21\cr \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 &18\cr \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 &12&00&18&12\cr \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 &18&00&12\cr \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 &24&36&30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4a.vii This is the power 7 matrix, $A^7$}. \vskip.5cm \hrule \vfill\eject \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&51&30&60&36&69&42&75&48&45&57&78&18&30&27&39&36&48&42&54\cr \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&66&6 &18&15&27&24&36&30&42\cr \+&30&18&00&24&36&39&48&48&54&57&60&63&63&51&51&69&24&18&33&27&42&36&48&48\cr \+&54&42&24&00&60&30&72&42&78&51&84&57&87&75&75&63&48&42&57&51&66&60&72&72\cr \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&72&12&24&21&33&30&42&36&48\cr \+&51&39&39&30&45&00&57&12&63&21&69&27&72&60&51&33&33&21&42&30&51&39&57&48\cr \+&30&30&48&72&12&57&00&48&6 &54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr \+&60&48&48&42&54&12&48&00&54&9 &60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr \+&36&36&54&78&18&63&6 &54&00&48&6 &54&12&24&36&57&30&42&21&33&15&27&21&33\cr \+&69&57&57&51&63&21&54&9 &48&00&54&6 &54&42&30&12&51&39&42&30&33&21&39&27\cr \+&42&42&60&84&24&69&12&60&6 &54&00&48&6 &18&30&51&36&48&27&39&21&33&15&27\cr \+&75&63&63&57&69&27&60&15&54&6 &48&00&48&36&24&6 &57&45&48&36&39&27&33&21\cr \+&48&45&63&87&30&72&18&63&12&54&6 &48&00&12&24&45&39&51&30&42&21&33&15&27\cr \+&45&33&51&75&39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 &15\cr \+&57&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr \+&78&66&69&63&72&33&63&21&57&12&51&6 &45&33&21&00&60&51&51&42&42&33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 &21&18&30&24&36\cr \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 &30&18&36&30\cr \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 &21&15&27\cr \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 &27&21\cr \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 &18\cr \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 &12&00&18&12\cr \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 &18&00&12\cr \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 &24&36&30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4a.viii This is the power 8 matrix, $A^8$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&51&30&60&36&69&42&75&48&45&57&78&18&30&27&39&36&48&42&54\cr \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&66&6 &18&15&27&24&36&30&42\cr \+&30&18&00&24&36&39&48&48&54&57&60&63&63&51&51&69&24&18&33&27&42&36&48&48\cr \+&54&42&24&00&60&30&72&42&78&51&84&57&87&75&75&63&48&42&57&51&66&60&72&72\cr \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&72&12&24&21&33&30&42&36&48\cr \+&51&39&39&30&45&00&57&12&63&21&69&27&72&60&51&33&33&21&42&30&51&39&57&48\cr \+&30&30&48&72&12&57&00&48&6 &54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr \+&60&48&48&42&54&12&48&00&54&9 &60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr \+&36&36&54&78&18&63&6 &54&00&48&6 &54&12&24&36&57&30&42&21&33&15&27&21&33\cr \+&69&57&57&51&63&21&54&9 &48&00&54&6 &54&42&30&12&51&39&42&30&33&21&39&27\cr \+&42&42&60&84&24&69&12&60&6 &54&00&48&6 &18&30&51&36&48&27&39&21&33&15&27\cr \+&75&63&63&57&69&27&60&15&54&6 &48&00&48&36&24&6 &57&45&48&36&39&27&33&21\cr \+&48&45&63&87&30&72&18&63&12&54&6 &48&00&12&24&45&39&51&30&42&21&33&15&27\cr \+&45&33&51&75&39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 &15\cr \+&57&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr \+&78&66&69&63&72&33&63&21&57&12&51&6 &45&33&21&00&60&51&51&42&42&33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 &21&18&30&24&36\cr \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 &30&18&36&30\cr \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 &21&15&27\cr \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 &27&21\cr \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 &18\cr \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 &12&00&18&12\cr \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 &18&00&12\cr \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 &24&36&30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4a.ix This is the power 9 matrix, $A^9$}. It is identical to the matrix in Figure 4a.viii and so the algorithm terminates. \vskip.5cm \hrule \vfill\eject \centerline{\bf Figures containing tables} \smallskip \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&* &* &18&* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* \cr \+&12&00&18&* &* &* &* &* &* &* &* &* &* &* &* &* &6 &* &* &* &* &* &* &* \cr \+&* &18&00&24&* &* &* &* &* &* &* &* &* &* &* &* &* &18&* &* &* &* &* &* \cr \+&* &* &24&00&* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* &* \cr \+&18&* &* &* &00&* &12&* & *&* &* &* &* &* &* &* &12&* &* &* &* &* &* &* \cr \+&* &* &* &* &* &00&* &12&* &* &* &* &* &* &* &* &* &21&* &* &* &* &* &* \cr \+&* &* &* &* &12&* &00&* &6 &* &* &* &* &* &* &* &* &* &15&* &* &* &* &* \cr \+&* &* &* &* &* &12&* &00&* &9 &* &* &* &* &* &* &* &* &* &21&* &* &* &* \cr \+&* &* &* &* &* &* &6 &* &00&* &6 &* &* &* &* &* &* &* &* &* &15&* &* &* \cr \+&* &* &* &* &* &* &* &9 &* &00&* &6 &* &* &* &* &* &* &* &* &* &21&* &* \cr \+&* &* &* &* &* &* &* &* &6 &* &00&* &6 &* &* &* &* &* &* &* &* &* &15&* \cr \+&* &* &* &* &* &* &* &* &* &6 &* &00&* &* &* &6 &* &* &* &* &* &* &* &21\cr \+&* &* &* &* &* &* &* &* &* &* &6 &* &00&12&* &* &* &* &* &* &* &* &* &* \cr \+&* &* &* &* &* &* &* &* &* &* &* &* &12&00&12&* &* &* &* &* &* &* &3 &* \cr \+&* &* &* &* &* &* &* &* &* &* &* &* &* &12&00&21&* &* &* &* &* &* &* &3 \cr \+&* &* &* &* &* &* &* &* &* &* &* &6 &* &* &21&00&* &* &* &* &* &* &* &* \cr \+&* &6 &* &* &12&* &* &* &* &* &* &* &* &* &* &* &00&12&9 &* &* &* &* &* \cr \+&* &* &18&* &* &21&* &* &* &* &* &* &* &* &* &* &12&00&* &9 &* &* &* &* \cr \+&* &* &* &* &* &* &15&* &* &* &* &* &* &* &* &* &9 &* &00&12&9 &* &* &* \cr \+&* &* &* &* &* &* &* &21&* &* &* &* &* &* &* &* &* &9 &12&00&* &9 &* &* \cr \+&* &* &* &* &* &* &* &* &15&* &* &* &* &* &* &* &* &* &9 &* &00&12&6 &* \cr \+&* &* &* &* &* &* &* &* &* &21&* &* &* &* &* &* &* &* &* &9 &12&00&* &12\cr \+&* &* &* &* &* &* &* &* &* &* &15&* &* &3 &* &* &* &* &* &* &6 &* &00&12\cr \+&* &* &* &* &* &* &* &* &* &* &* &21&* &* &3 &* &* &* &* &* &* &12&12&00\cr \vskip.5cm \noindent{\bf Figure 4b.i This is the initial matrix, $B$}. Figure 4b contains a set of nine tables (i to ix) illustrating the use of Hasse's algorithm on the LA freeway system following the earthquake of January 17, 1994. Travel times are in one-quarter minutes. An asterisk indicates that the travel time between locations is too large to enter the matrix. A double-zero indicates an entry of 0. \vskip.5cm \hrule \vfill\eject \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&* &18&* &30&* &* &* &* &* &* &* &* &* &18&* &* &* &* &* &* &* \cr \+&12&00&18&42&18&* &* &* &* &* &* &* &* &* &* &* &6 &18&15&* &* &* &* &* \cr \+&30&18&00&24&* &39&* &* &* &* &* &* &* &* &* &* &24&18&* &27&* &* &* &* \cr \+&* &42&24&00&* &* &* &* &* &* &* &* &* &* &* &* &* &42&* &* &* &* &* &* \cr \+&18&18&* &* &00&* &12&* &18&* &* &* &* &* &* &* &12&24&21&* &* &* &* &* \cr \+&* &* &39&* &* &00&* &12&* &21&* &* &* &* &* &* &33&21&* &30&* &* &* &* \cr \+&30&* &* &* &12&* &00&* &6 &* &12&* &* &* &* &* &24&* &15&27&21&* &* &* \cr \+&* &* &* &* &* &12&* &00&* &9 &* &15&* &* &* &* &* &30&33&21&* &30&* &* \cr \+&* &* &* &* &18&* &6 &* &00&* &6 &* &12&* &* &* &* &* &21&* &15&27&21&* \cr \+&* &* &* &* &* &21&* &9 &* &00&* &6 &* &* &* &12&* &* &* &30&33&21&* &27\cr \+&* &* &* &* &* &* &12&* &6 &* &00&* &6 &18&* &* &* &* &* &* &21&* &15&27\cr \+&* &* &* &* &* &* &* &15&* &6 &* &00&* &* &24&6 &* &* &* &* &* &27&33&21\cr \+&* &* &* &* &* &* &* &* &12&* &6 &* &00&12&24&* &* &* &* &* &* &* &15&* \cr \+&* &* &* &* &* &* &* &* &* &* &18&* &12&00&12&33&* &* &* &* &9 &* &3 &15\cr \+&* &* &* &* &* &* &* &* &* &* &* &24&24&12&00&21&* &* &* &* &* &15&15&3 \cr \+&* &* &* &* &* &* &* &* &* &12&* &6 &* &33&21&00&* &* &* &* &* &* &* &24\cr \+&18&6 &24&* &12&33&24&* &* &* &* &* &* &* &* &* &00&12&9 &21&18&* &* &* \cr \+&* &18&18&42&24&21&* &30&* &* &* &* &* &* &* &* &12&00&21&9 &* &18&* &* \cr \+&* &15&* &* &21&* &15&33&21&* &* &* &* &* &* &* &9 &21&00&12&9 &21&15&* \cr \+&* &* &27&* &* &30&27&21&* &30&* &* &* &* &* &* &21&9 &12&00&21&9 &* &21\cr \+&* &* &* &* &* &* &21&* &15&33&21&* &* &9 &* &* &18&* &9 &21&00&12&6 &18\cr \+&* &* &* &* &* &* &* &30&27&21&* &27&* &* &15&* &* &18&21&9 &12&00&18&12\cr \+&* &* &* &* &* &* &* &* &21&* &15&33&15&3 &15&* &* &* &15&* &6 &18&00&12\cr \+&* &* &* &* &* &* &* &* &* &27&27&21&* &15&3 &24&* &* &* &21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4b.ii. This is the power 2 matrix, $B^2$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&* &30&* &36&* &* &* &* &* &* &* &18&30&27&* &* &* &* &* \cr \+&12&00&18&42&18&39&30&* &* &* &* &* &* &* &* &* &6 &18&15&27&24&* &* &* \cr \+&30&18&00&24&36&39&* &48&* &* &* &* &* &* &* &* &24&18&33&27&* &36&* &* \cr \+&54&42&24&00&* &63&* &* &* &* &* &* &* &* &* &* &48&42&* &51&* &* &* &* \cr \+&18&18&36&* &00&45&12&* &18&* &24&* &* &* &* &* &12&24&21&33&30&* &* &* \cr \+&* &39&39&63&45&00&* &12&* &21&* &27&* &* &* &* &33&21&42&30&* &39&* &* \cr \+&30&30&* &* &12&* &00&48&6 &* &12&* &18&* &* &* &24&36&15&27&21&33&27&* \cr \+&* &* &48&* &* &12&48&00&* &9 &* &15&* &* &* &21&42&30&33&21&42&30&* &36\cr \+&36&* &* &* &18&* &6 &* &00&48&6 &* &12&24&* &* &30&* &21&33&15&27&21&33\cr \+&* &* &* &* &* &21&* &9 &48&00&* &6 &* &* &30&12&* &39&42&30&33&21&39&27\cr \+&* &* &* &* &24&* &12&* &6 &* &00&48&6 &18&30&* &* &* &27&* &21&33&15&27\cr \+&* &* &* &* &* &27&* &15&* &6 &48&00&* &36&24&6 &* &* &* &36&39&27&33&21\cr \+&* &* &* &* &* &* &18&* &12&* &6 &* &00&12&24&45&* &* &* &* &21&* &15&27\cr \+&* &* &* &* &* &* &* &* &24&* &18&36&12&00&12&33&* &* &18&* &9 &21&3 &15\cr \+&* &* &* &* &* &* &* &* &* &30&30&24&24&12&00&21&* &* &* &24&21&15&15&3 \cr \+&* &* &* &* &* &* &* &21&* &12&* &6 &45&33&21&00&* &* &* &* &* &33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&* &* &* &* &* &* &* &00&12&9 &21&18&30&24&* \cr \+&30&18&18&42&24&21&36&30&* &39&* &* &* &* &* &* &12&00&21&9 &30&18&* &30\cr \+&27&15&33&* &21&42&15&33&21&42&27&* &* &18&* &* &9 &21&00&12&9 &21&15&27\cr \+&* &27&27&51&33&30&27&21&33&30&* &36&* &* &24&* &21&9 &12&00&21&9 &27&21\cr \+&* &24&* &* &30&* &21&42&15&33&21&39&21&9 &21&* &18&30&9 &21&00&12&6 &18\cr \+&* &* &36&* &* &39&33&30&27&21&* &27&* &21&15&33&30&18&21&9 &12&00&18&12\cr \+&* &* &* &* &* &* &27&* &21&39&15&33&15&3 &15&36&24&* &15&27&6 &18&00&12\cr \+&* &* &* &* &* &* &* &36&33&27&27&21&27&15&3 &24&* &30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4b.iii This is the power 3 matrix, $B^3$}. \vskip.5cm \hrule \vfill\eject \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&51&30&* &36&* &42&* &* &* &* &* &18&30&27&39&36&* &* &* \cr \+&12&00&18&42&18&39&30&48&36&* &* &* &* &* &* &* &6 &18&15&27&24&36&30&* \cr \+&30&18&00&24&36&39&48&48&* &57&* &* &* &* &* &* &24&18&33&27&42&36&* &48\cr \+&54&42&24&00&60&63&* &72&* &* &* &* &* &* &* &* &48&42&57&51&* &60&* &* \cr \+&18&18&36&60&00&45&12&54&18&* &24&* &30&* &* &* &12&24&21&33&30&42&36&* \cr \+&51&39&39&63&45&00&57&12&* &21&* &27&* &* &* &33&33&21&42&30&51&39&* &48\cr \+&30&30&48&* &12&57&00&48&6 &54&12&* &18&30&* &* &24&36&15&27&21&33&27&39\cr \+&* &48&48&72&54&12&48&00&54&9 &* &15&* &* &39&21&42&30&33&21&42&30&48&36\cr \+&36&36&* &* &18&* &6 &54&00&48&6 &54&12&24&36&* &30&42&21&33&15&27&21&33\cr \+&* &* &57&* &* &21&54&9 &48&00&54&6 &* &42&30&12&51&39&42&30&33&21&39&27\cr \+&42&* &* &* &24&* &12&* &6 &54&00&48&6 &18&30&51&36&* &27&39&21&33&15&27\cr \+&* &* &* &* &* &27&* &15&54&6 &48&00&48&36&24&6 &* &45&48&36&39&27&33&21\cr \+&* &* &* &* &30&* &18&* &12&* &6 &48&00&12&24&45&* &* &30&* &21&33&15&27\cr \+&* &* &* &* &* &* &30&* &24&42&18&36&12&00&12&33&27&* &18&30&9 &21&3 &15\cr \+&* &* &* &* &* &* &* &39&36&30&30&24&24&12&00&21&* &33&30&24&21&15&15&3 \cr \+&* &* &* &* &* &33&* &21&* &12&51&6 &45&33&21&00&* &* &* &42&42&33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&51&36&* &* &27&* &* &00&12&9 &21&18&30&24&36\cr \+&30&18&18&42&24&21&36&30&42&39&* &45&* &* &33&* &12&00&21&9 &30&18&36&30\cr \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&* &9 &21&00&12&9 &21&15&27\cr \+&39&27&27&51&33&30&27&21&33&30&39&36&* &30&24&42&21&9 &12&00&21&9 &27&21\cr \+&36&24&42&* &30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 &18\cr \+&* &36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 &12&00&18&12\cr \+&* &30&* &* &36&* &27&48&21&39&15&33&15&3 &15&36&24&* &15&27&6 &18&00&12\cr \+&* &* &48&* &* &48&39&36&33&27&27&21&27&15&3 &24&* &30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4b.iv This is the power 4 matrix, $B^4$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&51&30&60&36&* &42&* &48&* &* &* &18&30&27&39&36&48&42&* \cr \+&12&00&18&42&18&39&30&48&36&57&42&* &* &33&* &* &6 &18&15&27&24&36&30&42\cr \+&30&18&00&24&36&39&48&48&54&57&* &63&* &* &51&* &24&18&33&27&42&36&48&48\cr \+&54&42&24&00&60&63&72&72&* &81&* &* &* &* &* &* &48&42&57&51&66&60&* &72\cr \+&18&18&36&60&00&45&12&54&18&63&24&* &30&39&* &* &12&24&21&33&30&42&36&48\cr \+&51&39&39&63&45&00&57&12&63&21&* &27&* &* &51&33&33&21&42&30&51&39&57&48\cr \+&30&30&48&72&12&57&00&48&6 &54&12&60&18&30&42&* &24&36&15&27&21&33&27&39\cr \+&60&48&48&72&54&12&48&00&54&9 &60&15&* &51&39&21&42&30&33&21&42&30&48&36\cr \+&36&36&54&* &18&63&6 &54&00&48&6 &54&12&24&36&57&30&42&21&33&15&27&21&33\cr \+&* &57&57&81&63&21&54&9 &48&00&54&6 &54&42&30&12&51&39&42&30&33&21&39&27\cr \+&42&42&* &* &24&* &12&60&6 &54&00&48&6 &18&30&51&36&48&27&39&21&33&15&27\cr \+&* &* &63&* &* &27&60&15&54&6 &48&00&48&36&24&6 &57&45&48&36&39&27&33&21\cr \+&48&* &* &* &30&* &18&* &12&54&6 &48&00&12&24&45&39&* &30&* &21&33&15&27\cr \+&* &33&* &* &39&* &30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 &15\cr \+&* &* &51&* &* &51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr \+&* &* &* &* &* &33&* &21&57&12&51&6 &45&33&21&00&* &51&51&42&42&33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&* &00&12&9 &21&18&30&24&36\cr \+&30&18&18&42&24&21&36&30&42&39&48&45&* &39&33&51&12&00&21&9 &30&18&36&30\cr \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 &21&15&27\cr \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 &27&21\cr \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 &18\cr \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 &12&00&18&12\cr \+&42&30&48&* &36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 &18&00&12\cr \+&* &42&48&72&48&48&39&36&33&27&27&21&27&15&3 &24&36&30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4b.v