%----->>>>VOLUME 1, NUMBER 2 OF SOLSTICE. WINTER, 1990; 10:07 PM 12/12/90. %----->>>>HAPPY HOLIDAYS!!!! Sandy Arlinghaus. %----->>>>IF YOU DO not WISH TO CONTINUE YOUR COMPLIMENTARY SUBSCRIPTION, %----->>>>FOR 1991, PLEASE WRITE Solstice@UMICHUM AND SO INDICATE. %----->>>>THANK YOU FOR PARTICIPATING IN GEOGRAPHY'S FIRST E-JOURNAL. %----->>>>HARD-COPY OF VOLUME I IS AVAILABLE AS MONOGRAPH 13 IN THE % IMaGe MONOGRAPH SERIES. \hsize = 6.5 true in %THIS FILE CONTAINS TYPESETTING CODE, ONLY. %ADD IT TO THE BEGINNING OF EACH OF THE FOLLOWING FILES TO TYPESET THEM. %ALSO ADD \bye TO CLOSE EACH FILE TO BE TYPESET. \input fontmac %delete to download, except on Univ. Mich. (MTS) equipment. \setpointsize{12}{9}{8}%same as previous 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 Winter, 1990 \hfil} \def\ref{\noindent\hang} \font\big = cmbx17 \font\tn = cmr10 \font\nn = cmr9 %The code has been kept simple to facilitate reading as e-mail \font\tbf = cmbx12 \outer\def\heading#1\par{\vskip 0pt plus .1\vsize \penalty-50 \vskip 0pt plus -.1\vsize \medskip\vskip\parskip \centerline{\tbf#1}\nobreak\medskip\noindent} \outer\def\section#1\par{\vskip 0pt plus .1\vsize \penalty-50 \vskip 0pt plus -.1\vsize \medskip\vskip\parskip \leftline{\tbf#1}\nobreak\smallskip\noindent} \outer\def\subsection#1\par{\vskip 0pt plus .1\vsize \penalty-50 \vskip 0pt plus -.1\vsize \medskip\vskip\parskip \leftline{\sl#1}\nobreak\smallskip\noindent} \def\pmb#1{\setbox0=\hbox{#1}% \kern-.025em\copy0\kern-\wd0 \kern.05em\copy0\kern-\wd0 \kern-.025em\raise.0433em\box0 } \centerline{\big SOLSTICE:} \vskip.5cm \centerline{\bf AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS} \vskip5cm \centerline{\bf WINTER, 1990} \vskip12cm \centerline{\bf Volume I, Number 2} \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} \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 (College of Architecture and Urban Planning).} \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. Information Systems Laboratory. \hfil} \smallskip \line{{\bf Business} \hfil} \line{{\bf Robert F. Austin}, Director, Automated Mapping and Facilities Management, CDI. \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. \vskip2in \noindent {\bf Send all correspondence to:} \vskip.1cm \centerline{\bf Institute of Mathematical Geography} \centerline{\bf 2790 Briarcliff} \centerline{\bf Ann Arbor, MI 48105-1429} \vskip.1cm \centerline{\bf (313) 761-1231} \centerline{\bf IMaGe@UMICHUM} \vfill\eject 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\/} will be sold (contact IMaGe for price--{\sl Solstice\/} and will be priced to cover expenses of journal production); it is the desire of IMaGe to offer electronic copies to interested parties for free--as a kind of academic newsstand at which one might browse, prior to making purchasing decisions. Whether or not it will be feasible to continue distributing complimentary electronic files remains to be seen.} \vskip.5cm Copyright, December, 1990, Institute of Mathematical Geography. All rights reserved. \vskip1cm ISBN: 1-877751-44-8 \vfill\eject \centerline{\bf SUMMARY OF CONTENT} \smallskip Numbering given below corresponds to the number of the electronically transmitted file. \smallskip \noindent 1. Typesetting code; file of {\TeX} commands that may be inserted at the beginning of each file (or in front of the whole set run at once) in order to typeset the document. \smallskip \noindent 2. File of front matter, including this material! \smallskip \noindent 3 and 4. Reprint of John D. Nystuen from 1974. {\sl A city of strangers: Spatial aspects of alienation in the Detroit metropolitan region.} \smallskip Examines urban shift from ``people space" to ``machine space" (see R. Horvath, {\sl Geographical Review\/} April, 1974) in the context of the Detroit metropolitan region of 1974. As with Clifford's {\sl Postulates of the Science of Space\/}, reprinted in the last issue of {\sl Solstice\/}, note the timely quality of many of the observations. \smallskip \noindent 5. Sandra Lach Arlinghaus. {\sl Scale and dimension: Their logical harmony\/} \smallskip Linkage between scale and dimension is made using the Fallacy of Division and the Fallacy of Composition in a fractal setting. \smallskip \noindent 6 and 7. Sandra Lach Arlinghaus. {\sl Parallels between parallels.\/} A manuscript originally accepted by the now--defunct interdisciplinary journal, {\sl Symmetry}. \smallskip 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 tropical 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 \noindent 8. Sandra L. Arlinghaus, William C. Arlinghaus, and John D. Nystuen. {\sl The Hedetniemi matrix sum: A real--world application.\/} \smallskip 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 entries when the underlying adjacency pattern of the road network was altered by the 1989 earthquake that closed the San Francisco--Oakland Bay Bridge. \smallskip \noindent 9. Sandra Lach Arlinghaus. {\sl Fractal geometry of infinite pixel sequences: ``Super--definition" resolution?} \smallskip Comparison of space--filling qualities of square and hexagonal pixels. \smallskip \noindent 10. {\sl Construction Zone\/}. Feigenbaum's number; a triangular coordinatization of the Euclidean plane. \vfill\eject \centerline{\bf INDUSTRIAL WASTELAND RIVER} \centerline{\bf Photograph by John D. Nystuen; Rouge River, Detroit, 1974.} \centerline{\bf FRONTISPIECE: A City of Strangers.} Click here for FrontispieceClick here for Figure 1.\vfill\eject \centerline{\bf A CITY OF STRANGERS:} \centerline{\bf SPATIAL ASPECTS OF ALIENATION IN} \centerline{\bf THE DETROIT METROPOLITAN REGION.} \smallskip \centerline{\sl John D. Nystuen} \centerline{The University of Michigan, Ann Arbor} \smallskip \centerline{An invited address given in the conference:} \centerline{\it Detroit Metropolitan Politics: Decisions and Decision Makers} \centerline{Conference held at Henry Ford Community College} \centerline{April 29, 1974} \centerline{Dearborn, Michigan} \centerline{Comments added, 1990} Suburbanization at the edge of the metropolitan region and the destruction of homes in the inner city through ``urban renewal'' or expressway construction are the results of uncoordinated and decentralized decisions made by people remote from those directly affected. Unwanted transportation burdens are forced on us by changes in the location of population and jobs. There has been a shift, still continuing, from ``people space'' to ``machine space'' [5] in our cities which we seem powerless to stem. ``Machine spaces'' are those spaces dedicated to machines or to inter--regional facilities which present larger than human, impersonal and often hostile, aspects of society. We are alienated from our urban environment to the degree it has become machine space. We are alienated from land controlled by strangers. These strangers may be decision makers in institutions with metropolitan--wide jurisdictions such as transportation planning authorities, mortgage and banking firms, and the regional power company. The interests of people of this type are at least focused on the metropolis. Other decision makers affecting local land use are outlanders whose concerns are not exclusively local. One type of outlander is the decision maker at state and federal level, concerned with and responsible for general policy of some aspect of urban life but whose vision cannot be expected to distinguish variations in every neighborhood within his/her broad jurisdiction. Other outlanders are decision makers in multi--state or international corporations and institutions whose structures extend horizontally across many communities or even continents. Their aspirations and understanding of urban life are often incommensurate with local community objectives. Misunderstanding, alienation, and conflict easily result. \heading {The Cost of Victory over the ``Tyranny of Space"} From the geographical point of view these disturbing aspects or urban life today are the result of our victory over the ``tyranny of space [7]." Much of the technological achievement of our society has been improvement in transportation and communication. We made the oceans routes not barriers; achieved air and space flight; built power transmission lines to move energy, and sewer lines to carry off wastes. Innovations in communication are equally important. The invention of the alphabet was a great achievement in ancient times (history begins); the printing press followed in medieval times (information widely shared); today we have mini--computers made of inexpensive printed circuits. Electronic data processing (embracing complexity) is as revolutionary as the alphabet and the printing press. The change which will be forthcoming can be only dimly perceived. These inventions affect society by radically changing spatial and temporal limits within which we are confined. This freedom over space and linear time, while closely linked to the rise in our standard of living, now threatens us in other ways. Previously, local community organization and control processes developed relatively free of outside interference because of the friction of distance. Decisions about local land uses and activities had to be made locally because control at a distance was too inefficient. Freedom from the tyranny of space has made us subject to other tyrannies which may be worse. The opportunity to control at a distance which technology offers us may be seized by those who are indifferent to others' needs, selfish and unscrupulous in their quest for power. Too often one man's gain is another man's loss. The unscrupulous become anonymous and unreachable by being hidden in vast institutional hierarchies. Traditional mechanisms of social control and the means to draw people to act for the good of the community are lost. The community is lost in the old geographical sense. We are a city of strangers. I do not advocate giving up our victory over space. Instead we must consider new means of association and control that will humanize the space around us once again. \heading Alienated Space Alienated land in the sense I am using it has two meanings. It is any place where humans are not welcome or may be in real danger; lands dedicated to machines are of this type. But it is also space controlled by strangers, perhaps pleasant places from which we are excluded by fences and ``no trespassing'' signs, or places we may enjoy but over which we have no control as to how they are to be used or changed; state and federal parks are examples. We may find ourselves excluded from many places, subject to regulations in others and even in that kingdom, our own home, denied the right to modify it as we see fit. Little wonder we feel a certain detachment and alienation. Loss of sense of community is the price for our victory over the tyranny of space. Machine space and control of community or neighborhood by strangers are the consequences. \heading {Machine space} Ron Horvath, in an article in the {\sl Geographical Review\/} entitled ``Machine Space,'' classified land parcels as ``machine space'' rather than ``people space'' depending upon ``who or what is given priority of use in the event of a conflict'' [Horvath, p. 169]. He then pointed out how much of our cities we have given up to machines, especially the automobile. He characterized this machine as the ``sacred cow'' in American culture. He said {\narrower{ \noindent In the minds of many Westerners, India's sacred cow has come to symbolize the lengths to which people will go to preserve a nonfunctional cultural trait. But India's sacred cow is downright rational in comparison to ours. Could an Indian imagine devoting 70 percent of downtown Delhi to cow trails and pasturage as we do for our automobiles in Detroit and Los Angeles. Every year nationally we sacrifice more than 50,000 Americans to our sacred cow in traffic accident fatalities (Figure 1) [2, p. 168].\par}} \topinsert \vskip11cm \noindent {\bf Figure 1.} ``Machine Space'' in downtown Detroit, ground level, 1971, by R. Horvath. Map reprinted with permission of The American Geographical Society, from ``Machine Space," R. Horvath, {\sl The Geographical Review\/}, April, 1974, p. 171.

\endinsert

\noindent Something like 20 percent of our gross national product is tied directly to manufacturing, servicing and fueling the automobile---twice the amount we spend on war machines, another more sinister genre of sacred cow machine to which we seem addicted. \heading {Vertical Control or Scale Transforms.} There are signs of a reaction setting in. Ralph Nader effectively pointed out that automobiles are ``unsafe at any speed." The solution called for is not crash proof cars. It is reduction of exposure by reducing passenger miles traveled by private automobiles. We can accomplish this in two very general ways: by developing mass transit systems and by reducing the number and length of trips taken. The latter calls for re-- ordering land use patterns or changing our life style by giving up some of our triumphs over space. Trends in the Detroit Metropolitan Area suggest otherwise. We are still in the process of completing an expressway system. The state has authorized one--half cent of the nine cent gasoline tax to be devoted to mass transit systems; a significant step but hardly a major re-- allocation of priorities. SEMTA, the state transportation authority for Southeast Michigan, has recently released its mass transportation plan calling for a 1990 completion date. If the experience of systems such as the San Francisco Bay Area's BART can be taken as an example, significant delays due to the operation of political processes will set that date further into the future, if indeed, the system is ever built. [As of 1990, the Southeastern Michigan Transportation Authority (SEMTA) is defunct. Their mass transit plan, released in 1975, called for a 1990 completion date (Figure 2). All that came of this plan was the elevated downtown Detroit People Mover, delayed, over budget, and out--of--control as the rest of the mass transportation plan was never implemented and doomed to go out of business. Too massive to tear down without great expense, it will remain a bizarre monument to inadequate planning and fragmented action. On the other hand, the Detroit expressway system is largely completed. A final link in the circumferential network, I-696, opened in 1989, twenty--five years after it was proposed. This stretch of expressway was met with determined opposition from an upper--middle class, politically effective neighborhood. The final links were modified to lessen impact on adjacent residents. Neighborhoods near downtown locations succumbed to the huge concrete corridors years ago. The expressways created huge barriers and the livable spaces between them proved too fragmented to sustain and are now abandoned.] \topinsert \vskip20cm \noindent {\bf Figure 2.} Map from 1974 suggests a network that was never built (as of 1990).Click here for Figure 2.

