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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
 between geography and mathematics.   Articles in which  elements
 of   one  discipline  are used to shed light on  the  other  are
 particularly sought.   Also welcome,  are original contributions
 that are purely geographical or purely mathematical.   These may
 be  prefaced  (by editor or author) with  commentary  suggesting
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 Individuals  wishing to submit articles,  either short or full--
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 suggest, and furnish material for, new regular features.
 \noindent {\bf Send all correspondence to:}
 \centerline{\bf Institute of Mathematical Geography}
 \centerline{\bf 2790 Briarcliff}
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 \centerline{\bf (313) 761-1231}
 \centerline{\bf IMaGe@UMICHUM}
       This  document is produced using the typesetting  program,
 {\TeX},  of Donald Knuth and the American Mathematical  Society.
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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. \vfill\eject \centerline{\bf SCALE AND DIMENSION: THEIR LOGICAL HARMONY} \smallskip \centerline{\sl Sandra Lach Arlinghaus} \smallskip \smallskip \centerline{\it ``Large streams from little fountains flow,} \centerline{\it Tall oaks from little acorns grow." } \smallskip \centerline{David Everett, {\sl Lines Written for a School Declamation\/}.} \smallskip \heading Introduction. Until recently, the concept of ``dimension" was one that brought ``integers" to mind to all but a handful of mathematicians [Mandelbrot, 1983]; a point has dimension 0, a line dimension 1, an area dimension 2, and a volume dimension 3 [Nystuen, 1963]. When a fourth dimension is added to these usual spatial dimensions, time can be included, as well. Indeed, much ``pure" mathematics takes place in abstract $n$--dimensional hypercubes, where $n$ is an integer. Geographic maps, globes (and other representations of part or of all of the earth), are traditionally bounded by these integral dimensions, as well; map scale is expressed in discrete, integral units. Often, however, it is the case in geography as it is in mathematics, that a change in scale, or in dimension, runs across a continuum of possible values. In either case, discrete regular steps are usual as benchmarks at which to consider what the continuing process looks like at varying stages of evolution. As fractal geometry suggests, however, this need not be the case. Within an integral view of scale or dimension, there are logical and perceptual difficulties in jumping from one integral vantage point to another: Edwin Abbott [1955] has commented on this in his classic abstract essay on ``Flatland," and more recently, Edward Tufte has done so in the real--world context of ``envisioning information" [1989]. Methods for dealing with these dimensional--jump difficulties abound, particularly in the arts [Barratt, 1980]. In a musical context Charles Wuorinen sees composition as a process of fitting ``large" musical forms with scaled--down, self--similar, equivalents of these larger components in order to introduce richness of detail to the theme [NY Times, 1990]. Maurits Escher, in his ``Circle Limit" series of tilings of the non-- Euclidean hyperbolic plane, uses tiles of successively smaller size to suggest a direction of movement---that of falling off an edge or of being engulfed in a central vortex. A gastronomic leap sees a Savarin as self--similar to a Baba au Rhum [Lach, 1974]; indeed, even more broadly, Savarin himself is purported to have said, ``You are what you eat." Rupert Brooke (in ``The Soldier") captured this notion poetically, in commenting on the possible fate of a soldier in a distant land: \centerline{``If I should die, think only this of me; } \centerline{ that there is some corner of a foreign field } \centerline{ that is forever England." } \noindent In the end, Brooke's ``Soldier" becomes `place'. The fractal concept of self--similarity can be employed to suggest one way to resolve difficulties in scale changes as one moves from dimension to dimension. At the theoretical level, symbolic logic classifies logical fallacies that may, or may not, emerge from scale shifts. When self--similarity is viewed in this sort of logic context, the outcome is a ``Scale Shift Law." What is presented here are the abstract