This is the power 5 matrix, $B^5$}. \vskip.5cm \hrule \vfill\eject \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&51&30&60&36&69&42&* &48&45&* &* &18&30&27&39&36&48&42&54\cr \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&* &6 &18&15&27&24&36&30&42\cr \+&30&18&00&24&36&39&48&48&54&57&60&63&* &51&51&69&24&18&33&27&42&36&48&48\cr \+&54&42&24&00&60&63&72&72&78&81&* &87&* &* &75&* &48&42&57&51&66&60&72&72\cr \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&* &12&24&21&33&30&42&36&48\cr \+&51&39&39&63&45&00&57&12&63&21&69&27&* &60&51&33&33&21&42&30&51&39&57&48\cr \+&30&30&48&72&12&57&00&48&6 &54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr \+&60&48&48&72&54&12&48&00&54&9 &60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr \+&36&36&54&78&18&63&6 &54&00&48&6 &54&12&24&36&57&30&42&21&33&15&27&21&33\cr \+&69&57&57&81&63&21&54&9 &48&00&54&6 &54&42&30&12&51&39&42&30&33&21&39&27\cr \+&42&42&60&* &24&69&12&60&6 &54&00&48&6 &18&30&51&36&48&27&39&21&33&15&27\cr \+&* &63&63&87&69&27&60&15&54&6 &48&00&48&36&24&6 &57&45&48&36&39&27&33&21\cr \+&48&45&* &* &30&* &18&63&12&54&6 &48&00&12&24&45&39&51&30&42&21&33&15&27\cr \+&45&33&51&* &39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 &15\cr \+&* &45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr \+&* &* &69&* &* &33&63&21&57&12&51&6 &45&33&21&00&60&51&51&42&42&33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 &21&18&30&24&36\cr \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 &30&18&36&30\cr \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 &21&15&27\cr \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 &27&21\cr \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 &18\cr \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 &12&00&18&12\cr \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 &18&00&12\cr \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 &24&36&30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4b.vi This is the power 6 matrix, $B^6$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&51&30&60&36&69&42&75&48&45&57&* &18&30&27&39&36&48&42&54\cr \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&66&6 &18&15&27&24&36&30&42\cr \+&30&18&00&24&36&39&48&48&54&57&60&63&63&51&51&69&24&18&33&27&42&36&48&48\cr \+&54&42&24&00&60&63&72&72&78&81&84&87&* &75&75&93&48&42&57&51&66&60&72&72\cr \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&72&12&24&21&33&30&42&36&48\cr \+&51&39&39&63&45&00&57&12&63&21&69&27&72&60&51&33&33&21&42&30&51&39&57&48\cr \+&30&30&48&72&12&57&00&48&6 &54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr \+&60&48&48&72&54&12&48&00&54&9 &60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr \+&36&36&54&78&18&63&6 &54&00&48&6 &54&12&24&36&57&30&42&21&33&15&27&21&33\cr \+&69&57&57&81&63&21&54&9 &48&00&54&6 &54&42&30&12&51&39&42&30&33&21&39&27\cr \+&42&42&60&84&24&69&12&60&6 &54&00&48&6 &18&30&51&36&48&27&39&21&33&15&27\cr \+&75&63&63&87&69&27&60&15&54&6 &48&00&48&36&24&6 &57&45&48&36&39&27&33&21\cr \+&48&45&63&* &30&72&18&63&12&54&6 &48&00&12&24&45&39&51&30&42&21&33&15&27\cr \+&45&33&51&75&39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 &15\cr \+&57&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr \+&* &66&69&93&72&33&63&21&57&12&51&6 &45&33&21&00&60&51&51&42&42&33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 &21&18&30&24&36\cr \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 &30&18&36&30\cr \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 &21&15&27\cr \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 &27&21\cr \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 &18\cr \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 &12&00&18&12\cr \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 &18&00&12\cr \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 &24&36&30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4b.vii This is the power 7 matrix, $B^7$}. \vskip.5cm \hrule \vfill\eject \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&51&30&60&36&69&42&75&48&45&57&78&18&30&27&39&36&48&42&54\cr \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&66&6 &18&15&27&24&36&30&42\cr \+&30&18&00&24&36&39&48&48&54&57&60&63&63&51&51&69&24&18&33&27&42&36&48&48\cr \+&54&42&24&00&60&63&72&72&78&81&84&87&87&75&75&93&48&42&57&51&66&60&72&72\cr \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&72&12&24&21&33&30&42&36&48\cr \+&51&39&39&63&45&00&57&12&63&21&69&27&72&60&51&33&33&21&42&30&51&39&57&48\cr \+&30&30&48&72&12&57&00&48&6 &54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr \+&60&48&48&72&54&12&48&00&54&9 &60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr \+&36&36&54&78&18&63&6 &54&00&48&6 &54&12&24&36&57&30&42&21&33&15&27&21&33\cr \+&69&57&57&81&63&21&54&9 &48&00&54&6 &54&42&30&12&51&39&42&30&33&21&39&27\cr \+&42&42&60&84&24&69&12&60&6 &54&00&48&6 &18&30&51&36&48&27&39&21&33&15&27\cr \+&75&63&63&87&69&27&60&15&54&6 &48&00&48&36&24&6 &57&45&48&36&39&27&33&21\cr \+&48&45&63&87&30&72&18&63&12&54&6 &48&00&12&24&45&39&51&30&42&21&33&15&27\cr \+&45&33&51&75&39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 &15\cr \+&57&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr \+&78&66&69&93&72&33&63&21&57&12&51&6 &45&33&21&00&60&51&51&42&42&33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 &21&18&30&24&36\cr \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 &30&18&36&30\cr \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 &21&15&27\cr \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 &27&21\cr \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 &18\cr \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 &12&00&18&12\cr \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 &18&00&12\cr \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 &24&36&30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4b.viii This is the power 8 matrix, $B^8$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&51&30&60&36&69&42&75&48&45&57&78&18&30&27&39&36&48&42&54\cr \+&12&00&18&42&18&39&30&48&36&57&42&63&45&33&45&66&6 &18&15&27&24&36&30&42\cr \+&30&18&00&24&36&39&48&48&54&57&60&63&63&51&51&69&24&18&33&27&42&36&48&48\cr \+&54&42&24&00&60&63&72&72&78&81&84&87&87&75&75&93&48&42&57&51&66&60&72&72\cr \+&18&18&36&60&00&45&12&54&18&63&24&69&30&39&51&72&12&24&21&33&30&42&36&48\cr \+&51&39&39&63&45&00&57&12&63&21&69&27&72&60&51&33&33&21&42&30&51&39&57&48\cr \+&30&30&48&72&12&57&00&48&6 &54&12&60&18&30&42&63&24&36&15&27&21&33&27&39\cr \+&60&48&48&72&54&12&48&00&54&9 &60&15&63&51&39&21&42&30&33&21&42&30&48&36\cr \+&36&36&54&78&18&63&6 &54&00&48&6 &54&12&24&36&57&30&42&21&33&15&27&21&33\cr \+&69&57&57&81&63&21&54&9 &48&00&54&6 &54&42&30&12&51&39&42&30&33&21&39&27\cr \+&42&42&60&84&24&69&12&60&6 &54&00&48&6 &18&30&51&36&48&27&39&21&33&15&27\cr \+&75&63&63&87&69&27&60&15&54&6 &48&00&48&36&24&6 &57&45&48&36&39&27&33&21\cr \+&48&45&63&87&30&72&18&63&12&54&6 &48&00&12&24&45&39&51&30&42&21&33&15&27\cr \+&45&33&51&75&39&60&30&51&24&42&18&36&12&00&12&33&27&39&18&30&9 &21&3 &15\cr \+&57&45&51&75&51&51&42&39&36&30&30&24&24&12&00&21&39&33&30&24&21&15&15&3 \cr \+&78&66&69&93&72&33&63&21&57&12&51&6 &45&33&21&00&60&51&51&42&42&33&36&24\cr \+&18&6 &24&48&12&33&24&42&30&51&36&57&39&27&39&60&00&12&9 &21&18&30&24&36\cr \+&30&18&18&42&24&21&36&30&42&39&48&45&51&39&33&51&12&00&21&9 &30&18&36&30\cr \+&27&15&33&57&21&42&15&33&21&42&27&48&30&18&30&51&9 &21&00&12&9 &21&15&27\cr \+&39&27&27&51&33&30&27&21&33&30&39&36&42&30&24&42&21&9 &12&00&21&9 &27&21\cr \+&36&24&42&66&30&51&21&42&15&33&21&39&21&9 &21&42&18&30&9 &21&00&12&6 &18\cr \+&48&36&36&60&42&39&33&30&27&21&33&27&33&21&15&33&30&18&21&9 &12&00&18&12\cr \+&42&30&48&72&36&57&27&48&21&39&15&33&15&3 &15&36&24&36&15&27&6 &18&00&12\cr \+&54&42&48&72&48&48&39&36&33&27&27&21&27&15&3 &24&36&30&27&21&18&12&12&00\cr \vskip.5cm \noindent{\bf Figure 4b.ix This is the power 9 matrix, $B^9$}. It is identical to the matrix in Figure 4b.viii and so the algorithm terminates. \vskip.5cm \hrule \vfill\eject \centerline{\bf Figure for Hasse algorithm in Pascal} \noindent program hasse(input,output); \noindent const max=999999; \qquad\qquad\qquad\qquad n=24; \noindent type hed=array[1..n,1..n] of integer; \noindent var a:array[1..n] of hed; \qquad done:boolean; \qquad i,j,k,num:integer; \noindent procedure print(matrix:hed); \noindent begin \qquad for i:=i to n do \qquad\qquad begin \qquad\qquad for j:=1 to n do \qquad\qquad\qquad if matrix[i,j]=max then write (' *') \qquad\qquad\qquad\qquad else write(matrix[i,j]:4); \qquad\qquad writeln \qquad\qquad end \noindent end; \noindent procedure hedsum(power, init:hed;var next:hed;var flag:boolean); \noindent var row,col,min,middle,temp:integer; \noindent begin \qquad flag:=true; \qquad for row:=1 to n do \qquad for col:=1 to n do \qquad begin \qquad\qquad min:=power[row,col]; \qquad\qquad for middle:=1 to n do \qquad\qquad begin \qquad\qquad\qquad temp:=power[row,middle]+init[middle,col]; \qquad\qquad\qquad if temppower[row,col] then flag:=false; \qquad end \noindent end; \noindent $\{$main program$\}$ %remark--$\{$ is { and $\}$ is } \noindent begin \qquad for i:=1 to n do for j:=1 to n do a[1][i,j]:=max; \qquad for i:=1 to n do a[1][i,i]:=0; \qquad repeat \qquad\qquad readln(i,j,num); \qquad\qquad a[1][i,j]:=num; \qquad\qquad a[1][j,i]:=num; \qquad until eof; \qquad page; writeln('this is the initial matrix');writeln; \qquad print(a[1]); \qquad k:=0; \qquad repeat \qquad\qquad k:=k+1; \qquad\qquad hedsum(a[k],a[1],a[k+1],done); \qquad\qquad page; writeln('this is power',k+1:5); writeln; \qquad\qquad print(a[k+1]); \qquad until (done) or (k=n-1); \qquad writeln; \qquad writeln('the number of steps was', k:5) \noindent end. \noindent {\bf Figure 5.} Computer program, written in Pascal, of W. C. Arlinghaus; originally presented on a poster by Arlinghaus, Arlinghaus, and Nystuen, ``Elements of Geometric Routing Theory--II" Association of American Geographers, National Meetings, Toronto, Ontario, April 1990. \vfill\eject \centerline{\bf Figures containing tables} \smallskip \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&* &* &18&* &* &* &* &* &* &* &* &* &* &* \cr \+&12&00&18&* &* &* &* &* &* &* &* &* &* &33&* &* \cr \+&* &18&00&24&* &* &* &* &* &* &* &* &* &* &45&* \cr \+&* &* &24&00&* &30&* &* &* &* &* &* &* &* &* &* \cr \+&18&* &* &* &00&42&12&* & *&* &* &* &* &* &* &* \cr \+&* &* &* &30&42&00&* &12&* &* &* &* &* &* &* &* \cr \+&* &* &* &* &12&* &00&45&6 &* &* &* &* &* &* &* \cr \+&* &* &* &* &* &12&45&00&* &9 &* &* &* &* &* &* \cr \+&* &* &* &* &* &* &6 &* &00&45&6 &* &* &* &* &* \cr \+&* &* &* &* &* &* &* &9 &45&00&* &6 &* &* &* &* \cr \+&* &* &* &* &* &* &* &* &6 &* &00&45&6 &* &* &* \cr \+&* &* &* &* &* &* &* &* &* &6 &45&00&* &* &* &6 \cr \+&* &* &* &* &* &* &* &* &* &* &6 &* &00&12&* &* \cr \+&* &33&* &* &* &* &* &* &* &* &* &* &12&00&12&* \cr \+&* &* &45&* &* &* &* &* &* &* &* &* &* &12&00&21 \cr \+&* &* &* &* &* &* &* &* &* &* &* &6 &* &* &21&00 \cr \vskip.5cm \noindent{\bf Figure 7a.i This is the initial matrix, $C$}. Figure 7a contains a set of seven tables (i to vii) illustrating the use of Hasse's algorithm on the LA freeway system and the limited access surface route network (Figure 6) prior to the earthquake of January 17, 1994. Travel times are in one-quarter minutes. An asterisk indicates that the travel time between locations is too large to enter the matrix. A double-zero indicates an entry of 0. \vskip.5cm \hrule \vfill\eject \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&* &18&60&30&* &* &* &* &* &* &45&* &* \cr \+&12&00&18&42&30&* &* &* &* &* &* &* &45&33&45&* \cr \+&30&18&00&24&* &54&* &* &* &* &* &* &* &51&45&66 \cr \+&* &42&24&00&72&30&* &36&* &* &* &* &* &* &69&* \cr \+&18&30&* &72&00&42&12&48&18&* &* &* &* &* &* &* \cr \+&60&* &54&30&42&00&51& 6&* &15&* &* &* &* &* &* \cr \+&30&* &* &* &12&51&00&45&6 &51&12&* &* &* &* &* \cr \+&* &* &* &36&48& 6&45&00&51&9 &* &15&* &* &* &* \cr \+&* &* &* &* &18&* &6 &51&00&45&6 &51&12&* &* &* \cr \+&* &* &* &* &* &15&51&9 &45&00&51&6 &* &* &* &12 \cr \+&* &* &* &* &* &* &12&* &6 &51&00&45&6 &18&* &51 \cr \+&* &* &* &* &* &* &* &15&51&6 &45&00&51&* &24&6 \cr \+&* &45&* &* &* &* &* &* &12&* &6 &51&00&12&24&* \cr \+&45&33&51&* &* &* &* &* &* &* &18&* &12&00&12&33 \cr \+&* &45&45&69&* &* &* &* &* &* &* &27&24&12&00&21 \cr \+&* &* &66&* &* &* &* &* &* &12&51&6 &* &33&21&00 \cr \vskip.5cm \noindent{\bf Figure 7a.ii. This is the power 2 matrix, $C^2$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&60&30&66&36&* &* &* &57&45&57&* \cr \+&12&00&18&42&30&72&42&* &* &* &51&* &45&33&45&66 \cr \+&30&18&00&24&48&54&* &60&* &* &* &72&63&51&45&66 \cr \+&54&42&24&00&72&30&81&36&* &45&* &* &* &75&69&90 \cr \+&18&30&48&72&00&42&12&48&18&57&24&* &* &63&* &* \cr \+&60&72&54&30&42&00&51& 6&57&15&* &21&* &* &99&* \cr \+&30&42&* &81&12&51&00&45&6 &51&12&57&18&* &* &* \cr \+&66&* &60&36&48& 6&45&00&51&9 &57&15&* &* &* &21 \cr \+&36&* &* &* &18&57&15&51&00&45&6 &51&12&24&* &57 \cr \+&* &* &* &45&57&15&51&9 &45&00&51&6 &57&* &33&12 \cr \+&* &51&* &* &24&* &12&57&6 &51&00&45&6 &18&30&51 \cr \+&* &* &72&* &* &21&57&15&51&6 &45&00&51&39&27&6 \cr \+&57&45&63&* &* &* &18&* &12&57&6 &51&00&12&24&45 \cr \+&45&33&51&75&63&* &* &* &24&* &18&39&12&00&12&33 \cr \+&57&45&45&69&* &99&* &* &* &33&30&27&24&12&00&21 \cr \+&* &66&66&90&* &* &* &21&57&12&51&6 &45&33&21&00 \cr \vskip.5cm \noindent{\bf Figure 7a.iii This is the power 3 matrix, $C^3$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&60&30&66&36&75&42&* &57&45&57&78 \cr \+&12&00&18&42&30&72&42&78&48&* &51&72&45&33&45&66 \cr \+&30&18&00&24&48&54&60&60&* &69&69&72&63&51&45&66 \cr \+&54&42&24&00&72&30&81&36&87&45&* &51&87&75&69&90 \cr \+&18&30&48&72&00&42&12&48&18&57&24&63&30&63&75&* \cr \+&60&72&54&30&42&00&51& 6&57&15&63&21&* &105&99&27\cr \+&30&42&60&81&12&51&00&45&6 &51&12&57&18&30&* &63 \cr \+&66&78&60&36&48& 6&45&00&51&9 &57&15&63&* &42&21 \cr \+&36&48&* &87&18&57&15&51&00&45&6 &51&12&24&36&57 \cr \+&75&* &69&45&57&15&51&9 &45&00&51&6 &57&45&33&12 \cr \+&42&51&69&* &24&63&12&57&6 &51&00&45&6 &18&30&51 \cr \+&* &72&72&51&63&21&57&15&51&6 &45&00&51&39&27&6 \cr \+&57&45&63&87&30&* &18&63&12&57&6 &51&00&12&24&45 \cr \+&45&33&51&75&63&105&30&* &24&45&18&39&12&00&12&33\cr \+&57&45&45&69&75&99&* &42&36&33&30&27&24&12&00&21 \cr \+&78&66&66&90&* &27&63&21&57&12&51&6 &45&33&21&00 \cr \vskip.5cm \noindent{\bf Figure 7a.iv This is the power 4 matrix, $C^4$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&60&30&66&36&75&42&81&48&45&57&78 \cr \+&12&00&18&42&30&72&42&78&48&78&51&72&45&33&45&66 \cr \+&30&18&00&24&48&54&60&60&66&69&69&72&63&51&45&66 \cr \+&54&42&24&00&72&30&81&36&87&45&93&51&87&75&69&57 \cr \+&18&30&48&72&00&42&12&48&18&57&24&63&30&63&75&69 \cr \+&60&72&54&30&42&00&51& 6&57&15&63&21&69&105&48&27\cr \+&30&42&60&81&12&51&00&45&6 &51&12&57&18&30&42&63 \cr \+&66&78&60&36&48& 6&45&00&51&9 &57&15&63&54&42&21 \cr \+&36&48&66&87&18&57&15&51&00&45&6 &51&12&24&36&57 \cr \+&75&78&69&45&57&15&51&9 &45&00&51&6 &57&45&33&12 \cr \+&42&51&69&93&24&63&12&57&6 &51&00&45&6 &18&30&51 \cr \+&81&72&72&51&63&21&57&15&51&6 &45&00&51&39&27&6 \cr \+&48&45&63&87&30&69&18&63&12&57&6 &51&00&12&24&45 \cr \+&45&33&51&75&63&105&30&54&24&45&18&39&12&00&12&33\cr \+&57&45&45&69&75&48&42&42&36&33&30&27&24&12&00&21 \cr \+&78&66&66&57&69&27&63&21&57&12&51&6 &45&33&21&00 \cr \vskip.5cm \noindent{\bf Figure 7a.v This is the power 5 matrix, $C^5$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&60&30&66&36&75&42&81&48&45&57&78 \cr \+&12&00&18&42&30&72&42&78&48&78&51&72&45&33&45&66 \cr \+&30&18&00&24&48&54&60&60&66&69&69&72&63&51&45&66 \cr \+&54&42&24&00&72&30&81&36&87&45&93&51&87&75&69&57 \cr \+&18&30&48&72&00&42&12&48&18&57&24&63&30&42&54&69 \cr \+&60&72&54&30&42&00&51& 6&57&15&63&21&69&60&48&27\cr \+&30&42&60&81&12&51&00&45&6 &51&12&57&18&30&42&63 \cr \+&66&78&60&36&48& 6&45&00&51&9 &57&15&63&54&42&21 \cr \+&36&48&66&87&18&57& 6&51&00&45&6 &51&12&24&36&57 \cr \+&75&78&69&45&57&15&51&9 &45&00&51&6 &57&45&33&12 \cr \+&42&51&69&93&24&63&12&57&6 &51&00&45&6 &18&30&51 \cr \+&81&72&72&51&63&21&57&15&51&6 &45&00&51&39&27&6 \cr \+&48&45&63&87&30&69&18&63&12&57&6 &51&00&12&24&45 \cr \+&45&33&51&75&42&60&30&54&24&45&18&39&12&00&12&33\cr \+&57&45&45&69&54&48&42&42&36&33&30&27&24&12&00&21 \cr \+&78&66&66&57&69&27&63&21&57&12&51&6 &45&33&21&00 \cr \vskip.5cm \noindent{\bf Figure 7a.vi This is the power 6 matrix, $C^6$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&60&30&66&36&75&42&81&48&45&57&78 \cr \+&12&00&18&42&30&72&42&78&48&78&51&72&45&33&45&66 \cr \+&30&18&00&24&48&54&60&60&66&69&69&72&63&51&45&66 \cr \+&54&42&24&00&72&30&81&36&87&45&93&51&87&75&69&57 \cr \+&18&30&48&72&00&42&12&48&18&57&24&63&30&42&54&69 \cr \+&60&72&54&30&42&00&51& 6&57&15&63&21&69&60&48&27\cr \+&30&42&60&81&12&51&00&45&6 &51&12&57&18&30&42&63 \cr \+&66&78&60&36&48& 6&45&00&51&9 &57&15&63&54&42&21 \cr \+&36&48&66&87&18&57& 6&51&00&45&6 &51&12&24&36&57 \cr \+&75&78&69&45&57&15&51&9 &45&00&51&6 &57&45&33&12 \cr \+&42&51&69&93&24&63&12&57&6 &51&00&45&6 &18&30&51 \cr \+&81&72&72&51&63&21&57&15&51&6 &45&00&51&39&27&6 \cr \+&48&45&63&87&30&69&18&63&12&57&6 &51&00&12&24&45 \cr \+&45&33&51&75&42&60&30&54&24&45&18&39&12&00&12&33\cr \+&57&45&45&69&54&48&42&42&36&33&30&27&24&12&00&21 \cr \+&78&66&66&57&69&27&63&21&57&12&51&6 &45&33&21&00 \cr \vskip.5cm \noindent{\bf Figure 7a.vii This is the power 7 matrix, $C^7$}. The algorithm terminates in six steps; this matrix is identical to $A^6$. \vskip.5cm \hrule \vfill\eject \centerline{\bf Figures containing tables} \smallskip \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&* &* &18&* &* &* &* &* &* &* &* &* &* &* \cr \+&12&00&18&* &* &* &* &* &* &* &* &* &* &33&* &* \cr \+&* &18&00&24&* &* &* &* &* &* &* &* &* &* &45&* \cr \+&* &* &24&00&* &* &* &* &* &* &* &* &* &* &* &* \cr \+&18&* &* &* &00&42&12&* & *&* &* &* &* &* &* &* \cr \+&* &* &* &* &42&00&* & 6&* &* &* &* &* &* &* &* \cr \+&* &* &* &* &12&* &00&45&6 &* &* &* &* &* &* &* \cr \+&* &* &* &* &* & 6&45&00&* &9 &* &* &* &* &* &* \cr \+&* &* &* &* &* &* &6 &* &00&45&6 &* &* &* &* &* \cr \+&* &* &* &* &* &* &* &9 &45&00&* &6 &* &* &* &* \cr \+&* &* &* &* &* &* &* &* &6 &* &00&45&6 &* &* &* \cr \+&* &* &* &* &* &* &* &* &* &6 &45&00&* &* &* &6 \cr \+&* &* &* &* &* &* &* &* &* &* &6 &* &00&12&* &* \cr \+&* &33&* &* &* &* &* &* &* &* &* &* &12&00&12&* \cr \+&* &* &45&* &* &* &* &* &* &* &* &* &* &12&00&21 \cr \+&* &* &* &* &* &* &* &* &* &* &* &6 &* &* &21&00 \cr \vskip.5cm \noindent{\bf Figure 7b.i This is the initial matrix, $D$}. Figure 7b contains a set of seven tables (i to vii) illustrating the use of Hasse's algorithm on the LA freeway system and the limited access surface route network (Figure 6) following the earthquake of January 17, 1994. Travel times are in one-quarter minutes. An asterisk indicates that the travel time between locations is too large to enter the matrix. A double-zero indicates an entry of 0. \vskip.5cm \hrule \vfill\eject \vskip.5cm \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&* &18&60&30&* &* &* &* &* &* &45&* &* \cr \+&12&00&18&42&30&* &* &* &* &* &* &* &45&33&45&* \cr \+&30&18&00&24&* &* &* &* &* &* &* &* &* &51&45&66 \cr \+&* &42&24&00&* &* &* &* &* &* &* &* &* &* &69&* \cr \+&18&30&* &* &00&42&12&48&18&* &* &* &* &* &* &* \cr \+&60&* &* &* &42&00&51& 6&* &15&* &* &* &* &* &* \cr \+&30&* &* &* &12&51&00&45&6 &51&12&* &* &* &* &* \cr \+&* &* &* &* &48& 6&45&00&51&9 &* &15&* &* &* &* \cr \+&* &* &* &* &18&* &6 &51&00&45&6 &51&12&* &* &* \cr \+&* &* &* &* &* &15&51&9 &45&00&51&6 &* &* &* &12 \cr \+&* &* &* &* &* &* &12&* &6 &51&00&45&6 &18&* &51 \cr \+&* &* &* &* &* &* &* &15&51&6 &45&00&51&* &27&6 \cr \+&* &45&* &* &* &* &* &* &12&* &6 &51&00&12&24&* \cr \+&45&33&51&* &* &* &* &* &* &* &18&* &12&00&12&33 \cr \+&* &45&45&69&* &* &* &* &* &* &* &27&24&12&00&21 \cr \+&* &* &66&* &* &* &* &* &* &12&51&6 &* &33&21&00 \cr \vskip.5cm \noindent{\bf Figure 7b.ii. This is the power 2 matrix, $D^2$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&60&30&66&36&* &* &* &57&45&57&* \cr \+&12&00&18&42&30&72&42&* &* &* &51&* &45&33&45&66 \cr \+&30&18&00&24&48&* &* &* &* &* &* &72&63&51&45&66 \cr \+&54&42&24&00&* &* &* &* &* &* &* &* &* &75&69&90 \cr \+&18&30&48&* &00&42&12&48&18&57&24&* &* &63&* &* \cr \+&60&72&* &* &42&00&51& 6&57&15&* &21&* &* &* &* \cr \+&30&42&* &* &12&51&00&45&6 &51&12&57&18&* &* &* \cr \+&66&* &* &* &48& 6&45&00&51&9 &57&15&* &* &* &21 \cr \+&36&* &* &* &18&57& 6&51&00&45&6 &51&12&24&* &57 \cr \+&* &* &* &* &57&15&51&9 &45&00&51&6 &57&* &33&12 \cr \+&* &51&* &* &24&* &12&57&6 &51&00&45&6 &18&30&51 \cr \+&* &* &72&* &* &21&57&15&51&6 &45&00&51&39&27&6 \cr \+&57&45&63&* &* &* &18&* &12&57&6 &51&00&12&24&45 \cr \+&45&33&51&75&63&* &* &* &24&* &18&39&12&00&12&33 \cr \+&57&45&45&69&* &* &* &* &* &33&30&27&24&12&00&21 \cr \+&* &66&66&90&* &* &* &21&57&12&51&6 &45&33&21&00 \cr \vskip.5cm \noindent{\bf Figure 7b.iii This is the power 3 matrix, $D^3$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54&18&60 &30&66&36&75&42&* &57&45 &57&78 \cr \+&12&00&18&42&30&72 &42&78&48&* &51&72&45&33 &45&66 \cr \+&30&18&00&24&48&90 &60&* &* &78&69&72&63&51 &45&66 \cr \+&54&42&24&00&72&* &* &* &* &* &* &96&87&75 &69&90 \cr \+&18&30&48&72&00&42 &12&48&18&57&24&63&30&63 &75&* \cr \+&60&72&90&* &42&00 &51& 6&57&15&63&21&* &105&* &27 \cr \+&30&42&60&* &12&51 &00&45&6 &51&12&57&18&30 &* &63 \cr \+&66&78&* &* &48& 6 &45&00&51&9 &57&15&63&* &42&21 \cr \+&36&48&* &* &18&57 & 6&51&00&45&6 &51&12&24 &36&57 \cr \+&75&* &78&* &57&15 &51&9 &45&00&51&6 &57&45 &33&12 \cr \+&42&51&69&* &24&63 &12&57&6 &51&00&45&6 &18 &30&51 \cr \+&* &72&72&96&63&21 &57&15&51&6 &45&00&51&39 &27&6 \cr \+&57&45&63&87&30&* &18&63&12&57&6 &51&00&12 &24&45 \cr \+&45&33&51&75&63&105&30&* &24&45&18&39&12&00 &12&33 \cr \+&57&45&45&69&75&* &* &42&36&33&30&27&24&12 &00&21 \cr \+&78&66&66&90&* &27 &63&21&57&12&51&6 &45&33 &21&00 \cr \vskip.5cm \noindent{\bf Figure 7b.iv This is the power 4 matrix, $D^4$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54 &18&60 &30&66&36&75 &42&81&48&45&57&78 \cr \+&12&00&18&42 &30&72 &42&78&48&78 &51&72&45&33&45&66 \cr \+&30&18&00&24 &48&90 &60&87&66&78 &69&72&63&51&45&66 \cr \+&54&42&24&00 &72&114&84&* &* &102&93&96&87&75&69&90 \cr \+&18&30&48&72 &00&42 &12&48&18&57 &24&63&30&42&75&69 \cr \+&60&72&90&114&42&00 &51& 6&57&15 &63&21&69&105&48&27\cr \+&30&42&60&84 &12&51 &00&45&6 &51 &12&57&18&30&42&63 \cr \+&66&78&87&* &48& 6 &45&00&51&9 &57&15&63&54&42&21 \cr \+&36&48&66&* &18&57 &6 &51&00&45 &6 &51&12&24&36&57 \cr \+&75&78&78&102&57&15 &51&9 &45&00 &51&6 &57&45&33&12 \cr \+&42&51&69&93 &24&63 &12&57&6 &51 &00&45&6 &18&30&51 \cr \+&81&72&72&96 &63&21 &57&15&51&6 &45&00&51&39&27&6 \cr \+&48&45&63&87 &30&69 &18&63&12&57 &6 &51&00&12&24&45 \cr \+&45&33&51&75 &42&105&30&54&24&45 &18&39&12&00&12&33 \cr \+&57&45&45&69 &75&48 &42&42&36&33 &30&27&24&12&00&21 \cr \+&78&66&66&90 &69&27 &63&21&57&12 &51&6 &45&33&21&00 \cr \vskip.5cm \noindent{\bf Figure 7b.v This is the power 5 matrix, $D^5$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54 &18&60 &30&66 &36&75 &42&81&48&45&57&78 \cr \+&12&00&18&42 &30&72 &42&78 &48&78 &51&72&45&33&45&66 \cr \+&30&18&00&24 &48&90 &60&87 &66&78 &69&72&63&51&45&66 \cr \+&54&42&24&00 &72&114&84&111&90&102&93&96&87&75&69&90 \cr \+&18&30&48&72 &00&42 &12&48 &18&57 &24&63&30&42&54&69 \cr \+&60&72&90&114&42&00 &51& 6 &57&15 &63&21&69&60&48&27 \cr \+&30&42&60&84 &12&51 &00&45 &6 &51 &12&57&18&30&42&63 \cr \+&66&78&87&111&48& 6 &45&00 &51&9 &57&15&63&54&42&21 \cr \+&36&48&66&90 &18&57 & 6&51 &00&45 &6 &51&12&24&36&57 \cr \+&75&78&78&102&57&15 &51&9 &45&00 &51&6 &57&45&33&12 \cr \+&42&51&69&93 &24&63 &12&57 &6 &51 &00&45&6 &18&30&51 \cr \+&81&72&72&96 &63&21 &57&15 &51&6 &45&00&51&39&27&6 \cr \+&48&45&63&87 &30&69 &18&63 &12&57 &6 &51&00&12&24&45 \cr \+&45&33&51&75 &42&60 &30&54 &24&45 &18&39&12&00&12&33 \cr \+&57&45&45&69 &54&48 &42&42 &36&33 &30&27&24&12&00&21 \cr \+&78&66&66&90 &69&27 &63&21 &57&12 &51&6 &45&33&21&00 \cr \vskip.5cm \noindent{\bf Figure 7b.vi This is the power 6 matrix, $D^6$}. \vskip.5cm \hrule \vfill\eject \hrule \vskip.5cm \settabs\+\qquad\qquad\qquad &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$ &00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&00$\,\,$&\cr %sample line \+&00&12&30&54 &18&60 &30&66 &36&75 &42&81&48&45&57&78 \cr \+&12&00&18&42 &30&72 &42&78 &48&78 &51&72&45&33&45&66 \cr \+&30&18&00&24 &48&90 &60&87 &66&78 &69&72&63&51&45&66 \cr \+&54&42&24&00 &72&114&84&111&90&102&93&96&87&75&69&90 \cr \+&18&30&48&72 &00&42 &12&48 &18&57 &24&63&30&42&54&69 \cr \+&60&72&90&114&42&00 &51& 6 &57&15 &63&21&69&60&48&27\cr \+&30&42&60&84 &12&51 &00&45 &6 &51 &12&57&18&30&42&63 \cr \+&66&78&87&111&48& 6 &45&00 &51&9 &57&15&63&54&42&21 \cr \+&36&48&66&90 &18&57 & 6&51 &00&45 &6 &51&12&24&36&57 \cr \+&75&78&78&102&57&15 &51&9 &45&00 &51&6 &57&45&33&12 \cr \+&42&51&69&93 &24&63 &12&57 &6 &51 &00&45&6 &18&30&51 \cr \+&81&72&72&96 &63&21 &57&15 &51&6 &45&00&51&39&27&6 \cr \+&48&45&63&87 &30&69 &18&63 &12&57 &6 &51&00&12&24&45 \cr \+&45&33&51&75 &42&60 &30&54 &24&45 &18&39&12&00&12&33 \cr \+&57&45&45&69 &54&48 &42&42 &36&33 &30&27&24&12&00&21 \cr \+&78&66&66&90 &69&27 &63&21 &57&12 &51&6 &45&33&21&00 \cr \vskip.5cm \noindent{\bf Figure 7b.vii This is the power 7 matrix, $D^7$}. The iteration terminates after 6 steps; this matrix is identical to $D^6$. \vskip.5cm \hrule \vfill\eject \centerline{\bf 5. SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE} \centerline{\bf BACK ISSUES OF {\sl SOLSTICE\/} ON A GOPHER} \noindent {\sl Solstice\/} is available on a GOPHER from the Department of Mathematics at Arizona State University: PI.LA.ASU.EDU port 70 \centerline{\bf BACK ISSUES OF {\sl SOLSTICE\/} AVAILABLE ON FTP} \noindent This section shows the exact set of commands that work to download {\sl Solstice\/} on The University of Michigan's Xerox 9700. Because different universities will have different installations of {\TeX}, this is only a rough guideline which {\sl might\/} be of use to the reader. (BACK ISSUES AVAILABLE using anonymous ftp to open um.cc.umich.edu, account GCFS; type cd GCFS after entering system; then type ls to get a directory; then type get solstice.190 (for example) and download it or read it according to local constraints.) Back issues will be available on this account; this account is ONLY for back issues; to write Solstice, send e-mail to Solstice@UMICHUM.bitnet or to Solstice@um.cc.umich.edu . Issues from this one forward are available on FTP on account IEVG (substitute IEVG for GCFS above). First step is to concatenate the files you received via bitnet/internet. Simply piece them together in your computer, one after another, in the order in which they are numbered, starting with the number, ``1." The files you have received are ASCII files; the concatenated file is used to form the .tex file from which the .dvi file (device independent) file is formed. They should run, possibly with a few harmless ``vboxes" over or under. \noindent ASSUME YOU HAVE SIGNED ON AND ARE AT THE SYSTEM PROMPT, \#. \smallskip \# create -t.tex \# percent-sign t from pc c:backslash words backslash solstice.tex to mts -t.tex char notab (this command sends my file, solstice.tex, which I did as a WordStar (subdirectory, ``words") ASCII file to the mainframe) \# run *tex par=-t.tex (there may be some underfull (or certain over) boxes that generally cause no problem; there should be no other ``error" messages in the typesetting--the files you receive were already tested.) \# run *dvixer par=-t.dvi \# control *print* onesided \# run *pagepr scards=-t.xer, par=paper=plain \vfill\eject \centerline{\bf 6. SOLSTICE--INDEX, VOLUMES I, II, III, IV} \smallskip \noindent{\bf Volume IV, Number 2, Winter, 1993} \smallskip \noindent {\bf 1.} Welcome to New Readers and Thank You Notes. \smallskip \noindent {\bf 2.} Press clippings, summary. \smallskip \noindent {\bf 3.} Article \smallskip Villages in Transition: Elevated Risk of Micronutrient Deficiency. \smallskip William D. Drake, S. Pak, I. Tarwotjo, Muhilal, J. Gorstein, R. Tilden. \smallskip Abstract; Moving from Traditional to Modern Village Life: Risks during Transition; Testing for Elevated Risks in Transition Villages; Testing for Risk Overlap within the Health Sector; Conclusions and Policy Implications. \noindent {\bf 4.} Downloading of Solstice \smallskip \noindent {\bf 5.} Index to Volumes I (1990), II (1991), III (1992), and IV.1 (1993) of Solstice. \smallskip \noindent {\bf 6.} Other Publications of IMaGe \smallskip \noindent {\bf 7.} Selected recent publications of interest involving Solstice Board members, and some goings on about Ann Arbor. %___________________________________________________________________ %___________________________________________________________________ \smallskip \noindent{\bf Volume IV, Number 1, Summer, 1993} \smallskip \noindent {\bf 1.} Welcome to New Readers. \smallskip \noindent {\bf 2.} Press clippings, summary. \smallskip \noindent {\bf 3.} Goings on about Ann Arbor--ESRI and IMaGe Gift \smallskip \noindent {\bf 4.