\endinsert Multi--million dollar transportation projects greatly affect land use patterns and are once--and--for--all investments. They come infrequently and permanently affect the geography of the region. The massive water and interceptor plan of the Detroit Water Board is a similar large scale project with more benign consequences. This brought water from Lake Huron via tunnel and aqueduct to a large portion of the metropolitan region. [It was also a planning error. In retrospect we see it was overbuilt due to the decline in heavy industry in the city and the exodus of people to the suburbs.] Decisions associated with large scale projects are examples of factors which are out of the hands of the ordinary citizen or even the large land developers working in the region. They impose important constraints on land use possibilities. They are decisions made by strangers and represent a loss of private or small community freedom of choice. Many gross forms in the Detroit metropolitan region are the consequence of decisions made many decades ago. Some individuals and communities try to resist the pressures of single large scale commitments. In the case of water procurement, this can be done by using local ground water wells and septic tanks or small municipal sewage plants. At low population densities these local devices may work fine and a decentralized system is probably best. At high densities, however, local environmental capacities are exceeded. Other public agencies, such as the County Health Departments, may then operate to pressure communities into the larger system. It is this hierarchical ordering of systems that removes local control from one aspect after another of urban life. When the problem condition in the environment enlarges previously separate problems begin to merge, the best institutional response we have yet devised is to establish a hierarchically ordered social process to address the larger problem. This change in scale may result in qualitatively different situations. Institutions operating at metropolitan levels may appear very inflexible and arbitrary from the point of view of a local authority, municipality, or private home owner. The need for standardization and routinization is absolutely crucial for such organizations. Alienation may develop between parties who view things at different scales without anyone being at fault. Politically, a metropolitan region is hierarchically organized by spatial jurisdictions. Local problems are most appropriately dealt with by local authority and regional problems by regional authorities. We have yet to devise a means of graciously transferring jurisdiction up or down the hierarchy to correspond to changes in scale in the nature of the problems. Our greatly increased capacity to overcome transportation and communication costs has led to changes in population density and locations of jobs which have often exacerbated local problems and called forth a scale transfer. The local community, no longer able to perform the service, loses jurisdiction over the problem to higher authorities. At a higher level, much of the loss of state power to the federal government has been a change of this sort. [To some extent deregulation efforts of recent years prior to 1990 have shifted responsibility back to local authorities, especially from Federal to State levels. Hierarchies need to be designed that set limits or levels of acceptable performance but remain tolerant of variation in local actions. State rules regarding equalization of county property taxes and local school performance are examples.] \heading {Horizontal Control.} Some institutions and corporations are cross--threaded in the fabric of society. Their interests and actions are uncoupled from the local community because they are interested in a single category of phenomena and not in the mix of all spatial categories at one location. The decision makers in these organizations are very likely to be outlanders; people who live in entirely different communities or even other nations, yet whose decisions may be controlling factors in a local situation. The ability of multi--plant firms to make long distance decisions is closely tied to the effectiveness of channels of control via communication and transportation facilities. As communication improves the management has the option to centralize decision making, thereby reducing the autonomy of each plant manager. In times of poorer communication major decisions regarding enlargement or closing of plants would have been made at the headquarters of the central management. A local community finds its fortunes very much in the hands of outlanders. Three subtle and disturbing aspects may characterize such a relationship. In the first place the central management may act in what it believes to be rational and moral purposes in closing least profitable facilities in favor of expansion in areas which promise higher returns. The overall result may be pernicious. A supermarket chain operating under such rules may end up closing all its stores in the inner city in favor of suburban stores. The internal firm reasons may make complete sense; close the oldest facilities on lots too small to accommodate the latest technologies, in neighborhoods which have declining populations and which do not yield high returns because of general low income levels. Inner city neighborhoods with older retired people and poverty stricken ethnic groups, losing population to urban renewal or expressway construction end up losing their local supermarket. They are the least able to afford the loss. The decision may be made in another city by outlanders unresponsive to the local peoples' problems and with no court of appeals available. A second difficulty for the local community with a plant owned by an international corporation is the policy of the corporation to keep its young and most talented management moving from place to place in order that they can learn the business and eventually be able to assume roles higher up in the corporate hierarchy. It is a perfectly reasonable policy with respect to the internal firm requirements. The consequence, however, is a cadre of talented nomads who show little or no interest in the local welfare of the community in which they are temporarily located. Nor would the community want to commit political resources to such people if they expressed an interest. They are simply removed from making a local community contribution which they might easily have pursued had they been permanently in the community. The only loyalty that makes sense to them is company loyalty. Higher corporate management is certainly not going to discourage this. A third tendency of horizontal cross--community control in society is the homogeneity of facilities and company policy. Hierarchies work best under standard operating procedures. Economies of scale are possible, substitution of material and personnel from one locality to another are facilitated if the installations are all the same. If disciplined standardization and routinization has been enforced top management can make broad, basic decisions secure in the knowledge that countless local exceptions will not subvert their intent during the implementation phase. But what happens when accommodation to local situations is required. You may get a machine answer, ``that request will not compute!'' or more likely the local manager will say, ``I sure would like to help you but my hands are tied by company policy." He may not be telling the truth. The impersonal corporate presence is an easy way to solve a problem by defining oneself out of any concern or responsibility. Of course, he may be telling the truth but be as powerless to change corporate policy as the outsider seeking accommodation. \heading {We Are the Enemy} Pogo said, ``We have met the enemy, and he is us'' [Kelly, 1972]. All metropolitan areas are complex. The Detroit region is no exception. There is no one to blame for the mess. We are the enemy; we are the city of strangers. There is no single leader or group, either evil or benign to blame. The land use pattern grows from our decentralized decision processes. The decisions which actually affect local land use extend over time and space well beyond the here and now. It is true the channels of control could be in the hands of evil doers and we could improve our lot by exposing and removing them. But I think we are not generally in the hands of the unscrupulous; not even in the hands of the stupid and insensitive. It just appears that way. Each decision or action is contingent upon conditions that are beyond the control of the individual or group making a particular choice. There is rarely an instance where these constraints are not present. The outcome often seems stupid or callous. Most deleterious outcomes are probably unanticipated. They are indirect effects not thought of by the decision makers. We need to understand our urban processes well enough to take action to avoid effects which cause discomfort or inequity to others. Constraints on decisions may be classed into three groups. There are institutional and legal policies. There are physical and natural environmental limitations which have to do with laws of nature and the technological capacities with which we may accommodate to those laws. And finally, there are limitations to our aspirations and goals, the imagined conditions that motivate our actions. These aspirations are not hampered by any finiteness of imagination in any single pursuit, for we all know flights of imagination are boundless. Rather limits appear because we harbor multiple needs which are often in conflict. We choose to restrain our objectives in one pursuit in order to achieve goals in other pursuits. For example we find it hard to have large lots and big lawns which provide us with seclusion and status and at the same time have many close and friendly neighbors which make available to us the pleasures and security of sharing a close community. Under most circumstances to gain one value is to lose the other. \heading {Scale Attributes of Value Systems} A definition of values is that they are an individual's feelings about and identification with things and people in his environment. Values have scale attributes. Another three fold classification is convenient. There are {\it individual/familial identification\/}, a commitment to proxemic space --- the space within which one touches, tastes and smells things. Secondly there is {\it community identification\/}, embracing the individual's feelings and concern for those with whom he or she lives and interacts, not in the same house, but in the vicinity or neighborhood. This is local space generally recognizable by sight and smell. Finally there is {\it political--cultural identification\/} which refers to ideals and concerns extending beyond the people and community with which the person has daily contact. This realm must be dealt with abstractly and through instruments, either mechanical or institutional for it is too large to be perceived by the senses directly. This is national or global space. Machine space and control by outlanders may be viewed as intrusions into our community space by organizations and facilities of this larger domain. How they look, sound or smell has not been taken into account in the design of such facilities. Examples include Edison power stations, the Lodge and Ford expressways, and Detroit Metropolitan Airport. We give up local community values for the benefits of the global mobility and interaction. Metropolitan life pushes us to scale extremes. We value individual rights and perogatives and mainline connections with the global culture over familial and community concerns. Intermediate spatial scale values suffer and the community declines along with them. The consequences are visual blight, noise pollution, reduced security, and injustice. Community values include concern for our fellow man, a sense of equity and humaneness. The mechanisms for enforcing a community code of ethics are ostracism, social pressure and the use of sense of humor to keep people responding to others as human beings. These mechanisms do not work well in a city of strangers and are not followed. They are particularly ineffective in those large impersonal machine spaces, the streets and expressways, bus stations, terminals and warehouse and factory districts. The urban code of ethics carefully preserves the privacy of individuals and tolerates eccentrics. A person has functional but fragmented value and is valued for specific tasks he or she can do. A major problem with the dehumanization and anonymity of urban life is that the unscrupulous are freed from social control along with the rest of us. We have distinct evidence that we are being ``ripped off" at both ends of the spatial scale of involvement. Corporations manipulate markets through advertisements thereby creating artificial shortages and rapid obsolescence of their products without fear of being called to account. Radical monopolies in the words of Ivan Illich. At the other extreme individuals, free of local control, satisfy their wants by committing violent criminal acts against others and then disappearing into the crowd. Ostracism and social pressure work between friends. They are meaningless to the corporate manipulator and street criminal. We are in a crisis of conflicting values when we attempt to reform the structure of society to eliminate these problems. We tend to throw the baby out with the bath water. Action against crime in the streets and the home is moving toward hardening our shelters, walling up windows, barring doors, hiring guards and guard dogs, and restricting access. Security guards in Detroit are big business. Even entering the Federal District Court in downtown Detroit now requires a personal search. These actions are destructive of community spirit. They are a falling back to greater individual isolation. Burglar proof apartments are more effective against neighbors than against burglars (Figure 3). \topinsert \vskip22cm \noindent {\bf Figure 3.} Photographs of Detroit scenes by John D. Nystuen, c. 1974.Click here for Figure 3.

\endinsert We have barely recognized the assault on our well being through manipulation by national corporations, let alone having devised counter measures. The major instruments of global firms are standardization and routinization. And Detroit is a symbol of giant multinational corporations and the Henry Ford--perfected assembly line. A defensive action of sorts is uncoupling part of one's life from the national distribution system. Making and using homemade products are countermeasures. The great rise in home crafts, community garden projects, potters' guilds, art fairs and galleries and counter--culture craft shops provide some vehicles for humanizing city space and reestablishing a sense of community. College youth are showing the way. Wearing old work clothes everywhere, worn and patched (whether needed or not) is a symbol of a society moving beyond mass consumption. Of course, as soon as old work clothes become {\it de rigueur\/} the agents of mass production can reassert themselves by selling pre--patched garments. Community values benefit most by seeking simple handmade products. The craft shop and modern craft guilds should be valued for their local community effect and should be supported because of their community value (Table 1). \midinsert \smallskip \hrule \smallskipClick here for Figure 4.

TYPESETTING FOR TABLE 1

\centerline{\bf TABLE 1.} \centerline{HUMAN VALUES CLASSED BY SPATIAL SCALE} \settabs\+\indent&individual--familial\qquad\qquad&global (national)\qquad \qquad&abstract via instruments\quad&\cr %sample line \+&{\bf Value}&{\bf Space}&{\bf How Sensed}\cr \smallskip \+&individual--familial&proxemic &see, hear, touch, smell \cr \+&communal &local &see, hear \cr \+&political--cultural &global (national)&abstract via instruments\cr \+&{} &{} &\quad and institutions \cr \smallskip \noindent Human values are an individual's feelings and sense of identification with people and things in the surrounding environment. \smallskip \hrule \smallskip \endinsertClick here for Figure 1.