arguments; it remains to test empirical content against these arguments. \heading Logical fallacies. A question of enduring interest in geography, and in other social sciences, is to consider what can be said about information concerning individuals of a group when given information only about characteristics of the group as a whole. When an attribute of the whole is {\bf erroneously} assigned to one or more of its parts, the logic of this assignment falters. In the social scientific literature, this is generally referred to as commission of the so--called ``ecological" fallacy; because the symphony played poorly does not necessarily mean that each, or indeed that any, individual musician did so. In this circumstance, it is simply not possible to assign any truth value, derived from principles of symbolic logic, to the quality of the performance of any subset of musicians (based only on the quality of the performance of the whole orchestra) [Engel, 1982]. It is natural, however, to look for a cause for the poor performance, and indeed to consider some ``middle" position that asks to what extent the performance of the orchestra is related to the performance of its individual members. It is this sort of search for finding and measuring the extent of relationship that is the hallmark of quantitative social scientific effort, much of which appears to have been guided [Upton, 1990], in varying degree, by an early effort to determine the extent to which race and literacy are related [Robinson, 1950]. A fallacy, in a lexicographic sense might be ``a false idea" or it might be of ``erroneous character" or ``an argument failing to satisfy the conditions of valid or correct inference" [Webster, 1965]. In a formal logic sense, a fallacy is ``a `natural' mistake in reasoning" [Copi, 1986, p. 4] or it is an argument that fails because its premisses do not imply its conclusion; it is an argument whose conclusion {\bf could} be (but is not necessarily) false even if all of its premisses are true [Copi, 1986, p. 90]. Viewed in this manner, the so--called ``ecological" fallacy is nothing different; it is merely a restatement of the ``fallacy of division" of classical elementary symbolic logic. The fallacy of division is committed by assigning, {\bf erroneously}, the attributes of the whole to one or more of its parts [Copi]. Thus, it may or may not be valid to make an inference about the nature of a part based on the nature of the whole. That is, sometimes the assignment of truth value from whole to part, in jumping across the dimensional scale from whole to part, is a reasonable practice, and sometimes it is not. The key is to determine when this practice is reasonable, when it is not, and when it simply does not apply. Commission of this fallacy is frequently the result of confusing terminology which refers to the whole (``collective" terms) with those which refer only to the parts (``distributive" terms) [Copi, 1986]. The fallacy of division exists within an abstract human system of reasoning based on the Law of the Excluded Middle: in this Law, a statement is true or false---not some of each. There is ``black" and ``white," but no ``gray" in this system. Statistical work that stems from this fallacy seeks, when it rests on finding correlations, relations that blend ``black" and ``white"---the foundation in ``logic" is thus ignored. This fallacy is examined, here, with an eye to understanding the logical circumstances under which such assignment might, or might not, be erroneous (when it applies). \heading Scale and dimension. To understand when the assignment of characteristics from whole to part (division), or from part to whole (the fallacy of composition---the string sections played well, therefore the symphony played well), might be erroneous, it is useful to consider what are the fundamental components composing these fallacies. The notion of scale is involved in the consideration of ``whole" and ``part." When is the individual a ``scaled--down" orchestra; or, when is the orchestra a ``scaled--up" individual? The notion of dimension is also involved. When does the zero--dimensional musician-- point spread out to fill the two--dimensional (or three--or more--dimensional) orchestra; or, when does the higher dimensional orchestra collapse, black--hole--like, into the single performer. The performing soloist can dominate the orchestra; the conductor perhaps does dominate the orchestra; yet, the orchestra itself