%----->>>>VOLUME 1, NUMBER 2 OF SOLSTICE. WINTER, 1990; 10:07 PM 12/12/90.  %----->>>>HAPPY HOLIDAYS!!!! Sandy Arlinghaus. %----->>>>IF YOU DO not WISH TO CONTINUE YOUR COMPLIMENTARY SUBSCRIPTION, %----->>>>FOR 1991, PLEASE WRITE Solstice@UMICHUM AND SO INDICATE. %----->>>>THANK YOU FOR PARTICIPATING IN GEOGRAPHY'S FIRST E-JOURNAL. %----->>>>HARD-COPY OF VOLUME I IS AVAILABLE AS MONOGRAPH 13 IN THE %         IMaGe MONOGRAPH SERIES. \hsize = 6.5 true in %THIS FILE CONTAINS TYPESETTING CODE, ONLY.  %ADD IT TO THE BEGINNING OF EACH OF THE FOLLOWING FILES TO TYPESET THEM.  %ALSO ADD \bye TO CLOSE EACH FILE TO BE TYPESET. \input fontmac %delete to download, except on Univ. Mich. (MTS) equipment. \setpointsize{12}{9}{8}%same as previous line; set font for 12 point type. \parskip=3pt \baselineskip=14 pt \mathsurround=1pt \headline = {\ifnum\pageno=1 \hfil \else {\ifodd\pageno\righthead   \else\lefthead\fi}\fi} \def\righthead{\sl\hfil SOLSTICE } \def\lefthead{\sl Winter, 1990 \hfil} \def\ref{\noindent\hang} \font\big = cmbx17 \font\tn = cmr10 \font\nn = cmr9 %The code has been kept simple to facilitate reading as e-mail \font\tbf = cmbx12  \outer\def\heading#1\par{\vskip 0pt plus .1\vsize \penalty-50   \vskip 0pt plus -.1\vsize \medskip\vskip\parskip   \centerline{\tbf#1}\nobreak\medskip\noindent} \outer\def\section#1\par{\vskip 0pt plus .1\vsize \penalty-50   \vskip 0pt plus -.1\vsize \medskip\vskip\parskip   \leftline{\tbf#1}\nobreak\smallskip\noindent} \outer\def\subsection#1\par{\vskip 0pt plus .1\vsize \penalty-50   \vskip 0pt plus -.1\vsize \medskip\vskip\parskip   \leftline{\sl#1}\nobreak\smallskip\noindent} \def\pmb#1{\setbox0=\hbox{#1}%     \kern-.025em\copy0\kern-\wd0     \kern.05em\copy0\kern-\wd0     \kern-.025em\raise.0433em\box0 } \centerline{\big SOLSTICE:} \vskip.5cm \centerline{\bf  AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS} \vskip5cm \centerline{\bf WINTER, 1990} \vskip12cm \centerline{\bf Volume I, Number 2} \smallskip \centerline{\bf Institute of Mathematical Geography} \vskip.1cm \centerline{\bf Ann Arbor, Michigan} \vfill\eject \hrule \smallskip \centerline{\bf SOLSTICE} \line{Founding Editor--in--Chief:  {\bf Sandra Lach Arlinghaus}. \hfil} \smallskip \centerline{\bf EDITORIAL BOARD} \smallskip \line{{\bf Geography} \hfil} \line{{\bf Michael Goodchild}, University of California, Santa Barbara. \hfil} \line{{\bf Daniel A. Griffith}, Syracuse University. \hfil} \line{{\bf Jonathan D. Mayer}, University of Washington; joint appointment  in School of Medicine.\hfil} \line{{\bf John D. Nystuen}, University of Michigan (College of  Architecture and Urban Planning).} \smallskip \line{{\bf Mathematics} \hfil} \line{{\bf William C. Arlinghaus}, Lawrence Technological University. \hfil} \line{{\bf Neal Brand}, University of North Texas. \hfil} \line{{\bf Kenneth H. Rosen}, A. T. \& T. Information Systems Laboratory.        \hfil} \smallskip \line{{\bf Business} \hfil} \line{{\bf Robert F. Austin}, Director, Automated Mapping and Facilities Management, CDI. \hfil} \smallskip \hrule \smallskip        The purpose of {\sl Solstice\/} is to promote  interaction between geography and mathematics.   Articles in which  elements of   one  discipline  are used to shed light on  the  other  are particularly sought.   Also welcome,  are original contributions that are purely geographical or purely mathematical.   These may be  prefaced  (by editor or author) with  commentary  suggesting directions  that  might  lead toward  the  desired  interaction. Individuals  wishing to submit articles,  either short or full-- length,  as well as contributions for regular  features,  should send  them,  in triplicate,  directly to the  Editor--in--Chief. Contributed  articles  will  be refereed by  geographers  and/or mathematicians.   Invited articles will be screened by  suitable members of the editorial board.  IMaGe is open to having authors suggest, and furnish material for, new regular features. \vskip2in \noindent {\bf Send all correspondence to:} \vskip.1cm \centerline{\bf Institute of Mathematical Geography} \centerline{\bf 2790 Briarcliff} \centerline{\bf Ann Arbor, MI 48105-1429} \vskip.1cm \centerline{\bf (313) 761-1231} \centerline{\bf IMaGe@UMICHUM} \vfill\eject        This  document is produced using the typesetting  program, {\TeX},  of Donald Knuth and the American Mathematical  Society. 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Currently,  the  text  is typeset using   {\TeX};  in that  way, mathematical formul{\ae} can be transmitted   as ASCII files and downloaded   faithfully   and   printed   out.    