\heading {Card Carrying Americans} My standard sized dictionary has a dozen meanings listed for the word {\it trust\/}. The first meaning of trust is that it is a confident reliance on the integrity, honesty, veracity or justice of another. It used to be that credit was a local community relationship. When you moved to a new town or new neighborhood you could gain credit by managing to buy some clothes or furniture on time and then making sure that you payed up in a timely fashion according to the agreed--upon terms. It was a way to establish trust with local merchants. Today large financial institutions and other multinational corporations such as petroleum companies have taken advantage of innovations in communication and information handling to make a space adjustment in extending credit which better fits their scale of operations. Credit cards make trust an abstract, formal relationship which operates nationwide or globally and which can be entrusted to machines for monitoring. But as with other abstractions, not all the original meaning of the word transfers to the new use. Justice fades. The new scale of operation provides a marvelous freedom for those who carry cards. Unfortunately it is easier for some people to get credit cards than it is for others. The poor and the young are often prevented from obtaining them at all. We have created two classes of Americans --- card carrying Americans and second class citizens who must pay cash. There is every reason to believe that in the future consumer exchanges will be increasingly handled by some type of credit transaction. The effect is pernicious in poor neighborhoods. In the past the local grocer or merchant often provided credit to local people whom they had come to trust. This service has become less common and the range of goods obtainable through local credit is shrinking as large corporations capture greater and greater share of the market. They deal in cash only or with credit cards. They do not maintain personal charge accounts. Typically in an urban renewal process a poor, ghettoed family is forced to move because their house is condemned by the ``improvement." They move to a new neighborhood where likely as not they must pay more for housing than they did previously and simultaneously they lose the credit relationship they had built with local merchants in the old neighborhood. Credit cards are typical of space adjusting developments which accomplish their purpose through abstracting and depersonalizing relations. Accounting for the full circumstances of an individual and making a judgment about his or her trustworthiness is not possible. Justice is lost in the transform and the word trust begins to mean something else. \heading {Mainlining Fantasy with the Television Tube} Just as surely as the automobile is the dominant anti-- neighborhood transportation device, television is the dominant anti--community communication device. Think of the products sold on television: standardized balms and salves for our bodies, stomachs and minds; automobiles to speed us into exotic landscapes; miracle materials to clean our homes without effort; and corporate images to make us all like the firms which deliver these products. Television is a device for mainlining messages directly from national and global organizations to individuals: to millions of individuals. The messages must necessarily be abstract, standardized and unreal. There is a certain lack of trust in the transmission. Value priorities and the meaning of common English words used in ads do not resemble the values and common usage used in face to face communications. The verbiage is exaggerated; hyperbole employed to describe mundane products. Cliches are strung together one after another. If one of these advertising images came alive in our living room and we tried to have a conversation we would find the person indeed odd. From the point of view of community values television messages have several bad features. First and foremost there is no way to clarify or challenge a point because the communication is one way. Secondly it is difficult to compete with the siren songs of the national product distributors. A message meant for millions is worth purchasing the best possible creative talent to deliver it. Corporations that can afford national TV time are selling standardization and routinization nationwide. They gain economies of scale in doing so. This often means they have a price advantage over local competition or worse, they convince people the national product is a superior albeit more expensive item than a local one. Countermeasures for this assault are to substitute handmade items for mass produced ones. Another step is to consume less. Seeking satisfaction in other than materialistic pursuits will often mean turning to local, community--level activities. It hardly need be said that the images projected by television are fantasies that mirror reality through very strange glasses. They glorify individualism and vilify community forces. Nature is also often depicted as implacable, hostile and competitive. This view requires that the individual seek some inner strength in order to prevail when threatened by the environment. Other views in which nature and society are more benign and cooperative are possible but they do not provide the excitement which seem to attract viewers. This hostile approach to the fantasy environment apparently affects people's evaluation of the real environment. There is evidence that people who watch television extensively are more fearful of crime than people who seldom watch it. Large communication systems affect perception apart from the fantasy content. In reporting news in a metropolitan area the size of Detroit with nearly five million people in the ``community" many bizarre crimes are avidly reported by telecasters and other media sources. Upon hearing such reports people think, ``What a terrible thing right here in our city." The populace of metropolitan areas of half a million will not hear such stories about their town with nearly the same frequency because there is an order of magnitude difference in the base population. This is not to make light of the crime rate in Detroit which is large on a {\it per capita\/} basis or by almost any measure. But the scale effect is present in addition to the hard facts of the high crime rates in Detroit. Further technological innovation may deliver us from some of the worst effects of the current revolution in transportation and communications devices. It is becoming more feasible to handle great complexity in large systems through information control. The likely consequence is greater individual freedom of choice while still permitting participation in a large system. The automobile assembly line is again an example. Henry Ford provided Model T and Model A Fords in the colors of your choice --- so long as that choice was black. Modern auto manufacturers now deliver autos of many styles, in scores of colors, streaming from assembly lines in a complex sequence which matches the week by week flow of customer orders coming in from throughout the country. This is achieved through computer control of parts scheduling on the assembly line. Cable TV promises multiple channels, possible two way communication, and tapes and libraries of past broadcasts, and narrow casting in which programs and exchanges are limited to specified audiences. These developments might provide such a great range of choices to the viewer that the current monopolizing of television by outlander interest, as with major news networks, could be weakened. Capacity to handle an order of magnitude greater complexity through effective information processing could serve a broader range of values. But, as with credit cards, who will be served by the greater freedom? Freedom will go to those with the knowledge and money to use the services. Justice need not be served. Community values could regain some lost ground under such developments but only if concerted and careful efforts in support of local values is brought to bear on decisions as to how the new technology is to be used. \heading {Strategies for Local Control} Our message is that the decline in quality of urban life is due in part to loss of community values in competition with individual and outlander values which were better served by advances in transportation and communication. Our goal should be to restore balance in our lives by restoring some community commitments. In general, as temporal and spatial constraints are lifted institutional and legal parameters need to be erected to avoid abuse and pathologies in our social processes. This is easier said than done. The first problem is to recognize a problem when we see it. We have been slow to see that the automobile is actually taking over the spaces of our cities as if it were becoming a biologically dominant species. Bunge and Bordessa suggest that we concentrate on improving and enlarging the spaces devoted to children in our cities as a first priority in ordering city space. They show that much benefit flows to the entire society through such strategies. People space gains at the expense of machine space. If the long distance transportation facilities and other sinews of the large metropolitan systems are channelized and confined to corridors and special locations the spatial cells created will be available for local uses. But priorities must be correct. We live in the local cells. We only temporarily exist in the transportation channels at which times we suspend normal civilities and common courtesy. The life cells (neighborhoods) should be the objects, not the residuals, of the urban form. Bunge and Bordessa [3] suggest mapping local and non--local land use in urban neighborhoods. The simple facts of that division will reveal the extent of outlander control of a community. I repeat, you have to see a problem before you can deal with it. Professional planners, academics and citizen groups should develop the concepts and generate the data which highlight the areas that are directly and humanly used rather than those spaces that are indirectly, abstractly used through machines. Hierarchies are necessary for the operation of large systems but the tendency for imposing standardization and routinization in control hierarchies should be resisted. This can be done by incorporating the rapidly increasing capacity to handle complex information flows. Great metropolitan--wide hierarchies to deal with water supply, traffic control and crime suppression are possible if these large structures are robust enough to allow local variation and still retain an overall integrity. The goals should be always to allow maximum freedom of choice at local levels but with that choice constrained by considerations of equity relative to other elements in the system. Promoting local initiative, self--respect and autonomy would tend to create a heterogeneous urban landscape. But freedom and equity can be conflicting values. We must strive to make the heterogeneity healthy. We would do well to give first consideration to local people space rather than to machine space. Once our attention is so directed we should make certain that no living space in the city is mere residual left from the process of carving the urban landscape into machine space and space for the outlander and the powerful. I wager that the reader is probably viewing the metropolis at full regional scales. I will close with a word of advice. If you are active in trying to make Detroit a better place in which to live you may well be viewed as an outlander by most of those with whom you interact. There may be a conflict of interest between local community and regional views. I believe your strategy should be to encourage local initiative to enlarge and to improve the quality of neighborhood people--space while at the same time being careful that such actions are not at the expense of other neighborhoods. The achieving of equity is the responsibility of those with regionwide vision. Value, understand, and encourage heterogeneity in living spaces but strive to prevent any living area from falling too far behind in the quest for quality neighborhoods. That will insure integrity of the whole while affording maximum freedom to the parts. \heading {References and Suggestions for Related Readings} \ref 1. Abler, Ronald F., ``Monoculture or Miniculture? The Impact of Communications Media on Culture in Space," in D. A. Lanegran and Risa Palm, {\sl An Invitation to Geography\/}. New York: McGraw Hill, 1973. \ref 2. Boulding, Kenneth E., {\sl Beyond Economics: Essays on Society, Religion and Ethics\/}. Ann Arbor, Michigan: University of Michigan Press, 1970. \ref 3. Bunge, W. W. and Bordessa, R. {\sl The Canadian Alternative: Survival, Expeditions, and Urban Change\/}, Geographical Monograph No. 2, Department of Geography, York University, Toronto, Intario, Canada, 1975. \ref 4. Gerber, George and Larry Gross. ``The Scary World of TV's Heavy Viewer," {\sl Psychology Today\/}, v. 9 no. 11 (April, 1976): 41-45. \ref 5. Horvath, Ronald, ``Machine Space," {\sl The Geographical Review\/}, v. 64 (1974): 167-188. \ref 6. Kelly, Walt, {\sl We Have Met the Enemy and He Is Us\/}. New York: Simon and Schuster, 1972. \ref 7. Little, Charles E., ``Urban Renewal in Atlanta Is Working Because More Power Is Being Given the the Neighborhood Citizens," {\sl Smithsonian\/} v. 7 no. 4 (July 1976):100-107. \ref 8. Warntz, William, ``Global Science and the Tyranny of Space," {\sl Papers\/}, Regional Science Association, v. 19 (1967): 7-19. \ref 9. Webber, Melvin M., ``Order in Diversity: Community Without Propinquity." In Lowdon Wingo, Jr. (editor), {\sl Cities and Space -- The Future Use of Urban Land\/}. Baltimore, Maryland: Johns Hopkins Press, 1963, pp. 23-54. \vfill\eject \centerline{\bf SCALE AND DIMENSION: THEIR LOGICAL HARMONY} \smallskip \centerline{\sl Sandra Lach Arlinghaus} \smallskip \smallskip \centerline{\it ``Large streams from little fountains flow,} \centerline{\it Tall oaks from little acorns grow." } \smallskip \centerline{David Everett, {\sl Lines Written for a School Declamation\/}.} \smallskip \heading Introduction. Until recently, the concept of ``dimension" was one that brought ``integers" to mind to all but a handful of mathematicians [Mandelbrot, 1983]; a point has dimension 0, a line dimension 1, an area dimension 2, and a volume dimension 3 [Nystuen, 1963]. When a fourth dimension is added to these usual spatial dimensions, time can be included, as well. Indeed, much ``pure" mathematics takes place in abstract $n$--dimensional hypercubes, where $n$ is an integer. Geographic maps, globes (and other representations of part or of all of the earth), are traditionally bounded by these integral dimensions, as well; map scale is expressed in discrete, integral units. Often, however, it is the case in geography as it is in mathematics, that a change in scale, or in dimension, runs across a continuum of possible values. In either case, discrete regular steps are usual as benchmarks at which to consider what the continuing process looks like at varying stages of evolution. As fractal geometry suggests, however, this need not be the case. Within an integral view of scale or dimension, there are logical and perceptual difficulties in jumping from one integral vantage point to another: Edwin Abbott [1955] has commented on this in his classic abstract essay on ``Flatland," and more recently, Edward Tufte has done so in the real--world context of ``envisioning information" [1989]. Methods for dealing with these dimensional--jump difficulties abound, particularly in the arts [Barratt, 1980]. In a musical context Charles Wuorinen sees composition as a process of fitting ``large" musical forms with scaled--down, self--similar, equivalents of these larger components in order to introduce richness of detail to the theme [NY Times, 1990]. Maurits Escher, in his ``Circle Limit" series of tilings of the non-- Euclidean hyperbolic plane, uses tiles of successively smaller size to suggest a direction of movement---that of falling off an edge or of being engulfed in a central vortex. A gastronomic leap sees a Savarin as self--similar to a Baba au Rhum [Lach, 1974]; indeed, even more broadly, Savarin himself is purported to have said, ``You are what you eat." Rupert Brooke (in ``The Soldier") captured this notion poetically, in commenting on the possible fate of a soldier in a distant land: \centerline{``If I should die, think only this of me; } \centerline{ that there is some corner of a foreign field } \centerline{ that is forever England." } \noindent In the end, Brooke's ``Soldier" becomes `place'. The fractal concept of self--similarity can be employed to suggest one way to resolve difficulties in scale changes as one moves from dimension to dimension. At the theoretical level, symbolic logic classifies logical fallacies that may, or may not, emerge from scale shifts. When self--similarity is viewed in this sort of logic context, the outcome is a ``Scale Shift Law." What is presented here are the abstract arguments; it remains to test empirical content against these arguments. \heading Logical fallacies. A question of enduring interest in geography, and in other social sciences, is to consider what can be said about information concerning individuals of a group when given information only about characteristics of the group as a whole. When an attribute of the whole is {\bf erroneously} assigned to one or more of its parts, the logic of this assignment falters. In the social scientific literature, this is generally referred to as commission of the so--called ``ecological" fallacy; because the symphony played poorly does not necessarily mean that each, or indeed that any, individual musician did so. In this circumstance, it is simply not possible to assign any truth value, derived from principles of symbolic logic, to the quality of the performance of any subset of musicians (based only on the quality of the performance of the whole orchestra) [Engel, 1982]. It is natural, however, to look for a cause for the poor performance, and indeed to consider some ``middle" position that asks to what extent the performance of the orchestra is related to the performance of its individual members. It is this sort of search for finding and measuring the extent of relationship that is the hallmark of quantitative social scientific effort, much of which appears to have been guided [Upton, 1990], in varying degree, by an early effort to determine the extent to which race and literacy are related [Robinson, 1950]. A fallacy, in a lexicographic sense might be ``a false idea" or it might be of ``erroneous character" or ``an argument failing to satisfy the conditions of valid or correct inference" [Webster, 1965]. In a formal logic sense, a fallacy is ``a `natural' mistake in reasoning" [Copi, 1986, p. 4] or it is an argument that fails because its premisses do not imply its conclusion; it is an argument whose conclusion {\bf could} be (but is not necessarily) false even if all of its premisses are true [Copi, 1986, p. 90]. Viewed in this manner, the so--called ``ecological" fallacy is nothing different; it is merely a restatement of the ``fallacy of division" of classical elementary symbolic logic. The fallacy of division is committed by assigning, {\bf erroneously}, the attributes of the whole to one or more of its parts [Copi]. Thus, it may or may not be valid to make an inference about the nature of a part based on the nature of the whole. That is, sometimes the assignment of truth value from whole to part, in jumping across the dimensional scale from whole to part, is a reasonable practice, and sometimes it is not. The key is to determine when this practice is reasonable, when it is not, and when it simply does not apply. Commission of this fallacy is frequently the result of confusing terminology which refers to the whole (``collective" terms) with those which refer only to the parts (``distributive" terms) [Copi, 1986]. The fallacy of division exists within an abstract human system of reasoning based on the Law of the Excluded Middle: in this Law, a statement is true or false---not some of each. There is ``black" and ``white," but no ``gray" in this system. Statistical work that stems from this fallacy seeks, when it rests on finding correlations, relations that blend ``black" and ``white"---the foundation in ``logic" is thus ignored. This fallacy is examined, here, with an eye to understanding the logical circumstances under which such assignment might, or might not, be erroneous (when it applies). \heading Scale and dimension. To understand when the assignment of characteristics from whole to part (division), or from part to whole (the fallacy of composition---the string sections played well, therefore the symphony played well), might be erroneous, it is useful to consider what are the fundamental components composing these fallacies. The notion of scale is involved in the consideration of ``whole" and ``part." When is the individual a ``scaled--down" orchestra; or, when is the orchestra a ``scaled--up" individual? The notion of dimension is also involved. When does the zero--dimensional musician-- point spread out to fill the two--dimensional (or three--or more--dimensional) orchestra; or, when does the higher dimensional orchestra collapse, black--hole--like, into the single performer. The performing soloist can dominate the orchestra; the conductor perhaps does dominate the orchestra; yet, the orchestra itself is composed of numerous single performers who do not dominate. \heading Self--similarity and scale shift. Integral dimensions, with discrete spacing separating them, might be viewed as simply a set of positions marking intervals along a continuum of fractional dimensions [Mandelbrot, 1983]. When the discrete set of integral dimensions is replaced by the ``dense" set of fractional dimensions (between any two fractional dimensions there is another one), what happens to our various relative vantage points and to scale problems associated with them? Abstractly, the relationship is not difficult to tie to logic, under the following fundamental assumption. \smallskip \line{\bf Fundamental Assumption.\hfil } \smallskip When two views of the same phenomenon at different scales are self--similar one can properly divide or compose these views to shift scale. \smallskip \noindent The whole can be divided ``continuously" through a ``dense" stream of fractional dimensions until the part is reached (and in reverse). Self--similarity suggests a sort of dimensional stability of the characteristic or phenomenon in question. One commits the Fallacy of Division (``Ecological" Fallacy) when the attributes (terminological or otherwise) of the whole are assigned to the parts that are {\bf not} self--similar to the whole. One commits the Fallacy of Composition when the attributes of the parts are assigned to a whole that is {\bf not} self--similar to these parts. This notion is evident in the many animated graphic displays of the Mandelbrot (and other) sets in which zooming in on some detail presents some sort of repetitive sequence of views (in the case of self--similarity, this sequence has length 1). More formally, this idea may be cast as a ``Law." \smallskip \line{\bf Scale Shift Law \hfil} \smallskip Suppose that the attributes of the whole (part) are assigned to the part (whole). \item{1.} If the whole and the part {\bf are not} self-- similar, then that assignment {\bf is} erroneous; and, conversely (inversely, actually), \item{2.} If the whole and the part {\bf are} self--similar, then that assignment {\bf is not\/} erroneous. \smallskip \noindent This is one way to look at the ``part--whole" dichotomy; physicists wonder about splitting the latest ``fundamental" particle; philosophers search for fundamental units of the self [Leibniz, monadology, in Thompson, 1956; Nicod, 1969]; topologists worry about what properties a topological subspace can inherit from its containing topological space [Kelley, 1955]. \heading References. \ref Abbot, Edwin A. (1956) ``Flatland." reprinted in {\sl The World of Mathematics\/}, James R. Newman, editor. New York: Simon and Schuster. \ref Barratt, Krome (1980) {\sl Logic and Design: The Syntax of Art, Science, and Mathematics\/}. Westfield, NJ: Eastview Editions, 1980. \ref Copi, Irving M. (1986) {\sl Introduction to Logic\/}. Seventh Edition. New York: Macmillan Publishing Company, (first edition, 1953). \ref Engel, S. Morris (1982) {\sl With Good Reason: An Introduction to Informal Fallacies\/}. Second Edition. New York: St. Martins Press. \ref Kelley, John L. (1963) {\sl General Topology\/}. Princeton: D. Van Nostrand. \ref Lach, Alma S. (1974) {\sl The Hows and Whys of French Cooking\/}, Chicago: The University of Chicago Press. \ref Mandelbrot, Benoit (1983) {\sl The Fractal Geometry of Nature\/}. San Francisco: Freeman. \ref Nicod, Jean (1969) {\sl Geometry and Induction: Containing `Geometry in the Sensible World' and `The Logical Problem of Induction' with Prefaces by Roy Harrod, Bertrand Russell, and Andre Lalande\/}. London: Routledge and Kegan Paul, New translation. \ref Nystuen, John D. (1963) ``Identification of some fundamental spatial concepts." {\sl Papers of Michigan Academy of Letters, Sciences, and Arts\/}. 48: 373-384. \ref Robinson, W. (1950) Ecological correlations and the behavior of individuals, {\sl American Sociological Review\/}. 15: 351-357. \ref Rockwell, John (1990) ``Fractals: A Mystery Lingers." Review/Music, {\sl The New York Times\/}, Thursday, April 26. \ref Thompson, D'Arcy Wentworth (1956) ``On Magnitude." In {\sl The World of Mathematics\/}, James R. Newman, Editor. New York: Simon and Schuster. \ref Tufte, Edward (1989) {\sl Envisioning Information\/}. Cheshire, CT. \ref Upton, Graham J. G. (1990) ``Information from Regional Data," in {\sl Spatial Statistics: Past, Present, and Future\/}, edited by Daniel A. Griffith. IMaGe Monograph, \#12. Ann Arbor: Michigan Document Services. \ref {\sl Webster's Seventh New Collegiate Dictionary\/} (1965) Springfield, MA: G. and C. Merriam Company. \vfill\eject \centerline{\bf PARALLELS BETWEEN PARALLELS} \smallskip \centerline{\sl Sandra Lach Arlinghaus} \smallskip \smallskip \centerline{\it ``I have a little shadow that goes in and out with me,} \centerline{\it And what can be the use of him is more than I can see."} \smallskip \centerline{\sl Robert Louis Stevenson } \centerline{``My Shadow" in {\sl A Child's Garden of Verses}} {\narrower\smallskip{\bf Abstract}: 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 tropical 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} \heading 1. Introduction. Subtle influences shape our perceptions of the world. The breadth of a world--view is a function not only of ``real"--world experience, but also of the ``abstract"--world context within which that experience can be structured. As William Kingdon Clifford asked in his Postulates of the Science of Space [3], how can one recognize flatness when magnification of the landscape merely reveals new wrinkles to traverse? Geometry is a ``source of form" not only in mathematics [10], but also in the ``real" world [2]. Street patterns are geometric; architectural designs are geometric; and, diffusion patterns are geometric. In this study, the geometric notion of parallelism is examined in relation to the manner in which the sun's trajectory in the earth's sky is observed by inhabitants at various latitudinal positions: from north and south of the tropics to between the tropical parallels of latitude. A fundamental geometrical notion is thus aligned with fundamental geographical and astronomical relationships; this alignment is interpreted in cultural contexts ranging from the design of rooflines to the design of board games. \heading 2. Basic Geometric Background. To understand how geometry might guide the perception of form, it is therefore important to understand what ``geometry" might be. Projective geometry is totally symmetric and possesses a completely ``dual" vocabulary: ``points" and ``lines," ``collinear" and ``concurrent," and a host of others, are interchangeable terms [6]. Indeed, a Principle of Duality serves as a linguistic axis, or mirror, halving the difficulty of proving theorems. Thus, because ``two points determine a line" is true, it follows, dually, that ``two lines determine a point" is also true. The corresponding situation does not hold in the Euclidean plane: two lines do not necessarily determine a point because parallel lines do not determine a point [6]. Coxeter classifies other geometries as specializations of projective geometry based on the notion of parallelism, depending on whether a geometry admits zero, one, or more than one lines parallel to a given line, through a point not on the given line [6]. In the ``elliptic" geometry of Riemann, there are no parallel lines, much as there are none in the geometry of the sphere that includes great circles as the only lines, any two of which intersect at antipodal points. In ``affine" geometry, there is exactly one line parallel to a given line, through a point not on that line. Affine geometry is further subdivided into Euclidean and Minkowskian geometries. Finally, in the ``hyperbolic" geometry of Lobachevsky, there are at least two lines parallel to a given line through a point not on that line. To visualize, intuitively, the possibility of more than one line parallel to a given line it is helpful to bend the lines, sacrificing ``straightness" in order to retain the non-- intersecting character of parallel lines. Thus, two upward-- bending lines $m$ and $m'$ passing through a point $P$ not on a given line $\ell$ never intersect $\ell$; they are divergently parallel to $\ell$ (Figure 1.a). Or, one might imagine lines $m$ and $m'$ that are asymptotically parallel to $\ell$ (Figure 1.b) [8]. \topinsert \vskip15cm