is composed of numerous single performers who do not dominate. \heading Self--similarity and scale shift. Integral dimensions, with discrete spacing separating them, might be viewed as simply a set of positions marking intervals along a continuum of fractional dimensions [Mandelbrot, 1983]. When the discrete set of integral dimensions is replaced by the ``dense" set of fractional dimensions (between any two fractional dimensions there is another one), what happens to our various relative vantage points and to scale problems associated with them? Abstractly, the relationship is not difficult to tie to logic, under the following fundamental assumption. \smallskip \line{\bf Fundamental Assumption.\hfil } \smallskip When two views of the same phenomenon at different scales are self--similar one can properly divide or compose these views to shift scale. \smallskip \noindent The whole can be divided ``continuously" through a ``dense" stream of fractional dimensions until the part is reached (and in reverse). Self--similarity suggests a sort of dimensional stability of the characteristic or phenomenon in question. One commits the Fallacy of Division (``Ecological" Fallacy) when the attributes (terminological or otherwise) of the whole are assigned to the parts that are {\bf not} self--similar to the whole. One commits the Fallacy of Composition when the attributes of the parts are assigned to a whole that is {\bf not} self--similar to these parts. This notion is evident in the many animated graphic displays of the Mandelbrot (and other) sets in which zooming in on some detail presents some sort of repetitive sequence of views (in the case of self--similarity, this sequence has length 1). More formally, this idea may be cast as a ``Law." \smallskip \line{\bf Scale Shift Law \hfil} \smallskip Suppose that the attributes of the whole (part) are assigned to the part (whole). \item{1.} If the whole and the part {\bf are not} self-- similar, then that assignment {\bf is} erroneous; and, conversely (inversely, actually), \item{2.} If the whole and the part {\bf are} self--similar, then that assignment {\bf is not\/} erroneous. \smallskip \noindent This is one way to look at the ``part--whole" dichotomy; physicists wonder about splitting the latest ``fundamental" particle; philosophers search for fundamental units of the self [Leibniz, monadology, in Thompson, 1956; Nicod, 1969]; topologists worry about what properties a topological subspace can inherit from its containing topological space [Kelley, 1955]. \heading References. \ref Abbot, Edwin A. (1956) ``Flatland." reprinted in {\sl The World of Mathematics\/}, James R. Newman, editor. New York: Simon and Schuster. \ref Barratt, Krome (1980) {\sl Logic and Design: The Syntax of Art, Science, and Mathematics\/}. Westfield, NJ: Eastview Editions, 1980. \ref Copi, Irving M. (1986) {\sl Introduction to Logic\/}. Seventh Edition. New York: Macmillan Publishing Company, (first edition, 1953). \ref Engel, S. Morris (1982) {\sl With Good Reason: An Introduction to Informal Fallacies\/}. Second Edition. New York: St. Martins Press. \ref Kelley, John L. (1963) {\sl General Topology\/}. Princeton: D. Van Nostrand. \ref Lach, Alma S. (1974) {\sl The Hows and Whys of French Cooking\/}, Chicago: The University of Chicago Press. \ref Mandelbrot, Benoit (1983) {\sl The Fractal Geometry of Nature\/}. San Francisco: Freeman. \ref Nicod, Jean (1969) {\sl Geometry and Induction: Containing `Geometry in the Sensible World' and `The Logical Problem of Induction' with Prefaces by Roy Harrod, Bertrand Russell, and Andre Lalande\/}. London: Routledge and Kegan Paul, New translation. \ref Nystuen, John D. (1963) ``Identification of some fundamental spatial concepts." {\sl Papers of Michigan Academy of Letters, Sciences, and Arts\/}. 48: 373-384. \ref Robinson, W. (1950) Ecological correlations and the behavior of individuals, {\sl American Sociological Review\/}. 15: 351-357. \ref Rockwell, John (1990) ``Fractals: A Mystery Lingers." Review/Music, {\sl The New York Times\/}, Thursday, April 26. \ref Thompson, D'Arcy Wentworth (1956) ``On Magnitude." In {\sl The World of Mathematics\/}, James R. Newman, Editor. New York: Simon and Schuster. \ref Tufte, Edward (1989) {\sl Envisioning Information\/}. Cheshire, CT. \ref Upton, Graham J. G. (1990) ``Information from Regional Data," in {\sl Spatial Statistics: Past, Present, and Future\/}, edited by Daniel A. Griffith. IMaGe Monograph, \#12. Ann Arbor: Michigan Document Services. \ref {\sl Webster's Seventh New Collegiate Dictionary\/} (1965) Springfield, MA: G. and C. Merriam Company. \vfill\eject \centerline{\bf PARALLELS BETWEEN PARALLELS} \smallskip \centerline{\sl Sandra Lach Arlinghaus} \smallskip \smallskip \centerline{\it ``I have a little shadow that goes in and out with me,} \centerline{\it And what can be the use of him is more than I can see."