} Articles \smallskip Electronic Journals: Observations Based on Actual Trials, 1987-Present, by Sandra L. Arlinghaus and Richard H. Zander. Headings: Abstract; Content issues; Production issues; Archival issues; References. \smallskip Wilderness As Place, by John D. Nystuen. Headings: Visual paradoxes; Wilderness defined; Conflict or synthesis; Wilderness as place; Suggested readings; Sources; Visual illusion authors \smallskip The Earth Isn't Flat. And It Isn't Round Either: Some Significant and Little Known Effects of the Earth's Ellipsoidal Shape, by Frank E. Barmore. reprinted from the {\sl Wisconsin Geographer\/}. Headings: Abstract; Introduction; The Qibla problem; The geographic center; The center of population; Appendix; References. \smallskip Microcell Hex-nets? by Sandra L. Arlinghaus Headings: Introduction; Lattices; Microcell hex-nets; References. \smallskip Sum Graphs and Geographic Information, by Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary. Headings: Abstract; Sum graphs; Sum graph unification: construction; Cartographic application of sum graph unification; Sum graph unification: theory; Logarithmic sum graphs; Reversed sum graphs; Augmented reversed logarithmic sum graphs; Cartographic application of ARL sum graphs; Summary \smallskip \noindent{\bf 5.} Downloading of {\sl Solstice\/}. \smallskip \noindent{\bf 6.} Index. \smallskip \noindent{\bf 7.} Other publications of IMaGe. %---------------------------------------------------------------- %---------------------------------------------------------------- \smallskip \noindent {\bf Volume III, Number 2, Winter, 1992} \smallskip \noindent {\bf 1.} A Word of Welcome from A to U. \smallskip \noindent {\bf 2.} Press clippings--summary. \smallskip \noindent {\bf 3.} Reprints: \smallskip \noindent {\bf A.} What Are Mathematical Models and What Should They Be? by Frank Harary, reprinted from {\sl Biometrie - Praximetrie\/}. \smallskip \noindent {\sl 1. What Are They? 2. Two Worlds: Abstract and Empirical 3. Two Worlds: Two Levels 4. Two Levels: Derviation and Selection 5. Research Schema 6. Sketches of Discovery 7. What Should They Be? \/} \smallskip \noindent {\bf B.} Where Are We? Comments on the Concept of Center of Population, by Frank E. Barmore, reprinted from {\sl The Wisconsin Geographer\/}. \smallskip \noindent {\sl 1. Introduction 2. Preliminary Remarks 3. Census Bureau Center of Population Formul{\ae} 4. Census Bureau Center of Population Description 5. Agreement Between Description and Formul{\ae} 6. Proposed Definition of the Center of Population 7. Summary 8. Appendix A 9. Appendix B 10. References \/} \smallskip \noindent {\bf 4.} Article: \smallskip The Pelt of the Earth: An Essay on Reactive Diffusion, by Sandra L. Arlinghaus and John D. Nystuen. \smallskip \noindent {\sl 1. Pattern Formation: Global Views 2. Pattern Formation: Local Views 3. References Cited 4. Literature of Apparent Related Interest. \/} \smallskip \noindent {\bf 5.} Feature Meet new{\sl Solstice\/} Board Member William D. Drake; comments on course in Transition Theory and listing of student-produced monograph. \smallskip \noindent {\bf 6.} Downloading of Solstice. \smallskip \noindent {\bf 7.} Index to Solstice. \smallskip \noindent {\bf 8.} Other Publications of IMaGe. \smallskip %---------------------------------------------------------------- %---------------------------------------------------------------- \noindent {\bf Volume III, Number 1, Summer, 1992} \smallskip \noindent{\bf 1. ARTICLES.} \smallskip\noindent {\bf Harry L. Stern}. \smallskip\noindent {\bf Computing Areas of Regions With Discretely Defined Boundaries}. \smallskip\noindent 1. Introduction 2. General Formulation 3. The Plane 4. The Sphere 5. Numerical Example and Remarks. Appendix--Fortran Program. \smallskip \noindent{\bf 2. NOTE } \smallskip\noindent {\bf Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg}. \smallskip\noindent {\bf The Quadratic World of Kinematic Waves} \smallskip \noindent{\bf 3. SOFTWARE REVIEW} \smallskip RangeMapper$^{\hbox{TM}}$ --- version 1.4. Created by {\bf Kenelm W. Philip}, Tundra Vole Software, Fairbanks, Alaska. Program and Manual by {\bf Kenelm W. Philip}. \smallskip Reviewed by {\bf Yung-Jaan Lee}, University of Michigan. \smallskip \noindent{\bf 4. PRESS CLIPPINGS} \smallskip \noindent{\bf 5. INDEX to Volumes I (1990) and II (1991) of {\sl Solstice}.} \smallskip %---------------------------------------------------------------- %---------------------------------------------------------------- \noindent {\bf Volume II, Number 2, Winter, 1991} \smallskip \noindent 1. REPRINT Saunders Mac Lane, ``Proof, Truth, and Confusion." Given as the Nora and Edward Ryerson Lecture at The University of Chicago in 1982. Republished with permission of The University of Chicago and of the author. I. The Fit of Ideas. II. Truth and Proof. III. Ideas and Theorems. IV. Sets and Functions. V. Confusion via Surveys. VI. Cost-benefit and Regression. VII. Projection, Extrapolation, and Risk. VIII. Fuzzy Sets and Fuzzy Thoughts. IX. Compromise is Confusing. \noindent 2. ARTICLE Robert F. Austin. ``Digital Maps and Data Bases: Aesthetics versus Accuracy." I. Introduction. II. Basic Issues. III. Map Production. IV. Digital Maps. V. Computerized Data Bases. VI. User Community. \noindent 3. FEATURES Press clipping; Word Search Puzzle; Software Briefs. \smallskip %---------------------------------------------------------------- %---------------------------------------------------------------- \noindent {\bf Volume II, Number 1, Summer, 1991} \smallskip \noindent 1. ARTICLE Sandra L. Arlinghaus, David Barr, John D. Nystuen. {\sl The Spatial Shadow: Light and Dark --- Whole and Part\/} This account of some of the projects of sculptor David Barr attempts to place them in a formal, systematic, spatial setting based on the postulates of the science of space of William Kingdon Clifford (reprinted in {\sl Solstice\/}, Vol. I, No. 1.). \smallskip \noindent 2. FEATURES \item{i} Construction Zone --- The logistic curve. \item{ii.} Educational feature --- Lectures on ``Spatial Theory" \smallskip %---------------------------------------------------------------- %---------------------------------------------------------------- \noindent{\bf Volume I, Number 2, Winter, 1990} \smallskip \noindent 1. REPRINT John D. Nystuen (1974), {\sl A City of Strangers: Spatial Aspects of Alienation in the Detroit Metropolitan Region\/}. This paper examines the urban shift from ``people space" to ``machine space" (see R. Horvath, {\sl Geographical Review\/}, April, 1974) in the Detroit metropolitan region of 1974. As with Clifford's {\sl Postulates\/}, reprinted in the last issue of {\sl Solstice\/}, note the timely quality of many of the observations. \noindent 2. ARTICLES Sandra Lach Arlinghaus, {\sl Scale and Dimension: Their Logical Harmony\/}. Linkage between scale and dimension is made using the Fallacy of Division and the Fallacy of Composition in a fractal setting. \smallskip Sandra Lach Arlinghaus, {\sl Parallels between Parallels\/}. The earth's sun introduces a symmetry in the perception of its trajectory in the sky that naturally partitions the earth's surface into zones of affine and hyperbolic geometry. The affine zones, with single geometric parallels, are located north and south of the geographic parallels. The hyperbolic zone, with multiple geometric parallels, is located between the geographic tropical parallels. Evidence of this geometric partition is suggested in the geographic environment --- in the design of houses and of gameboards. \smallskip Sandra L. Arlinghaus, William C. Arlinghaus, and John D. Nystuen. {\sl The Hedetniemi Matrix Sum: A Real-world Application\/}. In a recent paper, we presented an algorithm for finding the shortest distance between any two nodes in a network of $n$ nodes when given only distances between adjacent nodes [Arlinghaus, Arlinghaus, Nystuen, {\sl Geographical Analysis\/}, 1990]. In that previous research, we applied the algorithm to the generalized road network graph surrounding San Francisco Bay. Here, we examine consequent changes in matrix entires when the underlying adjacency pattern of the road network was altered by the 1989 earthquake that closed the San Francisco --- Oakland Bay Bridge. \smallskip Sandra Lach Arlinghaus, {\sl Fractal Geometry of Infinite Pixel Sequences: ``Su\-per\--def\-in\-i\-tion" Resolution\/}? Comparison of space-filling qualities of square and hexagonal pixels. \smallskip \noindent 3. FEATURES \item{i.