{\bf Figure 1.} The hyperbolic plane. \item{a.} Two lines $m$ and $m'$ (passing through $P$) are divergently parallel to line $\ell$. \item{b.} Two lines $m$ and $m'$ (passing through $P$) are asymptotically parallel to line $\ell$. \endinsert Elliptic geometry, with no parallels, and associated great--circle charts and maps have long been used as the basis for finding routes to traverse the surface of the earth. The suggestion here is that affine geometry, with single geometric parallels, captures fundamental elements of the earth--sun system outside the tropical parallels of latitude, and that hyperbolic geometry, with multiple geometric parallels does so between the tropical parallels of latitude. \heading 3. Geographic and Geometric ``Parallels". As the Principle of Duality is a ``meta" concept about symmetry in relation to projective geometry, so too is the earth--sun system in relation to terrestrial space. The changing seasons and the passing from daylight into darkness are straightforward facts of life on earth, often taken for granted. Some individuals appear to be more sensitive to observing this broad relationship, and to deriving information from it, than do others. Shadows may serve as markers of orientation as well as of the passing of time. \section 3.1 North and south of the tropical parallels. Individuals north of $23.5^{\circ}$ N. latitude and those south of $23.5^{\circ}$ S. latitude always look in the same direction for the path of the sun: either to the south, or to the north (not both). Shadows give them linear information only, as to whether it is before or after noon; shadows never lie on the south side of an object north of the Tropic of Cancer. The perceived path of the sun in the sky does not intersect the expanse of the observer's habitat, from horizon to horizon. Thus, it is ``parallel" to that habitat. North and South of the tropics there is but one such parallel, corresponding to the one basic direction an individual must look to follow the sun's trajectory across the sky. \section 3.2 Between the tropical parallels. Between the tropics, however, the situation is entirely different. On the equator, for example, one must look half the year to the north and half the year to the south to follow the path of the sun. Thus, there are two distinct (asymptotic) parallels for the path of the sun through the observer's point of perception. Shadows can lie in any direction, providing a full compass--rose of straightforward information as to time of day as well as to time of year: apparently a broader ``use" of shadow than Stevenson envisioned! This population is thus surrounded, in its perception of the external environment of earth--sun relations, by the multiple parallel notion. (Those accustomed to primarily an Euclidean earth--sun trajectory might find this disconcerting.) This hyperbolic ``vision" of the earth--sun system, suggests a consistency, for tropical inhabitants only, established in a natural correspondence of the perception of the external environment and the internal environment of the brain. For, it is the contention of R. K. Luneberg that hyperbolic geometry is the natural geometry of the mapping of visual images onto the brain [9]. \heading 4. The Poincar\'e Model of the Hyperbolic Plane. To see how this variation in perception of the earth--sun system might be reflected in real--world settings, and to compare such settings between and outside the tropical parallels, it is necessary to understand one of these geometries in terms of the other. Both Euclidean and hyperbolic geometries are single, complete mathematical systems. They are not, themselves, composed of multiple subgeometries, nor can one of them be deduced from the other: they have the mathematical attributes of being categorical and consistent [6]. A mathematical system is categorical if all possible (mathematical) models of the system are structurally equivalent to one another (isomorphic) [13]; these models are, by definition, Euclidean and are therefore useful as tools of visualization. Because the hyperbolic plane is a categorical system, all models of it are isomorphic. Therefore, it will suffice to understand but a single one, and that one will then serve as an Euclidean model of the hyperbolic plane. Henri Poincar\'e's conformal disk model (in the Euclidean plane) of the hyperbolic plane [8], was inspired by considering the path of a light ray (in a circle) whose velocity at an arbitrary point in the circle is equal to the distance of the point from the circular perimeter [4]. To understand how the model works, a ``dictionary" that aligns basic shapes in the hyperbolic plane with corresponding Euclidean objects is useful (Table 1, Figure 2) [8].

\topinsert \vskip11cm \smallskip \hrule \smallskip \centerline{\bf Table 1:} \centerline{The Poincar\'e conformal model of the hyperbolic plane} \centerline{(referenced to Figure 2---after Greenberg)} \smallskip \hrule \smallskip \settabs\+\indent&Term in hyperbolic \quad & in the Poincar\'e model \quad&\cr \+&Term in hyperbolic&Corresponding term \cr \+&geometry &in the Poincar\'e model\cr \+&{} &in the Euclidean \cr \+&{} &plane \cr \smallskip \hrule \smallskip \+&Hyperbolic plane &A disk, $D$, interior to a \cr \+&{} &Euclidean circle, $C$ \cr \smallskip \+&Point &Point, $P$, in the disk, $D$.\cr \smallskip \+&Line &\item{1.} Disk diameter, $\ell$, not \cr \+&{} &including endpoints on $C$); or \cr \+&{} &\item{2.} Arcs, $m$, $m'$, in $D$ of circles \cr \+&{} &orthogonal to $C$ (tangent lines \cr \+&{} &at points of intersection are \cr \+&{} &mutually perpendicular). \cr \smallskip \hrule \smallskip \endinsert \topinsert \vskip15cm {\bf Figure 2.} The Poincar\'e Disk Model of the hyperbolic plane.Click here for Figure 2.

\item{a.} The diameter, $\ell$, is a Poincar\'e line of the model, as are arcs $m$ and $m'$ which are orthogonal to the boundary $C$. The Poincar\'e lines $\ell$ and $m$ are parallel (do not intersect); the lines $\ell$ and $m'$ are not parallel (do intersect). \item{b.} The sum of the angles of $\Delta OPQ$ is less than $180^{\circ}$. The triangle is formed by sides $\ell$, $m$, $n$; the Poincar\'e lines $\ell$ and $m$ are diameters, and the Poincar\'e line $n$ is an arc of a circle orthogonal to C. \item{c.} A Lambert quadrilateral with three right angles and one acute angle $(PRQ)$. Pairs of opposite sides are parallel. \endinsert The hyperbolic plane is represented as the disk, $D$, interior to an Euclidean circle $C$. Because the bounding circle, $C$, is not included, the notion of infinity is suggested by choosing points of $D$ closer and closer to this unreachable boundary. Points in the hyperbolic plane correspond to points in $D$. Lines in the hyperbolic plane correspond to diameters of $D$ or to arcs of circles orthogonal to $C$. These arcs and diameters are referred to as ``Poincar\'e" lines. Because $C$ is not included in the model, the endpoints of the Poincar\'e lines are not included, suggesting the notion of two points at infinity. Two Poincar\'e lines $\ell$ and $m$ are parallel if and only if they have no common point. Thus, the disk diameter $\ell$ and the circular arc, $m$, orthogonal to $C$ are parallel because they do not intersect; however, the disk diameter $\ell$ and the circular arc, $m'$, orthogonal to $C$ are not parallel because they do intersect (Figure 2a). \heading 5. Hyperbolic Triangles and Quadrilaterals. Any triangle in the hyperbolic plane is such that the sum of its angles is less than $180^{\circ}$. When a triangle is drawn in the Poincar\'e model this becomes quite believable; draw Poincar\'e lines $\ell$ and $m$ as disk diameters and draw Poincar\'e line $n$ as an arc of a circle orthogonal to the disk boundary (Figure 2b) [8]. The triangle formed in this manner has one side that has ``caved--in" suggesting how it happens that the angle sum can be less than $180^{\circ}$ (note that three diameters cannot intersect in a triangle because all diameters are concurrent at the center of the disk). Triangles formed from more than one Poincar\'e line that is an arc of a circle would become even more concave. Because all triangles have angle sum less than $180^{\circ}$, there can be no rectangles (quadrilaterals with four right angles) in the hyperbolic plane. The idea that corresponds to that of a rectangle is a quadrilateral with three right angles, one acute angle, and pairs of opposite sides parallel (in the hyperbolic sense). The sides, $OP$, $OQ$, $PR$, and $RQ$, of this quadrilateral are drawn on Poincar\'e lines that are segments of disk diameters or arcs of circles orthogonal to the outer circle (Figure 2c; $OQ$ is parallel to $PR$ and $RQ$ is parallel to $PO$). This quadrilateral is called a Lambert quadrilateral after Johann Heinrich Lambert [8], creator of the ``Lambert" azimuthal equal area map projection (among others) [12]. When such a quadrilateral is drawn in the Poincar\'e model, the acute angle at $R$ can be drawn to suggest that its sides are divergent, asymptotic, or intersecting. Here, these sides have been drawn to intersect (Figure 2c) and to evidently compress the angle at $R$ as a suggestion of the angular compression [12] present in azimuthal map projections (including those of Lambert) around the projection center. \heading 6. Tiling the Hyperbolic Plane. If one views a map grid as a tiling by quadrilaterals of a portion of the Euclidean plane, then it might be instructive to consider a tiling of the ``map" of the Poincar\'e disk model by Lambert and other quadrilaterals [5]. Gluing quadrilaterals together along Poincar\'e lines produces a variety of quadrilaterals (Figure 3). All have pairs of opposite sides parallel; Poincar\'e lines represented as arcs are orthogonal to the outer circle. Naturally, the tiling can never completely cover the disk, because the disk boundary is not included. Thus, tilings of this map have quadrilaterals of shrinking dimensions as the outer circle is approached. This permits hyperbolic ``tilings" to suggest the infinite; indeed, they have served as artistic inspiration for the ``limitless" art of M. C. Escher [7]. \midinsert \vskip11cmClick here for Figure 3.