} \smallskip \centerline{\sl Robert Louis Stevenson } \centerline{``My Shadow" in {\sl A Child's Garden of Verses}} {\narrower\smallskip{\bf Abstract}: The earth's sun introduces a symmetry in the perception of its trajectory in the sky that naturally partitions the earth's surface into zones of affine and hyperbolic geometry. The affine zones, with single geometric parallels, are located north and south of the geographic tropical parallels. The hyperbolic zone, with multiple geometric parallels, is located between the geographic tropical parallels. Evidence of this geometric partition is suggested in the geographic environment---in the design of houses and of gameboards.\smallskip} \heading 1. Introduction. Subtle influences shape our perceptions of the world. The breadth of a world--view is a function not only of ``real"--world experience, but also of the ``abstract"--world context within which that experience can be structured. As William Kingdon Clifford asked in his Postulates of the Science of Space [3], how can one recognize flatness when magnification of the landscape merely reveals new wrinkles to traverse? Geometry is a ``source of form" not only in mathematics [10], but also in the ``real" world [2]. Street patterns are geometric; architectural designs are geometric; and, diffusion patterns are geometric. In this study, the geometric notion of parallelism is examined in relation to the manner in which the sun's trajectory in the earth's sky is observed by inhabitants at various latitudinal positions: from north and south of the tropics to between the tropical parallels of latitude. A fundamental geometrical notion is thus aligned with fundamental geographical and astronomical relationships; this alignment is interpreted in cultural contexts ranging from the design of rooflines to the design of board games. \heading 2. Basic Geometric Background. To understand how geometry might guide the perception of form, it is therefore important to understand what ``geometry" might be. Projective geometry is totally symmetric and possesses a completely ``dual" vocabulary: ``points" and ``lines," ``collinear" and ``concurrent," and a host of others, are interchangeable terms [6]. Indeed, a Principle of Duality serves as a linguistic axis, or mirror, halving the difficulty of proving theorems. Thus, because ``two points determine a line" is true, it follows, dually, that ``two lines determine a point" is also true. The corresponding situation does not hold in the Euclidean plane: two lines do not necessarily determine a point because parallel lines do not determine a point [6]. Coxeter classifies other geometries as specializations of projective geometry based on the notion of parallelism, depending on whether a geometry admits zero, one, or more than one lines parallel to a given line, through a point not on the given line [6]. In the ``elliptic" geometry of Riemann, there are no parallel lines, much as there are none in the geometry of the sphere that includes great circles as the only lines, any two of which intersect at antipodal points. In ``affine" geometry, there is exactly one line parallel to a given line, through a point not on that line. Affine geometry is further subdivided into Euclidean and Minkowskian geometries. Finally, in the ``hyperbolic" geometry of Lobachevsky, there are at least two lines parallel to a given line through a point not on that line. To visualize, intuitively, the possibility of more than one line parallel to a given line it is helpful to bend the lines, sacrificing ``straightness" in order to retain the non-- intersecting character of parallel lines. Thus, two upward-- bending lines $m$ and $m'$ passing through a point $P$ not on a given line $\ell$ never intersect $\ell$; they are divergently parallel to $\ell$ (Figure 1.a). Or, one might imagine lines $m$ and $m'$ that are asymptotically parallel to $\ell$ (Figure 1.b) [8]. \topinsert \vskip15cm