} Construction Zone --- Feigenbaum's number; a triangular coordinatization of the Euclidean plane. \item{ii.} A three-axis coordinatization of the plane. \smallskip %---------------------------------------------------------------- %---------------------------------------------------------------- \noindent{\bf Volume I, Number 1, Summer, 1990} \noindent 1. REPRINT William Kingdon Clifford, {\sl Postulates of the Science of Space\/} This reprint of a portion of Clifford's lectures to the Royal Institution in the 1870's suggests many geographic topics of concern in the last half of the twentieth century. Look for connections to boundary issues, to scale problems, to self- similarity and fractals, and to non-Euclidean geometries (from those based on denial of Euclid's parallel postulate to those based on a sort of mechanical ``polishing"). What else did, or might, this classic essay foreshadow? \noindent 2. ARTICLES. Sandra L. Arlinghaus, {\sl Beyond the Fractal.} An original article. The fractal notion of self-similarity is useful for characterizing change in scale; the reason fractals are effective in the geometry of central place theory is because that geometry is hierarchical in nature. Thus, a natural place to look for other connections of this sort is to other geographical concepts that are also hierarchical. Within this fractal context, this article examines the case of spatial diffusion. When the idea of diffusion is extended to see ``adopters" of an innovation as ``attractors" of new adopters, a Julia set is introduced as a possible axis against which to measure one class of geographic phenomena. Beyond the fractal context, fractal concepts, such as ``compression" and ``space-filling" are considered in a broader graph-theoretic setting. \smallskip William C. Arlinghaus, {\sl Groups, Graphs, and God} \smallskip \noindent 3. FEATURES \smallskip \item{i.} Theorem Museum --- Desargues's Two Triangle Theorem from projective geometry. \item{ii.} Construction Zone --- a centrally symmetric hexagon is derived from an arbitrary convex hexagon. \item{iii.} Reference Corner --- Point set theory and topology. \item{iv.} Educational Feature --- Crossword puzzle on spices. \item{v.} Solution to crossword puzzle. \smallskip \noindent 4. SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE \smallskip \vfill\eject \centerline{\bf 7. OTHER PUBLICATIONS OF IMaGe} \centerline{\bf MONOGRAPH SERIES} \centerline{Scholarly Monographs--Original Material, refereed} Prices exclusive of shipping and handling; payable in U.S. funds on a U.S. bank, only. All monographs are \$15.95, except \#12 which is \$39.95. Monographs are printed by Gryphon Publishing 1. Sandra L. Arlinghaus and John D. Nystuen. Mathematical Geography and Global Art: the Mathematics of David Barr's ``Four Corners Project,'' 1986. 2. Sandra L. Arlinghaus. Down the Mail Tubes: the Pressured Postal Era, 1853-1984, 1986. 3. Sandra L. Arlinghaus. Essays on Mathematical Geography, 1986. 4. Robert F. Austin, A Historical Gazetteer of Southeast Asia, 1986. 5. Sandra L. Arlinghaus, Essays on Mathematical Geography--II, 1987. 6. Pierre Hanjoul, Hubert Beguin, and Jean-Claude Thill, Theoretical Market Areas Under Euclidean Distance, 1988. (English language text; Abstracts written in French and in English.) 7. Keith J. Tinkler, Editor, Nystuen---Dacey Nodal Analysis, 1988. 8. James W. Fonseca, The Urban Rank--size Hierarchy: A Mathematical Interpretation, 1989. 9. Sandra L. Arlinghaus, An Atlas of Steiner Networks, 1989. 10. Daniel A. Griffith, Simulating $K=3$ Christaller Central Place Structures: An Algorithm Using A Constant Elasticity of Substitution Consumption Function, 1989. 11. Sandra L. Arlinghaus and John D. Nystuen, Environmental Effects on Bus Durability, 1990. 12. Daniel A. Griffith, Editor. Spatial Statistics: Past, Present, and Future, 1990. 13. Sandra L. Arlinghaus, Editor. Solstice --- I, 1990. 14. Sandra L. Arlinghaus, Essays on Mathematical Geography --- III, 1991. 15. Sandra L. Arlinghaus, Editor, Solstice --- II, 1991. 16. Sandra L. Arlinghaus, Editor, Solstice --- III, 1992. 17. Sandra L. Arlinghaus, Editor, Solstice --- IV, 1993. %---------------------------------------------------------------- %---------------------------------------------------------------- \smallskip DISCUSSION PAPERS--ORIGINAL Editor, Daniel A. Griffith Professor of Geography Syracuse University 1. Spatial Regression Analysis on the PC: Spatial Statistics Using Minitab. 1989. Cost: \$12.95, hardcopy. %---------------------------------------------------------------- %---------------------------------------------------------------- \smallskip DISCUSSION PAPERS--REPRINTS Editor of MICMG Series, John D. Nystuen Professor of Geography and Urban Planning The University of Michigan 1. Reprint of the Papers of the Michigan InterUniversity Community of Mathematical Geographers. Editor, John D. Nystuen. Cost: \$39.95, hardcopy. Contents--original editor: John D. Nystuen. 1. Arthur Getis, ``Temporal land use pattern analysis with the use of nearest neighbor and quadrat methods." July, 1963 2. Marc Anderson, ``A working bibliography of mathematical geography." September, 1963. 3. William Bunge, ``Patterns of location." February, 1964. 4. Michael F. Dacey, ``Imperfections in the uniform plane." June, 1964. 5. Robert S. Yuill, A simulation study of barrier effects in spatial diffusion problems." April, 1965. 6. William Warntz, ``A note on surfaces and paths and applications to geographical problems." May, 1965. 7. Stig Nordbeck, ``The law of allometric growth." June, 1965. 8. Waldo R. Tobler, ``Numerical map generalization;" and Waldo R. Tobler, ``Notes on the analysis of geographical distributions." January, 1966. 9. Peter R. Gould, ``On mental maps." September, 1966. 10. John D. Nystuen, ``Effects of boundary shape and the concept of local convexity;" Julian Perkal, ``On the length of empirical curves;" and Julian Perkal, ``An attempt at objective generalization." December, 1966. 11. E. Casetti and R. K. Semple, ``A method for the stepwise separation of spatial trends." April, 1968. 12. W. Bunge, R. Guyot, A. Karlin, R. Martin, W. Pattison, W. Tobler, S. Toulmin, and W. Warntz, ``The philosophy of maps." June, 1968. %---------------------------------------------------------------- %---------------------------------------------------------------- Reprints of out-of-print textbooks. 1. Allen K. Philbrick. This Human World. 2. John F. Kolars and John D. Nystuen. Human Geography. \bye