{\bf Figure 3.} A partial tiling of the Poincar\'e Disk Model by quadrilaterals bounded by Poincar\'e lines. Quadrilateral $(OPQR)$ is a Lambert quadrilateral with two sides drawn asymptotic to each other. \endinsert \heading 7. Triangles, Quadrilaterals, and Tilings Between the Tropics. Concern with home and family are universal human values. Typical American houses exhibit Euclidean cross sections: a rectangular one from a side view and a pentagonal one, as a triangular roofline atop a square base, from a head--on view. Western Sumatran Minangkabau house-- types fit more naturally into a non--Euclidean framework than they do into the Euclidean one, exhibiting hyperbolic cross sections as a Saccheri quadrilateral (two Lambert quadrilaterals glued together along a ``straight" edge (Figure 4a) [8]) when viewed from the side, and as a concave, hyperbolic, triangle atop a (possibly Euclidean) quadrilateral when viewed from the front (Figure 4b). \topinsert \vskip18cm {\bf Figure 4.} Click here for Figure 4.

\item{a.} A Saccheri quadrilateral, formed from two Lambert quadrilaterals. It has two right angles and two acute angles. Pairs of opposite sides are parallel, as drawn in the Poincar\'e Disk Model. \item{b.} West Sumatran Minangkabau house. Roofline is suggestive of a Saccheri quadrilateral. Photograph by John D. Nystuen. \endinsert \vfill\eject Games children play often reveal deeper traditions of an entire society. As the sun moves through its entire range of possible positions, shadows dance across the full range of compass positions on Indonesian soil and come alive, as ``shadow puppets," in Indonesian theatrical productions. Elegant cut--outs traced on goat skins and other hides are mounted on sticks and dance in a plane of light between a single point--source and a screen, casting their filigreed, shadowy outlines high enough for all to see. The motions of the Indonesian puppetteer are regulated by the world of projective geometry, with shadows stretching out diffuse arms toward the infinite. A commonly played Indonesian board game is ``Sodokan," a variant of checkers [1]. Two people play until all of an opponent's ten pieces, arranged initially on the intersection points of the last two lines of a $5\times 5$ board (Figure 5a), have been captured. Pieces move across the board horizontally, vertically, or diagonally, one square at a time. What is unusual is the method of capture; to take an opponent's marker requires a ``surprise" attack along the loops outside the apparent natural grid of the gameboard. \topinsert \vskip11cm {\bf Figure 5.}Click here for Figure 5a.\item{a.} Sodokan game board in Euclidean space. Markers travel along lines separating regions of contrasting color and along circular loops at the corners. \endinsert For example, with just two pieces remaining (so that there are no intervening pieces), black may capture white (Figure 5b). To do so, black must traverse at least one loop; in the act of capture, black can slide across as many open grid intersections as required to gain entry to a loop. Then, still in the same turn, black slides around the loop, re--enters the game board, and continues to slide across grid intersections and loops until an opponent's marker is reached, and therefore captured. \midinsert \vskip11cm {\bf Figure 5.}Click here for Figure 5b.\item{b.} Sample of capture. Black captures white---a single move. \endinsert The name, ``Sodokan," means ``push out." Its name seems to apply only loosely to the $5\times 5$ Euclidean game board (Figure 5a) because the loops are not, themselves, ``pushed out" from the natural gameboard grid. If they were, the corners of the Euclidean grid would disappear. However, when the game board is drawn on a grid in the Poincar\'e disk model of the hyperbolic plane (Figure 5c), the loops appear naturally from grid intersections outside the circular boundary. A marker engaged in a capture on this non--Euclidean (hyperbolic) board traverses the entire hyperbolic plane (``universe"), passes across the infinite and is provided a natural avenue within the system for return to the universe. The loops are naturally ``pushed out" of the underlying grid, tiled partially by Lambert quadrilaterals; they might suggest paths along which gods [11], skipping across space, interrupt (sacrifice) elements within the predictable universe of the life--space in the disk. However, independent of speculation as to what such paths might mean, the fact remains that it is within the hyperbolic geometric framework, only, that this game board emerges as a part of a natural grid system. Thus, capture is no longer a mysterious event from ``outside" the system; the change in theoretical framework, from an Euclidean to an hyperbolic viewpoint, made it a logical occurence. \topinsert \vskip20cm {\bf Figure 5.} \item{c.}Click here for Figure 5c.Sodokan game board drawn on the Poincar\'e Disk Model of the hyperbolic plane. The four central quadrilaterals are Lambert quadrilaterals---the intersecting versions of quadrilateral $(OPQR)$ in Figure 3. When their sides are extended, the gameboard loops are formed naturally by these grid lines and their intersection points. \endinsert A change in the underlying symmetry introduced order. The ``meta" earth--sun system, when viewed as that which introduces a symmetric partition of the earth according to bands of sun--delivered affine and hyperbolic geometry, offered order in understanding roofline and gameboard shape where none had been apparent. Sources of evidence for other similar interpretations are plentiful: from Indonesian calendars based on a nested hierarchy of cycles, to the loops within loops creating the syncopated forms characteristic of Indonesian gamelan music. Perhaps Indonesians and other between--the--parallels dwellers have escaped the asymmetric confines of Euclidean thought, enabling them to include a comfortable vision of infinity as part of the underlying symmetry of their daily circle of life. \vfill\eject \heading 8. References. \ref 1. R. C. Bell, {\sl The Boardgame Book\/} Open Court, New York, 1983. \ref 2. William Wheeler Bunge, {\sl Theoretical Geography\/} Lund Studies in Geography, ser. C, no. 1, Lund, 1966. \ref 3. William Kingdon Clifford, The postulates of the science of space, 1873. Reprinted in {\sl The World of Mathematics\/} ed. J. R. Newman, 552-567, Simon and Schuster, New York, 1956. [Portions also reprinted in {\sl Solstice\/}, Vol. I, No. 1, Summer, 1990.] \ref 4. Richard Courant and Herbert Robbins, {\sl What Is Mathematics?\/} Oxford University Press, London, 1941. \ref 5. H. S. M. Coxeter, {\sl Introduction to Geometry\/} Wiley, New York, 1961. \ref 6. H. S. M. Coxeter, {\sl Non--Euclidean Geometry\/} University of Toronto Press, Toronto, 1965. \ref 7. Maurits C. Escher, Circle Limit IV (Heaven and Hell), woodcut, 1960. \ref 8. Marvin J. Greenberg, {\sl Euclidean and Non--Euclidean Geometries: Development and History\/} W. H. Freeman, San Francisco, 1974. \ref 9. R. K. Luneburg, {\sl Mathematical Analysis of Binocular Vision\/} Princeton University Press, Princeton, 1947. \ref 10. Saunders Mac Lane, {\sl Mathematics: Form and Function\/} Springer, New York, 1986. \ref 11. John D. Nystuen, Personal communication, 1989. \ref 12. J. A. Steers, {\sl An Introduction to the Study of Map Projections\/} London University Press, London, 1962. \ref 13. Raymond L. Wilder, {\sl Introduction to the Foundations of Mathematics\/} New York: Wiley, New York, 1961. \heading Acknowledgment The author wishes to thank John D. Nystuen for his kindness in sharing information, concerning various aspects of Indonesian culture, gathered in field work. Nystuen pointed out the connection between West Sumatran, Minangkabau house--types and Saccheri quadrilaterals, and taught the author and others to play the board game he had learned of in Indonesia. The photograph of the West Sumatran house was taken by Nystuen and appears here with his permission. She also wishes to thank Istv\'an Hargittai of the Hungarian Academy of Sciences and Arthur Loeb of Harvard University for earlier efforts with this manuscript; this paper was originally accepted by {\sl Symmetry\/}---Dr. Hargittai was Editor of that journal and Professor Loeb was the Board member of that now defunct journal who communicated this work to Hargittai. The paper appears here exactly as it was communicated to {\sl Symmetry\/}. \vfill\eject \centerline{\bf THE HEDETNIEMI MATRIX SUM: A REAL--WORLD APPLICATION} \smallskip \centerline{\sl Sandra L. Arlinghaus, William C. Arlinghaus, John D. Nystuen.} \smallskip 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, 1990(b)]. In that previous research, we applied the algorithm to the generalized road network graph surrounding San Francisco Bay. The resulting matrices are repeated here (Figure 1), in order to examine consequent changes in matrix entries when the underlying adjacency pattern of the road network was altered by the 1989 earthquake that closed the San Francisco--Oakland Bay Bridge. Thus, we test the algorithm against a changed adjacency configuration and interpret the results with the benefit of hindsight from an actual event. Figure 1 shows a graph, with edges weighted with time--distances, representing the general expressway linkage pattern joining selected cities surrounding San Francisco Bay. The matrix $A$ displays these time--distances in tabular form; an asterisk indicates that there is no direct linkage between corresponding entries. Thus, an asterisk in entry $a_{13}$ indicates that there is no single edge of the graph linking San Francisco and San Jose (all paths have 2 or more edges). Higher powers of the matrix $A$ count numbers of paths of longer length---$A^2$ counts paths of 2 edges as well as those of one edge. Thus, one expects in $A^2$ to see a number measuring time--distance between San Francisco and San Jose; indeed, there are two such paths, one of length 30+50=80, and one of length 30+25=55. The Hedetniemi matrix operator always selects the shortest. Readers wishing to understand the mechanics of this algorithm should refer to the other references related to this topic in the list at the end [Arlinghaus, Arlinghaus, and Nystuen; W. Arlinghaus]. It is sufficieint here simply to understand generally how the procedure works, as described above. When a recent earthquake caused a disastrous collapse of a span on the San Francisco--Oakland Bay Bridge, forcing the closing of the bridge, municipal authorities managed to keep the city moving using a well--balanced combination of added ferry boats, media messages urging people to stay off the roads, and dispersal of information concerning alternate route strategies. National telecasts showed a city on the move, albeit slowly, although outside forecasters of doom were predicting a massive grid--lock that never occured. What would the Hedetniemi algorithm have forecast in this situation? To find out, we compare the matrices of Figure 1 to those of Figure 2, derived from the graph of Figure 1 with the link between San Francisco and Oakland removed; that is, the edge linking vertex 4 to vertex 1 is removed --- the results show in the matrix entries $a_{14}$ and $a_{41}$. Thus in Figure 2, the adjacency matrix $A$, describing 1--step edge linkages differs from that of Figure 1 only in the $a_{14}$ ($a_{41}$) position. The value of * replaces the time--distance of 30 minutes in that graph because the bridge connection was destroyed. When 2--edge paths are counted, there is spread of increased time--distances across these paths, as well. What used to take 30 minutes, under conditions of normal traffic, to go from San Francisco to Oakland now takes 70 minutes, under conditions of normal traffic, going by way of San Mateo. The trip from San Francisco to Walnut Creek had been possible along a 2--edge path passing through Oakland (and taking a total of 60 minutes); the asterisk in $A^2$ in the $a_{15}$ entry indicates that that path no longer exists. The journey from San Francisco to Richmond, along a 2--edge path, increased in time--distance from 50 to 60 minutes---going around the ``longer" side of the rectangle. Note that what is being evaluated here is change in trip--time under ``normal" circumstances, according to whether or not routing exists; congestion fluctuates but actual road lengths do not (once in place). These values therefore form a set of benchmarks against which to measure time--distance changes resulting from more variable quantities, such as increased congestion. When three--edged paths are brought into the system, in $A^3$ (Figure 2), the trip from San Francisco to Walnut Creek now becomes possible, but takes 100 rather than 60 minutes. Also, the trip from San Francisco to Vallejo now becomes possible (in both pre-- and post--earthquake systems) although it takes 10 minutes longer with removal of the bridge. When paths of length four are introduced, no changes occur in these entries; the system is stable and the effects are confined to locations ``close'' to the bridge that was removed. The relatively small number of changes in the basic underlying route choices, forced by the removal of the Bay Bridge, suggest {\bf why} it was possible, with swift action by municipal authorities and citizens to control congestion, to avert a situation that appeared destined to lead to gridlock. What if the Golden Gate Bridge had been removed rather than the San Francisco--Oakland Bay Bridge? Figure 3 shows that the same sort of clustered, localized results follow. When both bridges are removed (Figure 4), the position of affected matrix entries is identical to the union of the positions of entries in Figures 1 and 2, but the magnitude of time--distances has been magnified by the combined removal. With hindsight, the test seems to be reasonable. One direction for a larger application might therefore be to consider historical evidence in which bridge bombing (or some such) was critical to associated circulation patterns. When large data sets are entered into a computer, and manipulated using the Hedetniemi matrix algorithm, previously unnoticed historical associations might emerge and maps showing alternate possibilities could be produced. In short, this might serve as a tool useful in historical discovery. Other important directions for application of the Hedetniemi algorithm involve those in a discrete mathematical setting that focus on tracing actual paths [W. Arlinghaus, 1990---includes program for algorithm], and those using the Hedetniemi algorithm in the computer architecture of parallel processing [Romeijn and Smith]. \vfill\ejectClick here for Figure 1, graph.Click here for Figure 1, matrix.

TYPESETTING THAT PRODUCED FIGURE 1.

\centerline{SAN FRANCISCO BAY AREA; GRAPH OF TIME--DISTANCES} \centerline{(in minutes)} \centerline{LEGEND: numeral attached to city is its node number in} \centerline{the corresponding, underlying, graph.} \line{1. SAN FRANCISCO \hfil} \line{2. SAN MATEO COUNTY \hfil} \line{3. SAN JOSE \hfil} \line{4. OAKLAND \hfil} \line{5. WALNUT CREEK \hfil} \line{6. RICHMOND \hfil} \line{7. VALLEJO \hfil} \line{8. NOVATO \hfil} \line{9. SAN RAFAEL (MARIN COUNTY) \hfil} $$ A = \pmatrix{ 0& 30& *& 30& *& *& *& *&40 \cr 30& 0&25& 40& *& *& *& *& * \cr *& 25& 0& 50& *& *& *& *& * \cr 30& 40&50& 0&30&20& *& *& * \cr *& *& *& 30& 0& *&25& *& * \cr *& *& *& 20& *& 0&20& *&20 \cr *& *& *& *&25&20& 0&25& * \cr *& *& *& *& *& *&25& 0&20 \cr 40& *& *& *& *&20& *&20& 0 \cr} $$ $$ A^2 = \pmatrix{ 0& 30&55& 30&60&50& *&60&40\cr 30& 0&25& 40&70&60& *& *&70\cr 55& 25& 0& 50&80&70& *& *& *\cr 30& 40&50& 0&30&20&40& *&40\cr 60& 70&80& 30& 0&45&25&50& *\cr 50& 60&70& 20&45& 0&20&40&20\cr *& *& *& 40&25&20& 0&25&40\cr 60& *& *& *&50&40&25& 0&20\cr 40& 70& *& 40& *&20&40&20& 0\cr} $$ $$ A^3 = \pmatrix{ 0& 30&55& 30&60&50&70&60&40\cr 30& 0&25& 40&70&60&80&90&70\cr 55& 25& 0& 50&80&70&90& *&90\cr 30& 40&50& 0&30&20&40&60&40\cr 60& 70&80& 30& 0&45&25&50&65\cr 50& 60&70& 20&45& 0&20&40&20\cr 70& 80&90& 40&25&20& 0&25&40\cr 60& 90& *& 60&50&40&25& 0&20\cr 40& 70&90& 40&65&20&40&20& 0\cr} $$ $$ A^4 = \pmatrix{ 0& 30&55& 30&60&50&70&60&40\cr 30& 0&25& 40&70&60&80&90&70\cr 55& 25& 0& 50&80&70&90&110&90\cr 30& 40&50& 0&30&20&40&60&40\cr 60& 70&80& 30& 0&45&25&50&65\cr 50& 60&70& 20&45& 0&20&40&20\cr 70& 80&90& 40&25&20& 0&25&40\cr 60& 90&110& 60&50&40&25& 0&20\cr 40& 70&90& 40&65&20&40&20& 0\cr} $$ $$ A^5 = \pmatrix{ 0& 30&55& 30&60&50&70&60&40\cr 30& 0&25& 40&70&60&80&90&70\cr 55& 25& 0& 50&80&70&90&110&90\cr 30& 40&50& 0&30&20&40&60&40\cr 60& 70&80& 30& 0&45&25&50&65\cr 50& 60&70& 20&45& 0&20&40&20\cr 70& 80&90& 40&25&20& 0&25&40\cr 60& 90&110& 60&50&40&25& 0&20\cr 40& 70&90& 40&65&20&40&20& 0\cr} $$ $$ A^4 = A^5 = \ldots = A^9 $$ {\bf Figure 1}. Pre--earthquake matrix sequence. \vfill\eject

Click here for Figure 2, graph.