The     reader inexperienced  in the use of {\TeX} should note that  this    is not  a what--you--see--is--what--you--get"  display;  however, we  hope  that  such readers find {\TeX} easier to  learn  after exposure to {\sl Solstice\/}'s e-files written using {\TeX}!}        {\nn  Copyright  will  be taken out in  the  name  of  the Institute of Mathematical Geography, and authors are required to transfer  copyright  to  IMaGe as a  condition  of  publication. There are no page charges; authors will be given  permission  to make reprints from the electronic file,  or to have IMaGe make a single master reprint for a nominal fee dependent on  manuscript length.   Hard  copy of {\sl Solstice\/}  will be sold  (contact IMaGe  for  price--{\sl Solstice\/} and will be priced to  cover expenses  of journal production);  it is the desire of IMaGe  to offer  electronic  copies to  interested parties for free--as  a kind of academic newsstand at which one might browse,  prior  to making purchasing decisions.  Whether or not it will be feasible to  continue distributing complimentary electronic files remains to be seen.} \vskip.5cm Copyright, December, 1990, Institute of Mathematical Geography. All rights reserved. \vskip1cm ISBN: 1-877751-44-8 \vfill\eject \centerline{\bf SUMMARY OF CONTENT} \smallskip      Numbering  given  below  corresponds  to the number  of the electronically transmitted file. \smallskip \noindent 1.  Typesetting code; file of {\TeX} commands that may be inserted at the beginning of each file  (or in front  of  the whole set run at once) in order to typeset the document. \smallskip \noindent 2.  File of front matter, including this material! \smallskip \noindent 3 and 4.  Reprint of John D. Nystuen from 1974.  {\sl A city of strangers:   Spatial aspects of alienation in the Detroit metropolitan region.} \smallskip    Examines urban shift from people space" to machine space" (see R. Horvath, {\sl Geographical Review\/} April, 1974) in the context  of  the  Detroit  metropolitan region of 1974.  As with Clifford's {\sl Postulates of the Science of Space\/}, reprinted in the last issue of {\sl Solstice\/}, note the  timely  quality of many of the observations. \smallskip \noindent 5.  Sandra Lach Arlinghaus.  {\sl Scale and dimension: Their logical harmony\/} \smallskip      Linkage between scale  and  dimension  is  made  using  the   Fallacy of Division and the Fallacy of Composition in a  fractal setting. \smallskip \noindent 6 and 7.  Sandra Lach Arlinghaus.  {\sl  Parallels  between parallels.\/}  A manuscript  originally  accepted  by  the  now--defunct  interdisciplinary  journal, {\sl Symmetry}. \smallskip      The earth's sun introduces a symmetry in  the  perception  of its trajectory in the sky that naturally  partitions  the  earth's surface into zones of affine and hyperbolic geometry.  The  affine zones, with single geometric  parallels,  are  located  north  and south of the geographic tropical parallels.  The hyperbolic  zone, with  multiple  geometric  parallels,  is  located   between   the geographic  tropical  parallels.   Evidence  of   this   geometric partition is  suggested in  the  geographic  environment---in  the design of houses and of gameboards. \smallskip \noindent 8.   Sandra L. Arlinghaus,  William C. Arlinghaus,  and John D. Nystuen. {\sl The Hedetniemi  matrix  sum:  A real--world application.\/} \smallskip In  a  recent paper,  we presented an algorithm for  finding  the  shortest distance between any two nodes in a network of $n$ nodes  when  given only distances between  adjacent  nodes  [Arlinghaus,  Arlinghaus,  Nystuen,  {\sl  Geographical Analysis, 1990\/}].  In  that  previous  research,    we  applied  the  algorithm  to  the generalized  road network graph surrounding  San  Francisco  Bay.  Here, we examine consequent changes  in  matrix  entries when the underlying adjacency pattern of  the road  network was altered by  the 1989 earthquake that closed  the  San Francisco--Oakland  Bay Bridge.   \smallskip \noindent 9.  Sandra Lach Arlinghaus. {\sl Fractal geometry of infinite pixel sequences:   Super--definition" resolution?} \smallskip Comparison of space--filling qualities of square and hexagonal pixels. \smallskip \noindent 10.  {\sl Construction Zone\/}.  Feigenbaum's number; a triangular coordinatization of the Euclidean plane. \vfill\eject \centerline{\bf INDUSTRIAL WASTELAND RIVER} \centerline{\bf Photograph by John D. Nystuen; Rouge River, Detroit, 1974.} \centerline{\bf FRONTISPIECE:  A City of Strangers.}
\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}

