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\centerline{\big SOLSTICE:}
\centerline{\bf WINTER, 1990}
\centerline{\bf Volume I, Number 2}
\centerline{\bf Institute of Mathematical Geography}
\centerline{\bf Ann Arbor, Michigan}
\centerline{\bf SOLSTICE}
\line{Founding Editor--in--Chief: {\bf Sandra Lach Arlinghaus}. \hfil}
\centerline{\bf EDITORIAL BOARD}
\line{{\bf Geography} \hfil}
\line{{\bf Michael Goodchild}, University of California, Santa Barbara.
\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).}
\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.
\line{{\bf Business} \hfil}
\line{{\bf Robert F. Austin},
Director, Automated Mapping and Facilities Management, CDI. \hfil}

The purpose of {\sl Solstice\/} is to promote interaction
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Copyright, December, 1990, Institute of Mathematical Geography.
All rights reserved.
ISBN: 1-877751-44-8
\centerline{\bf SUMMARY OF CONTENT}
Numbering given below corresponds to the number of the
electronically transmitted file.
\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.
\noindent 2. File of front matter, including this material!
\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.}

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.
\noindent 5. Sandra Lach Arlinghaus. {\sl Scale and dimension:
Their logical harmony\/}

Linkage between scale and dimension is made using the
Fallacy of Division and the Fallacy of Composition in a fractal
\noindent 6 and 7. Sandra Lach Arlinghaus.
{\sl Parallels between parallels.\/} A manuscript originally
accepted by the now--defunct interdisciplinary journal,
{\sl Symmetry}.

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.
\noindent 8. Sandra L. Arlinghaus, William C. Arlinghaus, and
John D. Nystuen. {\sl The Hedetniemi matrix sum: A real--world

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
\noindent 9. Sandra Lach Arlinghaus.
{\sl Fractal geometry of infinite pixel sequences:
``Super--definition" resolution?}

Comparison of space--filling qualities of square and hexagonal
\noindent 10. {\sl Construction Zone\/}. Feigenbaum's number;
a triangular coordinatization of the Euclidean plane.
\centerline{\bf Photograph by John D. Nystuen; Rouge River, Detroit, 1974.}
\centerline{\bf FRONTISPIECE: A City of Strangers.}
Click here for Frontispiece

\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.

Click here for Figure 1.


  \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.

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

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).
\noindent {\bf Figure 3.}
Photographs of Detroit scenes by John D. Nystuen, c. 1974.
Click here for Figure 3.

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).
Click here for Figure 4.

\centerline{\bf TABLE 1.}
\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
\+&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
\noindent Human values are an individual's feelings and sense of
identification with people and things in the surrounding environment.

\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.
\centerline{\sl Sandra Lach Arlinghaus}
\centerline{\it ``Large streams from little fountains flow,}
\centerline{\it Tall oaks from little acorns grow." }
\centerline{David Everett, {\sl Lines Written for a School Declamation\/}.}

\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.
\centerline{\sl Sandra Lach Arlinghaus}
\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."}
\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)
\topinsert \vskip15cm
Click here for Figure 1.

 {\bf Figure 1.}  The hyperbolic plane.
Two lines $m$ and $m'$ (passing through $P$) are divergently
parallel to line $\ell$.
Two lines $m$ and $m'$ (passing through $P$) are asymptotically
parallel to line $\ell$.

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

\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
 \centerline{\bf Table 1:}
 \centerline{The Poincar\'e conformal model of the hyperbolic plane}
 \centerline{(referenced to Figure 2---after Greenberg)}
 \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
 \+&Hyperbolic plane &A disk, $D$, interior to a \cr
 \+&{}               &Euclidean circle, $C$      \cr
 \+&Point            &Point, $P$, in the disk, $D$.\cr
 \+&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
 \topinsert \vskip15cm
 {\bf Figure 2.}  The Poincar\'e Disk Model of the hyperbolic plane.
Click here for Figure 2.
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

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.

A Lambert quadrilateral with three right angles and one acute
angle $(PRQ)$. Pairs of opposite sides are parallel.

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 \vskip11cm
Click 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.

\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.

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.

West Sumatran Minangkabau house. Roofline is suggestive of a
Saccheri quadrilateral. Photograph by John D. Nystuen.

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.
Sodokan game board in Euclidean space. Markers travel along
lines separating regions of contrasting color and along
circular loops at the corners.

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.

Sample of capture. Black captures white---a single move.

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.}
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.

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

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.

\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,

\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\/}.
\centerline{\sl Sandra L. Arlinghaus, William C. Arlinghaus, John D.

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].
Click here for Figure 1, graph.

Click here for Figure 1, matrix.

\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.

Click here for Figure 2, graph.

Click here for Figure 2, matrix.

\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.

Click here for Figure 3, graph.

Click here for Figure 3, matrix.

\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.

Click here for Figure 4, graph.

Click here for Figure 4, matrix.

\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.

\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.

\centerline{ \bf Table 1}
\centerline{(derived from a Table in Arlinghaus and Arlinghaus, 1989)}
\+&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

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.
\centerline{\bf CONSTRUCTION ZONE}
\centerline{readers might wish to construct figures to accompany}
\centerline{the electronic text as they read}
\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

\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).

\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-
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 $0$.

\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].
{\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.


\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 $c$.

\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."
{\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