Click here for Figure 2, matrix.

TYPESETTING THAT PRODUCED FIGURE 2 \centerline{POST--EARTHQUAKE SAN FRANCISCO BAY AREA} \centerline{SAN FRANCISCO--OAKLAND BAY BRIDGE IS REMOVED.} \centerline{GRAPH OF TIME--DISTANCES (in minutes)} \centerline{Adjustment is made for change in time--distance} \centerline{in a ``normal" situation--not for resultant fluctuation in congestion} $$ A = \pmatrix{ 0& 30& *& *& *& *& *& *&40\cr 30& 0&25& 40& *& *& *& *& *\cr *& 25& 0& 50& *& *& *& *& *\cr *& 40&50& 0&30&20& *& *& *\cr *& *& *& 30& 0& *&25& *& *\cr *& *& *& 20& *& 0&20& *&20\cr *& *& *& *&25&20& 0&25& *\cr *& *& *& *& *& *&25& 0&20\cr 40& *& *& *& *&20& *&20& 0\cr} $$ $$ A^2 = \pmatrix{0& 30&55& 70& *&60& *&60&40\cr 30& 0&25& 40&70&60& *& *&70\cr 55& 25& 0& 50&80&70& *& *& *\cr 70& 40&50& 0&30&20&40& *&40\cr *& 70&80& 30& 0&45&25&50& *\cr 60& 60&70& 20&45& 0&20&40&20\cr *& *& *& 40&25&20& 0&25&40\cr 60& *& *& *&50&40&25& 0&20\cr 40& 70& *& 40& *&20&40&20& 0\cr} $$ $$ A^3 =\pmatrix{0& 30&55& 70&100&60&80&60&40\cr 30& 0&25& 40&70&60&80&90&70\cr 55& 25& 0& 50&80&70&90& *&90\cr 70& 40&50& 0&30&20&40&60&40\cr 100& 70&80& 30& 0&45&25&50&65\cr 60& 60&70& 20&45& 0&20&40&20\cr 80& 80&90& 40&25&20& 0&25&40\cr 60& 90& *& 60&50&40&25& 0&20\cr 40& 70&90& 40&65&20&40&20& 0\cr} $$ $$ A^4 =\pmatrix{0& 30&55& 70&100&60&80&60&40\cr 30& 0&25& 40&70&60&80&90&70\cr 55& 25& 0& 50&80&70&90&110&90\cr 70& 40&50& 0&30&20&40&60&40\cr 100& 70&80& 30& 0&45&25&50&65\cr 60& 60&70& 20&45& 0&20&40&20\cr 80& 80&90& 40&25&20& 0&25&40\cr 60& 90&110& 60&50&40&25& 0&20\cr 40& 70&90& 40&65&20&40&20& 0\cr} $$ $$ A^5 =\pmatrix{0& 30&55& 70&100&60&80&60&40\cr 30& 0&25& 40&70&60&80&90&70\cr 55& 25& 0& 50&80&70&90&110&90\cr 70& 40&50& 0&30&20&40&60&40\cr 100& 70&80& 30& 0&45&25&50&65\cr 60& 60&70& 20&45& 0&20&40&20\cr 80& 80&90& 40&25&20& 0&25&40\cr 60& 90&110& 60&50&40&25& 0&20\cr 40& 70&90& 40&65&20&40&20& 0\cr} $$ $$ A^4 = A^5 = \ldots = A^9 $$ {\bf Figure 2}. Matrix sequence with San Francisco--Oakland Bay Bridge removed. \vfill\eject

Click here for Figure 3, graph.

Click here for Figure 3, matrix.

TYPESETTING THAT PRODUCED FIGURE 3 \centerline{POST--EARTHQUAKE SAN FRANCISCO BAY AREA} \centerline{GOLDEN GATE BRIDGE IS REMOVED.} \centerline{GRAPH OF TIME--DISTANCES (in minutes)} \centerline{Adjustment is made for change in time--distance} \centerline{in a ``normal" situation---not for resultant fluctuation in congestion} $$ A = \pmatrix{0& 30& *& 30& *& *& *& *& *\cr 30& 0&25& 40& *& *& *& *& *\cr *& 25& 0& 50& *& *& *& *& *\cr 30& 40&50& 0&30&20& *& *& *\cr *& *& *& 30& 0& *&25& *& * \cr *& *& *& 20& *& 0&20& *&20\cr *& *& *& *&25&20& 0&25& *\cr *& *& *& *& *& *&25& 0&20\cr *& *& *& *& *&20& *&20& 0\cr} $$ $$ A^2 = \pmatrix{0& 30&55& 30&60&50& *& *& *\cr 30& 0&25& 40&70&60& *& *& *\cr 55& 25& 0& 50&80&70& *& *& *\cr 30& 40&50& 0&30&20&40& *&40\cr 60& 70&80& 30& 0&45&25&50& * \cr 50& 60&70& 20&45& 0&20&40&20\cr *& *& *& 40&25&20& 0&25&40\cr *& *& *& *&50&40&25& 0&20\cr *& *& *& 40& *&20&40&20& 0\cr} $$ $$ A^3 = \pmatrix{0& 30&55& 30&60&50&70& *&70\cr 30& 0&25& 40&70&60&80& *&80\cr 55& 25& 0& 50&80&70&90& *&90\cr 30& 40&50& 0&30&20&40&60&40\cr 60& 70&80& 30& 0&45&25&50&65 \cr 50& 60&70& 20&45& 0&20&40&20\cr 70& 80&90& 40&25&20& 0&25&40\cr *& *& *& 60&50&40&25& 0&20\cr 70& 80&90& 40&65&20&40&20& 0\cr} $$ $$ A^4 = \pmatrix{0& 30&55& 30&60&50&70&90&70\cr 30& 0&25& 40&70&60&80&100&80\cr 55& 25& 0& 50&80&70&90&110&90\cr 30& 40&50& 0&30&20&40&60&40\cr 60& 70&80& 30& 0&45&25&50&65 \cr 50& 60&70& 20&45& 0&20&40&20\cr 70& 80&90& 40&25&20& 0&25&40\cr 90&100&110& 60&50&40&25& 0&20\cr 70& 80&90& 40&65&20&40&20& 0\cr} $$ $$ A^5 = \pmatrix{0& 30&55& 30&60&50&70&90&70\cr 30& 0&25& 40&70&60&80&100&80\cr 55& 25& 0& 50&80&70&90&110&90\cr 30& 40&50& 0&30&20&40&60&40\cr 60& 70&80& 30& 0&45&25&50&65 \cr 50& 60&70& 20&45& 0&20&40&20\cr 70& 80&90& 40&25&20& 0&25&40\cr 90&100&110& 60&50&40&25& 0&20\cr 70& 80&90& 40&65&20&40&20& 0\cr} $$ $$ A^4 = A^5 = \ldots = A^9 $$ {\bf Figure 3}. Matrix sequence with the Golden Gate Bridge removed. \vfill\eject

Click here for Figure 4, graph.

Click here for Figure 4, matrix.

TYPESETTING THAT PRODUCED FIGURE 4. \centerline{POST--EARTHQUAKE SAN FRANCISCO BAY AREA} \centerline{BAY BRIDGE AND GOLDEN GATE BRIDGE ARE BOTH REMOVED.} \centerline{GRAPH OF TIME--DISTANCES (in minutes)} \centerline{Adjustment is made for change in time--distance} \centerline{in a ``normal" situation---not for resultant fluctuation in congestion} $$ A = \pmatrix{0& 30& *& *& *& *& *& *& *\cr 30& 0&25& 40& *& *& *& *& *\cr *& 25& 0& 50& *& *& *& *& *\cr *& 40&50& 0&30&20& *& *& *\cr *& *& *& 30& 0& *&25& *& * \cr *& *& *& 20& *& 0&20& *&20\cr *& *& *& *&25&20& 0&25& *\cr *& *& *& *& *& *&25& 0&20\cr *& *& *& *& *&20& *&20& 0\cr} $$ $$ A^2 = \pmatrix{0& 30&55& 70& *& *& *& *& *\cr 30& 0&25& 40&70&60& *& *& *\cr 55& 25& 0& 50&80&70& *& *& *\cr 70& 40&50& 0&30&20&40& *&40\cr *& 70&80& 30& 0&45&25&50& * \cr *& 60&70& 20&45& 0&20&40&20\cr *& *& *& 40&25&20& 0&25&40\cr *& *& *& *&50&40&25& 0&20\cr *& *& *& 40& *&20&40&20& 0\cr} $$ $$ A^3 = \pmatrix{0& 30&55& 70&100&90& *& *& *\cr 30& 0&25& 40&70&60&80& *&80\cr 55& 25& 0& 50&80&70&90& *&90\cr 70& 40&50& 0&30&20&40&60&40\cr 100& 70&80& 30& 0&45&25&50&65 \cr 90& 60&70& 20&45& 0&20&40&20\cr *& 80&90& 40&25&20& 0&25&40\cr *& 80& *& 60&50&40&25& 0&20\cr *& *&90& 40&65&20&40&20& 0\cr} $$ $$ A^4 = \pmatrix{0& 30&55& 70&100&90&110&*&110\cr 30& 0&25& 40&70&60&80&100&80\cr 55& 25& 0& 50&80&70&90&110&90\cr 70& 40&50& 0&30&20&40&60&40\cr 100& 70&80& 30& 0&45&25&50&65 \cr 90& 60&70& 20&45& 0&20&40&20\cr 110& 80&90& 40&25&20& 0&25&40\cr *&100&110& 60&50&40&25& 0&20\cr 110& 80&90& 40&65&20&40&20& 0\cr} $$ $$ A^5 = \pmatrix{0& 30&55& 70&100&90&110&130&110\cr 30& 0&25& 40&70&60&80&100&80\cr 55& 25& 0& 50&80&70&90&110&90\cr 70& 40&50& 0&30&20&40&60&40\cr 100& 70&80& 30& 0&45&25&50&65 \cr 90& 60&70& 20&45& 0&20&40&20\cr 110& 80&90& 40&25&20& 0&25&40\cr 130&100&110&60&50&40&25& 0&20\cr 110& 80&90& 40&65&20&40&20& 0\cr} $$ $$ A^4 = A^5 = \ldots = A^9 $$ {\bf Figure 4}. Matrix sequence with both the Golden Gate and the Bay bridges removed. \vfill\eject

\heading References. \ref Arlinghaus, S. L.; W. C. Arlinghaus; J. D. Nystuen. 1990. Poster---{\sl Elements of Geometric Routing Theory--II\/}. Association of American Geographers, National Meetings, Toronto, Ontario, April. \ref Arlinghaus, S. L.; W. C. Arlinghaus; J. D. Nystuen. 1990. ``The Hedetniemi Matrix Sum: An Algorithm for Shortest Path and Shortest Distance." {\sl Geographical Analysis\/}. 22: 351-360. \ref Arlinghaus, W. C. ``Shortest Path Problems," invited chapter in {\sl Applications of Discrete Mathematics\/}, edited by Kenneth H. Rosen and John Michaels. March 11, 1990. In press, McGraw--Hill. \ref Romeijn, H. E. and R. L. Smith. ``Notes on Parallel Algorithms and Aggregation for Solving Shortest Path Problems." Unpublished, October, 1990. \vfill\eject \centerline{\bf FRACTAL GEOMETRY OF INFINITE PIXEL SEQUENCES:} \centerline{\bf ``SUPER--DEFINITION" RESOLUTION?} \centerline{\sl Sandra Lach Arlinghaus} \heading Introduction The fractal approach to the geometry of central place theory is particularly powerful because, among other things, it provides numerical proof that the subjective labels of ``marketing,'' ``transportation,'' and ``administration'' for the $K=3$, $K=4$, and $K=7$ hierarchies are indeed correct [Arlinghaus, 1985] and because it enables solution of all open geometric questions identified by Dacey, Marshall, and others in earlier research [Dacey; Marshall; Arlinghaus and Arlinghaus]. When the problem is wrapped back on itself and the nature of the original, underlying environment is altered---from urban to electronic---the same results, recast in a different light, suggest the degree of improvement in picture resolution that can come from decreasing pixel size. Curves on cathode ray tubes are formed from a sequence of pixels hooked together at their corners; font designers in word processors offer an easy opportunity to observe these pixel formations (Horstmann, 1986). The pixel sequence merely suggests the curve; it does not actually produce a ``correct" curve. Reducing the size of the pixel can improve the resolution of the image representing the curve. The material below uses established results from fractal geometry to evaluate the degree of success, in improving resolution in a raster environment, that results from decreasing pixel size. \heading Manhattan pixel arrangement When a square pixel is the fundamental unit, a sequence of pixels has boundaries separating pixels in Manhattan, ``city-- block" space. When smaller square pixels are introduced, more lines separating pixels are also introduced. The interior of the pixel is what carries the content---not the boundary of the pixel. Thus, it is significant to know what proportion of the space filled with pixels is filled with pixel boundary. Suppose that, in an effort to produce ``high-- definition" resolution, the number of square pixels used to cover a fixed area (a cathode ray tube) is substantially increased. One might be tempted to use even more pixels to produce even better resolution and even more beyond that. If the process is carried out infinitely, using a Manhattan grid, the pixel mesh has arbitrarily small cell size and the entire plane region is ``filled" with pixel boundary, only; the scale transformation of superimposing finer and finer square mesh on a fixed area has dimension $D=2$ (Mandelbrot, p. 63, 1983). In this situation, all pixel content is therefore lost. Clearly then, improvement in resolution does not continue, ad infinitum; there is some point at which the tradeoff between fineness in resolution and loss of information content is at its peak. Determining this point is an issue of difficulty and significance. Is this dilemma a universal situation that exists independent of the shape of the fundamental pixel unit? \heading Hexagonal pixel arrangement Consider instead an electronic environment in which the fundamental picture element is hexagonal in shape (Rosenfeld; Gibson and Lucas). Such a geometric environment has a number of well--documented advantages, centering on close--packing characteristics (Gibson and Lucas). This environment is examined here along the lines suggested above---to see if improvement in resolution can be carried out infinitely through pixel subdivision. When a bounded lattice of regular hexagons of uniform cell diameter (on a CRT) is refined as a similar lattice of smaller uniform cell diameter, improvement in resolution results. There are an infinite number of ways in which the lattice of smaller cell--size might be superimposed on the lattice of larger cell size. The geometry of central place theory describes these relative positions of layers. Independent of the orientation selected, when this transformation from larger to smaller cell lattice is iterated infinitely, the bounded space is once again filled (as in the rectangular pixel case) with hexagonal pixel boundary. Thus, in both the case of the rectangular pixel and the hexagonal pixel environments, infinite ``improvement" in resolution, brought about by decreasing pixel size, causes a black--hole--like collapse of the original, entire image. However, is this characteristic of the whole necessarily inherited by each of its parts? Any part that does not inherit this collapsing, space--filling characteristic is capable of infinite, ``super--definition'' resolution. Such a part is invariant (to some extent) under scale transformation. The fractal approach to central place theory shows that there do exist shapes in the hexagonal pixel environment which, when refined infinitely, do not fill a bounded piece of two dimensional space. Figure 1 shows a hexagon to which a fractal generator has been applied to produce a $K=4$ hierarchy. Infinite iteration of this self--similarity transformation produces a highly crenulated replacement which {\bf does not} fill a bounded two--dimensional space; in fact, it fills only 1.585 of a two--dimensional space. When the corrresponding self--similarity transformation is applied to a square pixel a highly crenulated shape is again the result of infinite iteration; this shape {\bf does} fill a bounded two-- dimensional space (Figure 2). The two fractal generators selected are parallel in structure: each is half of the boundary of the fundamental pixel shape. \topinsert\vskip19cm {\bf Figure 1.} K=4 hierarchy of hexagonal pixels generated fractally. \endinsert