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. \vfill\eject \centerline{\bf CONSTRUCTION ZONE} \smallskip \centerline{FIRST CONSTRUCTION;} \centerline{readers might wish to construct figures to accompany} \centerline{the electronic text as they read} \smallskip \centerline{\bf Feigenbaum's number: exposition of one case} \centerline{Motivated by queries from Michael Woldenberg,} \centerline{Department of Geography, SUNY Buffalo,} \centerline{during his visit to Ann Arbor, Summer, 1990.} Here is a description of how Feigenbaum's number arises from a graphical analysis of a simple geometric system [1]. Feigenbaum's original paper is clear and straightforward [1]; this construction is presented to serve as exposure prior to reading Feigenbaum's longer paper [1]. The construction is complicated although individual steps are not generally difficult. Following the construction, a suggestion will be offered as to how to select mathematical constraints within which to choose geographical systems for Feigenbaum--type analysis. \item{1.} Consider the family of parabolas $y=x^2 + c$, where $c$ is an integral constant. This is just the set of parabolas that are like $y=x^2$, slid up or down the $y$-axis. The smaller the value of $c$, the more the parabola opens up (otherwise a lower one would intersect a higher one, creating an algebraic impossibility such as $-1=0$) (Figure 1). \smallskip \item{2.} To begin, consider the particular parabola, $y=x^2 - 1$, obtained by setting $c = -1$. Graph this (Figure 2). Also draw the line $y=x$ on this graph. Now we're going to look at the ``orbit" of the value $x=1/2$ with respect to this parabola (function). By ``orbit" is meant simply the iteration string obtained by using $x=1/2$ as input into $y=x^2 -1$, then using that output as a new input into $y=x^2-1$, then using that output as a new input $\ldots $ and so forth. In this case, the orbit of $x=1/2$ is represented as follows, numerically. (Use $.5 \mapsto -0.75$ to mean that the input of $.5$ is mapped to the output value of $-0.75$ by the function $y=x^2- 1$.) $$ 0.5 \mapsto -0.75 \mapsto -0.4375 \mapsto -0.8085938 $$ $$ \mapsto -0.3461761 \mapsto -0.8801621 \mapsto -0.2253147 $$ $$ \mapsto -0.9492333 \mapsto -0.0989562 \mapsto -0.9902077 $$ $$ \mapsto -0.019488 \mapsto -0.9996202 \mapsto -0.0007595 $$ $$ \mapsto -0.9999994 \mapsto -0.0000012 \mapsto -1 $$ $$ \mapsto 0 \mapsto -1 \mapsto 0 \mapsto \ldots $$ Clearly the values bounce around for awhile, and then eventually settle down to the values, $-1$ and $0$. \smallskip \item{3.} Let's see what this particular iteration string means geometrically (Figure 3). Locate $x=0.5$ on the $x$--axis. Drop down to the parabola to read off the corresponding $y$--value (in the usual manner) $-0.75$. Now it is this $y$--value that is to be used as the next input in the iteration string. We could go back up to the $x$--axis and find it and drop back to the parabola, but we won't. Instead execute the following, equivalent transformation---THIS IS THE KEY POINT. Assume your penpoint is on the $y$--value $-0.75$; now slide horizontally over to the line $y=x$---you want to use the $y$--value in the role of the $x$--value. Thus, treat this point as the new input and drop to the parabola from it as you did in moving from the $x$--axis to the parabola. Then, with your penpoint on the parabola, slide horizontally back to the line $y=x$ and use this as the input; drop to the parabola and keep going. A glance at Figure 2 suggests why economists call this a ``cobweb" diagram (presumably looking at fluctuating supply and demand). Follow this diagram long enough, and you will see that eventually values for $x$ fluctuate between $0$ and $-1$, around a stationary square cycle. Looking at the ``dynamics" of a value, with respect to a function, in this geometrical manner is referred to as (Feigenbaum's) ``graphical analysis" [1]. \topinsert\vskip19cm {\bf Figure 1.} Parabolas of the form $y=x^2+c$.

Click here for Figure 1. 

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

Click here for Figure 2.

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

Click here for Figure 3.

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

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

Click here for Figure 5.

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

Figure 6.

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

Figure 7.

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

Click here for Figure 10.

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