Click here for Figure 1.\vfill\eject \topinsert\vskip8cm {\bf Figure 2.} K=4 type of hierarchy generated fractally from square initiators.

Click here for Figure 2.\endinsert If both geometric environments are then viewed as composed of these highly--crenulated elements (which do fit together to cover the plane), then the hexagonal environment is the one that permits infinite iteration without loss of all pixel content. This approach is akin to that of Barnsley, which stores sets of transformations that are used to drive image production. What is suggested here is a possible way to vastly improve image resolution corresponding, to some extent, to Barnsley's successful strategy to improve data compression (Barnsley). This approach is also similar, in general strategy to that employed by Hall and G\"okmen; both seek transformations, applied in an electronic environment, under which some properties are preserved. Hall and G\"okmen focus on transformations linking hexagonal and rectangular pixel space whereas the transformations employed here function entirely within a single type of geometric environment (using one on the other appears to be of interest). Additionally, this approach offers a systematic characterization, in the infinite, for the aggregate 7--kernels of hexagons, at various levels of aggregation, suggested only as finite sequences in Gibson and Lucas. Finally, Tobler's maps of Swiss migration patterns at three levels of spatial resolution suggest a methodological handle of an attractivity function to implement ideas involving spatial resolution in an electronic environment. Deeper analysis, of the sort represented in the works mentioned here, is beyond the scope of this particular short piece. Table 1 shows a set of fractal dimensions for selected L\"oschian numbers. \midinsert

Click here for Table 1.

TYPESETTING THAT PRODUCED TABLE 1.

\smallskip \hrule \smallskip \centerline{ \bf Table 1} \centerline{(derived from a Table in Arlinghaus and Arlinghaus, 1989)} \settabs\+&$K=3,\,D=1.262$;\quad&$K=12,\,D=1.116$;\quad&$K=27,\,D=1.087$;\quad &$K=49,\,D=1.074$&$\ldots$&\cr \+&K=3, D=1.262;&K=12, D=1.116;&K=27, D=1.087;&K=48, D=1.074;&$\ldots$\cr \+&K=7, D=1.129;&K=19, D=1.093;&K=37, D=1.078;&K=61, D=1.069;&$\ldots$\cr \+&K=4, D=1.585;&K=13, D=1.255;&K=28, D=1.168;&K=49, D=1.129;&$\ldots$\cr \smallskip \hrule \smallskip \endinsert

The line of L\"oschian numbers that begins with $K=4$, those that are organized according to an ``transportation" principle, are the ones that fill two dimensional space most thickly. Thus, when introducing smaller and smaller hexagonal cells to improve resolution in the quality of curve representation, or when ``zooming in," it would appear appropriate to let the orientation of successive layers of smaller and smaller cells correspond to the $K=4$ type of hierarchy. Clutter would not enter as fast as in the Manhattan environment, even in this densest arrangement. ``Super," rather than ``high," definition of resolution could therefore fall naturally from an underlying hexagonal pixel geometry with measures of clutter and information content determined using fractal dimensions. \heading Shortest paths At an even broader scale, one might also look for this sort of application in hooking computers together as parallel processing units. When ``central places" are thought of as central processing units, not of urban information, but rather of electronic information, then an underlying geometry for finding ``shortest'' paths through networks linking multiple points might emerge. For in an electronic environment with the hexagonal pixel as the fundamental unit, the $120^{\circ}$ intersection points would correspond exactly to the requirements for finding Steiner networks, as ``shortest" networks linking multiple locations. Steiner points in an electronic configuration might then correspond to locations at which to ``jump'' from one hexagonal lattice of fixed cell--size to another of different cell size (from one machine to another), where cell size is prescribed by ``lengths'' (in whatever metric) between ``transmission times'' between adjacent Steiner points. \heading References \ref Arlinghaus, S. (1985). Fractals take a central place. {\sl Geografiska Annaler\/}, 67B, 2, 83-88. \ref Arlinghaus, S. and Arlinghaus, W. (1989). The fractal theory of central place geometry: A Diophantine analysis of fractal generators for arbitrary L\"oschian numbers. {\sl Geographical Analysis\/} 21, 2, 103-121. \ref Barnsley, M. F. {\sl Fractals Everywhere\/}. San Diego: Academic Press, 1988. \ref Dacey, M. F. The geometry of central place theory. {\sl Geografiska Annaler\/}. 47: 111-124. \ref Gibson, L. and Lucas D., Vectorization of raster images using hierarchical methods. Paper: Interactive Systems Corporation, 5500 South Sycamore Street, Littleton, Colorado, 80120. \ref Hall, R. W. and M. G\"okmen. Rectangular/hexagonal tesselation transforms and parallel shrinking. Paper: Department of Electrical Engineering, University of Pittsburgh, Pittsburgh, PA 15261, TR-SP-90-004, June, 1990. Presented: Summer Conference on General Topology and Applications. Long Island University, 1990. \ref Horstmann, C. (1986). {\sl ChiWriter: the scientific/multifont word processor for the IBM-P.C. (and compatibles)\/}. Ann Arbor: Horstmann Software Design. \ref Mandelbrot, B. (1983). {\sl The Fractal Geometry of Nature\/}. San Francisco: W. H. Freeman. \ref Marshall, J. U. 1975. The L\"oschian numbers as a problem in number theory. {\sl Geographical Analysis\/}. 7: 421-426. \ref Rosenfeld, A. (1990). Session on Digital Topology, National meetings of the American Mathematical Society, Louiville, KY, January, 1990. \ref Tobler, W. R. Frame independent spatial analysis, in Goodchild, M. F. and Gopal, {\sl The Accuracy of Spatial Databases\/}. London: Taylor and Francis, 1990. \smallskip \smallskip $^*$ The author wishes to thank Michael Goodchild for constructive comments on a 1989 version of this paper. Much of this content content has been presented previously: before national meetings of the American Mathematical Society in August of 1990; before national meetings of the Association of American Geographers in April of 1990; and, before a classroom audience at The University of Michigan in the Winter Semester of 1989/90. \vfill\eject \centerline{\bf CONSTRUCTION ZONE} \smallskip \centerline{FIRST CONSTRUCTION;} \centerline{readers might wish to construct figures to accompany} \centerline{the electronic text as they read} \smallskip \centerline{\bf Feigenbaum's number: exposition of one case} \centerline{Motivated by queries from Michael Woldenberg,} \centerline{Department of Geography, SUNY Buffalo,} \centerline{during his visit to Ann Arbor, Summer, 1990.} Here is a description of how Feigenbaum's number arises from a graphical analysis of a simple geometric system [1]. Feigenbaum's original paper is clear and straightforward [1]; this construction is presented to serve as exposure prior to reading Feigenbaum's longer paper [1]. The construction is complicated although individual steps are not generally difficult. Following the construction, a suggestion will be offered as to how to select mathematical constraints within which to choose geographical systems for Feigenbaum--type analysis. \item{1.} Consider the family of parabolas $y=x^2 + c$, where $c$ is an integral constant. This is just the set of parabolas that are like $y=x^2$, slid up or down the $y$-axis. The smaller the value of $c$, the more the parabola opens up (otherwise a lower one would intersect a higher one, creating an algebraic impossibility such as $-1=0$) (Figure 1). \smallskip \item{2.} To begin, consider the particular parabola, $y=x^2 - 1$, obtained by setting $c = -1$. Graph this (Figure 2). Also draw the line $y=x$ on this graph. Now we're going to look at the ``orbit" of the value $x=1/2$ with respect to this parabola (function). By ``orbit" is meant simply the iteration string obtained by using $x=1/2$ as input into $y=x^2 -1$, then using that output as a new input into $y=x^2-1$, then using that output as a new input $\ldots $ and so forth. In this case, the orbit of $x=1/2$ is represented as follows, numerically. (Use $.5 \mapsto -0.75$ to mean that the input of $.5$ is mapped to the output value of $-0.75$ by the function $y=x^2- 1$.) $$ 0.5 \mapsto -0.75 \mapsto -0.4375 \mapsto -0.8085938 $$ $$ \mapsto -0.3461761 \mapsto -0.8801621 \mapsto -0.2253147 $$ $$ \mapsto -0.9492333 \mapsto -0.0989562 \mapsto -0.9902077 $$ $$ \mapsto -0.019488 \mapsto -0.9996202 \mapsto -0.0007595 $$ $$ \mapsto -0.9999994 \mapsto -0.0000012 \mapsto -1 $$ $$ \mapsto 0 \mapsto -1 \mapsto 0 \mapsto \ldots $$ Clearly the values bounce around for awhile, and then eventually settle down to the values, $-1$ and $0$. \smallskip \item{3.} Let's see what this particular iteration string means geometrically (Figure 3). Locate $x=0.5$ on the $x$--axis. Drop down to the parabola to read off the corresponding $y$--value (in the usual manner) $-0.75$. Now it is this $y$--value that is to be used as the next input in the iteration string. We could go back up to the $x$--axis and find it and drop back to the parabola, but we won't. Instead execute the following, equivalent transformation---THIS IS THE KEY POINT. Assume your penpoint is on the $y$--value $-0.75$; now slide horizontally over to the line $y=x$---you want to use the $y$--value in the role of the $x$--value. Thus, treat this point as the new input and drop to the parabola from it as you did in moving from the $x$--axis to the parabola. Then, with your penpoint on the parabola, slide horizontally back to the line $y=x$ and use this as the input; drop to the parabola and keep going. A glance at Figure 2 suggests why economists call this a ``cobweb" diagram (presumably looking at fluctuating supply and demand). Follow this diagram long enough, and you will see that eventually values for $x$ fluctuate between $0$ and $-1$, around a stationary square cycle. Looking at the ``dynamics" of a value, with respect to a function, in this geometrical manner is referred to as (Feigenbaum's) ``graphical analysis" [1]. \topinsert\vskip19cm {\bf Figure 1.} Parabolas of the form $y=x^2+c$.

Click here for Figure 1.{\bf Figure 2.} The parabola $y=x^2-1$ and $y=x$.

Click here for Figure 2.{\bf Figure 3.} Graphical analysis of $y=x^2-1$.

Click here for Figure 3.\endinsert \vfill\eject \item{4.} So, we have the numerical orbit and the graphical analysis for the value $x=0.5$ with respect to the function $y=x^2 - 1$. What about calculating these values for starting values of $x$ other than $x=0.5$. Consider $x=1.6$. Its orbit is as below, and the corresponding graphical analysis is given in Figure 4. $$ 1.6 \mapsto 1.56 \mapsto 1.4336 \mapsto 1.055209 $$ $$ \mapsto 0.1134659 \mapsto -0.9871255 \mapsto -0.0255833 $$ $$ \mapsto -0.9993455 \mapsto -0.0013086 \mapsto -0.9999983 $$ $$ \mapsto -0.0000034 \mapsto -1 \mapsto 0 \mapsto -1 \mapsto 0 \mapsto \ldots $$ The dynamics of $x=1.6$ are really very much the same as for $x=0.5$ with respect to the given function. Let's look at $x=1.7$. $$ 1.7 \mapsto 1.89 \mapsto 2.5721 \mapsto 5.6156984 $$ $$ \mapsto 30.536069 \mapsto 931.45149 \mapsto 867600.87 \mapsto \ldots to \infty . $$ Graphical analysis shows this clearly, geometrically, too (Figure 5). This shooting off to infinity is not ``interesting" in the way that the cobweb dynamics are. So, for what values of $x$ do you get ``interesting" dynamics? \topinsert\vskip19cm {\bf Figure 4.} Orbit of $x=1.6$.

{\bf Figure 5.} Orbit of $x=1.7$. Click here for Figure 4.

Click here for Figure 5.\endinsert \vfill\eject \item{5.} No doubt you will have noted from the graphical analyses in Figures 4 and 5 that the reason one iteration closes down into a cobweb and the other goes to infinity is that one initial value of $x$ lies to the left of the intersection point of the parabola and the line $y=x$, and the other lies to the right of that intersection point. You might therefore be tempted to guess that all initial values of $x$ that lie between the right hand intersection point (call it $p^+$) of the parabola and the line and the left hand intersection point (call it $p^-$) of the parabola and the line $y=x$, produce interesting dynamics. (The $x$--coordinates for $p^+$ and $p^-$ are found by solving $y=x$ and $y=x^2-1$ simultaneously---that is by solving $x^2-x- 1=0$---the quadratic formula yields $x =(1 \pm \sqrt 5)/2$, or $x = 1.618034$, $x= -0.618034$). Indeed, if you try a number of values intermediate between these you will find that to be the case. However, consider a value of $x$ to the left of $x=-0.62$. Try $x=-1.6$. $$ -1.6 \mapsto 1.56 \mapsto 1.4336 \mapsto 1.055209 $$ $$ \mapsto 0.1134659 \mapsto -0.9871255 \mapsto -0.0255833 $$ $$ \mapsto -0.9993455 \mapsto -0.0013086 \mapsto -0.9999983 $$ $$ \mapsto -0.000003 \mapsto -1 \mapsto 0 \mapsto -1 \mapsto 0 \mapsto \ldots $$ There is obvious bilateral (about the $y$--axis) symmetry in the iteration string, produced by squaring inputs. Clearly, the initial value of $-1.7$ will go to positive infinity, as above. So, the interval of values of $x$ that will produce interesting dynamics is NOT $[p^-, p^+]$, but rather $[-p^+, p^+]$. You might want to draw graphical analyses for $x=-1.6$ and $x=-1.7$ with respect to this function. Call the interval, $[-p^+, p^+]$ the ``critical" interval for any given system of parabola and $y=x$. In the case of the system $y=x$ and $y=x^2-1$ the critical interval has length $3.236068$. So, now we know something general about the dynamics of input values with respect to the function $y=x^2 - 1$. Recall that we got this function by picking one value, $c=-1$, from the family of parabolas $y=x^2 + c$. Let's see what happens for different values of $c$. \smallskip \item{6.} Consider $c=0.25$. For this value of $c$, the line $y=x$ and the parabola $y=x^2+0.25$ are tangent to each other. Values of $x$ to the left of the point of tangency (at ($0.5$, $0.25$)) have orbits that converge to $0.5$ (Figure 6) while values of $x$ to the right of the point of tangency have orbits that go to positive infinity. Initial inputs to the left of the point of tangency have orbits that are ``attracted" to the point of tangency, while initial inputs to the right of the point of tangency have orbits that are ``repelled" from the point of tangency. Here, you might view it that $p^+ = p^-$. When $c>0.25$, the line $y=x$ and the corresponding parabola do not intersect, and so all orbits go to infinity---the dynamics are not interesting (Figure 7). So, we should be looking at parabolas with $c$ less than or equal to $0.25$. Let's look at some, in regard to the notions of ``attracting" and ``repelling." \topinsert\vskip19cm {\bf Figure 6.} The case for $c=1/4$.

Figure 6.{\bf Figure 7.} The case for $c>1/4$.

Figure 7.\endinsert \vfill\eject \item{7.} Consider $c=0.24$---system: $y=x$, $y=x^2+0.24$ (Figure 8). Use graphical analysis to study the dynamics (Figure 8). An orbit of $0.5$ is $$ 0.5 \mapsto .3025 \mapsto .3315063 \mapsto .3498964 $$ $$ \mapsto .362427 \mapsto .3713537 \mapsto .3779036 $$ $$ \mapsto .3828111 \mapsto .3865443 \mapsto .3894165 $$ $$ \mapsto .3916452 \mapsto .393386 \mapsto .3947525 \mapsto \ldots. \mapsto 0.4. $$ The orbit converges to the $x$--value of $p^-$ which is found as $0.4$ by solving the system using the quadratic formula. Here, $p^-$ is an attracting fixed point of the system, and $p^+$ is a repelling fixed point of the system. There is convergence of orbits to a single value within the zone [$-p^+$, $p^+$]. Notice a kind of doubling effect as one moves from the system with $c=0.25$ to the one with $c=0.26$ (period--doubling). \smallskip \item{8.} Consider $c=-0.74$. The system is: $y=x$, $y=x^2- 0.74$. Graphical analysis (Figure 9) shows that this system behaves similarly to the one for $c=0.24$; $p^-$ is attracting and $p^+$ is repelling for all $x$ in [$-p^+$, $p^+$]. The values of $p^-$ and $p^+$ are respectively $-0.4949874$ and $1.4949874$. Look at the orbit of $0.5$, for example. $$ 0.5 \mapsto -0.49 \mapsto -0.4999 \mapsto -0.4901 $$ $$ \mapsto -0.499802 \mapsto -0.490198 \mapsto \ldots \mapsto -0.4949874 $$ \topinsert\vskip19cm {\bf Figure 8.} The case for $c=0.24$. {\bf Figure 9.} The case for $c=-0.74$. \endinsert \vfill\eject \item{9.} Consider $c=-0.75$. The system is: $y=x$, $y=x^2- 0.75$. This is not at all the same sort of system as those in 7 and 8 above. Here, $p^-$ and $p^+$ are respectively $-0.5$ and $1.5$. Consider the orbit of $0.5$. $$ 0.5 \mapsto -0.5 \mapsto -0.5 \mapsto -0.5 \mapsto \ldots $$ Consider the orbit of $0.1$: $$ 0.1 \mapsto -0.74 \mapsto -0.2024 \mapsto -0.7090342 $$ $$ \mapsto -0.2472704 \mapsto -.6888573 \mapsto -.2754756 $$ $$ \mapsto -.6741132 \mapsto -.2955714 \mapsto -.6626376 \mapsto -.3109115 \mapsto \ldots $$ here, one might see this closing in, from above and below, very slowly on $-0.5$. Or, there might be two points the orbit is fluctuating toward getting close to. Consider the orbit of $1.4$: $$ 1.4 \mapsto 1.21 \mapsto .7141 \mapsto -.2400612 \mapsto -.6923706 \mapsto \ldots $$ Again, the same sort of thing as above. The behavior of this system is suggestive of that of the tangent case when $c=0.25$. \smallskip \item{10.} So, we might suspect some sort of shift in the dynamics for values of $c$ less than $-0.75$. Indeed, we have already looked at the case $c=-1$. In that case, the point $p^-$ is repelling, rather than attracting (as it was for $0.25<c<- 0.75$). Also, the length of the period over which an orbit stabilizes has doubled --- lands on two values, instead of converging to one. Again, there is a sort of bifurcation of dynamical process at $c=-0.75$, much as there was at $c=0.25$. The next value of c at which there is bifurcation of process is at $c=-l.25$ (analysis not shown). Values of $c$ slightly less than $-1.25$ produce systems with orbits for initial $x$--values in the critical interval that settle down to fluctuating among four values; the point $p^-$, which had been repelling for $- 0.75<c<-1.25$ now becomes attracting. And so this continues--- another bifurcation near $1.37$, and another somewhere near $1.4$. The values for $c$ at which successive bifurcations occur come faster and faster. \item{11.} A summary of this material appears below. \smallskip Bifurcation values, $b$: $$ c=0.25 --- b=1 $$ $$ c=-0.75 --- b=2 $$ $$ c=-1.25 --- b=3 $$ $$ c=-1.37 --- b=4 $$ derived from empirical evidence of examining the orbit dynamics of the corresponding systems of parabolas and $y=x$. Lengths of critical intervals, $I_b$, [$-p^+$, $p^+$], associated with the system corresponding to each bifurcation value, $b$. \smallskip $c=0.25$; Solve: $y=x$, $y=x^2+.25$; use quadratic formula--- $x=(1 \pm \sqrt(1-4\times 0.25))/2 = 0.5$. Thus, $p^+=0.5$ so $$ I_1=2\times 0.5=1.0 $$ $c=-0.75$. Solve: $y=x$, $y=x^2-.75$. $x=(1 \pm \sqrt(1+4\times 0.75))/2=1.5$ or $-0.5$. Thus, $p^+=1.5$ so $$ I_2=2 \times 1.5=3.0 $$ $c=-1.25$. Solve: $y=x$, $y=x^2-1.25$. $x=(1 \pm \sqrt(1+4\times 1.25))/2= 1.7247449$ or $-0.7247449$. So, $$ I_3=3.4494898 $$ $c=-1.37$. Solve: $y=x$, $y=x^2-1.37$. $x=(1 \pm \sqrt(1+4\times 1.37))/2= 1.7727922$ or $-0.7727922$. So, $$ I_4=3.5455844 $$ Now, suppose we find the successive differences between these interval lengths: $$ D_1=I_2-I_1=3-1=2 $$ $$ D_2=I_3-I_2=3.4494898-3=0.4494898 $$ $$ D_3=I_4-I_3=3.5455844-3.4494898=0.0960946 $$ Then, form successive ratios of these differences, larger over smaller: $$ D_1/D_2=2/0.4494898=4.4494892 $$ $$ D_2/D_3=.4494898/.0960946=4.6775761 $$ This set of ratios converges to Feigenbaum's number, $4.6692016\ldots $ \smallskip \item{12.} Apparently, empirical evidence suggests that any parabola--like system exhibits the same sorts of dynamics and the corresponding sets of ratios converge to Feigenbaum's number. For example, this appears to be the case, from literature, for the system $y=x$ and $y=c(sin x)$ and for the system involving the logistic curve, $y=x$ and $y=cx(1-x)$ [1]. \smallskip \item{13.} However, when the curved piece of the system is not parabola--like, different constants may occur. (A different curve might be a parabola with the vertex squared off--- singularities are introduced---where the derivative is undefined) [1]. \smallskip \item{14.} Obviously, many geographical systems can be characterized by a curve with fluctuations that are somewhat parabolic. Of course, we often do not know the equation of the curve. But, Simpson's rule from calculus, that pieces together parabolic slabs to approximate the area under a curve, generally gives a good approximation to the area of such curves. Thus, geographic systems that give rise to curves for which Simpson's rule provides a good areal approximation are ones that might be reasonable to explore in connection with Feigenbaum's number. \smallskip \item{15.} Steps 1 to 11 show how Feigenbaum's ``universal" number can be generated. Steps 12 to 14 give a systematic way to select geographical systems to examine with respect to this constant. \smallskip \smallskip \centerline{REFERENCE} \ref Feigenbaum, Mitchell J. ``Universal behavior in non--linear systems." {\sl Los Alamos Science\/}, Summer, 1980, pp. 4-27. \vfill\eject \centerline{SECOND CONSTRUCTION} \smallskip \centerline{A three--axis coordinatization of the plane} \smallskip \centerline{Motivated by a question from Richard Weinand} \smallskip \centerline{Department of Computer Science, Wayne State University} \smallskip \item{1.} Triangulate the plane using equilateral triangles. Then, choose any triangle as a triangle of reference---this triangle is to serve as an ``origin" for a coordinate system (an area--origin rather than a conventional point--origin---this is like homogeneous coordinates in projective geometry {\it e.g.\/} H. S. M. Coxeter, {\sl The Real Projective Plane\/}). Each side of the triangle is an axis---$x=0$, $y=0$, $z=0$ (Figure 10--draw to match text). \topinsert\vskip19cm {\bf Figure 10.} Three--axis coordinate system for the plane.

Click here for Figure 10.\endinsert \vfill\eject \item{2.} Each vertex of a triangle has unique representation as an ordered triple with reference to the origin--triangle (but, not every ordered triple of integers corresponds to a lattice point--- there is no point $(x,x,x)$) (Figure 10). \item{3.} Assign an orientation (clockwise or counterclockwise) to the origin--triangle, and mark the edges of the triangle with arrowheads to correspond to this orientation. This then determines the orientation of all the remaining triangles. \item{4.} Now suppose that a triangle is picked out at random. Suppose it has orientation the same as the reference triangle (clockwise, say). The coordinates of its vertices, in general, will be (choosing $(x, y, z)$ to be the lower left--hand corner): $$ (x, y, z); (x+1, y, z-1); (x, y+1, z-1) $$ and those of triangles sharing a common edge with it (and of opposite orientation to it) will have coordinates: $$ \hbox{left}: (x, y, z); (x+1, y, z-1); (x+1, y-1, z) $$ $$ \hbox{right}: (x+1, y, z-1); (x, y+1, z-1); (x+1, y+1, z-2) $$ $$ \hbox{bottom}: (x, y+1, z-1); (x, y, z); (x-1, y+1, z) $$ Suppose the arbitrarily selected triangle has orientation opposite that of the reference triangle (counterclockwise). The coordinates of its vertices, in general, will be (choosing $(x, y, z)$ to be the upper left--hand corner): $$ (x, y, z); (x-1, y+1, z); (x, y+1, z-1) $$ and those of triangles sharing a common edge with it (and of opposite orientation to it (clockwise)) will have coordinates: $$ \hbox{left}: (x, y, z); (x-1, y+1, z); (x-1, y, z+1) $$ $$ \hbox{right}: (x-1, y+1, z); (x, y+1, z-1); (x-1, y+2, z-1) $$ $$ \hbox{top}: (x, y, z); (x+1, y, z-1); (x, y+1, z-1) $$ \smallskip \item{5.} Coordinates of triangles sharing a point--boundary (and of the same orientation as the arbitrarily selected triangle) might also be read off in a similar fashion. \smallskip \item{6.} Naturally, six of these triangles form a hexagon. So, this could be considered from the viewpoint of an hexagonal tesselation, as well. Choose an arbitrary hexagon and read off coordinates of adjacent hexagonal regions in a similar manner. \smallskip \item{7.} In a current {\sl College Mathematics Journal\/}, Vol 21, No. 4, September, 1990, there is an article by David Singmaster (of Rubik's Cube fame) which also employs triangular coordinates of the sort mentioned above (pages 278-285--- ``Triangles with integer sides and sharing barrels"). \smallskip \item{8.} This strategy would seem to work for any developable surface (cylinder, torus, M\"obius strip, Klein bottle---all can be cut apart into a plane). Triangles were chosen because procedure involving them might be extended to simplicial complexes (triangle=simplex). \smallskip \item{9.} One way to triangulate a sphere is to project an icosahedron, inscribed in the sphere, onto the surface of the sphere (conversation with Jerrold Grossman, Dep't. of Mathematics, Oakland University). This procedure will produce 20 triangular regions of equal size (under suitable transformation). But, more triangles may be desirable. Alternately, one might subdivide the triangular faces of the icosahedron into, say, three triangles of equal area, and project the point that produces this subdivision (a barycentric subdivision, for example) onto the sphere (using gnomonic projection (from the sphere's center)). (Subdividing all of them a second time would produce 180 triangles of equal area and shape covering the sphere.) Subdivision centers on opposite sides of the icosahedron appear to lie on a single diameter of the sphere; therefore, when their images are projected onto the sphere they will be antipodal points. In that event, a coordinate system similar to the one described for developable surfaces might work. \bye