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.
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
consider new means of association and control that will  humanize
the space around us once again.

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.

Ron  Horvath,  in  an  article  in  the  {\sl  Geographical
Review\/} entitled Machine Space,'' classified land parcels  as
machine  space''   rather than people space'' depending  upon
who  or  what  is  given priority of use  in  the  event  of  a
conflict'' [Horvath,  p.  169].   He then pointed out how much of
our  cities  we  have  given  up  to  machines,   especially  the
automobile.  He characterized this machine as the  sacred cow''
in American culture.  He said

{\narrower{ \noindent
In the minds of many Westerners,  India's sacred cow has come to
symbolize  the  lengths  to which people will go  to  preserve  a
nonfunctional   cultural  trait.    But  India's  sacred  cow  is
downright  rational  in  comparison to  ours.   Could  an  Indian
imagine devoting 70 percent of downtown  Delhi to cow trails  and
pasturage as we do for our automobiles in Detroit and Los Angeles.
Every year nationally we sacrifice more than 50,000 Americans  to
our sacred cow in traffic accident fatalities (Figure 1)
[2, p. 168].\par}}

\topinsert \vskip11cm
\noindent {\bf Figure 1.}
Machine Space'' in downtown Detroit, ground level, 1971, by R.
Horvath.  Map reprinted with permission of The American
Geographical Society, from Machine Space," R. Horvath, {\sl The
Geographical Review\/}, April, 1974, p. 171.

\endinsert

  \noindent   Something  like  20 percent of  our  gross  national   product is tied directly to manufacturing,  servicing and fueling  the  automobile---twice  the  amount we spend  on  war  machines,  another  more  sinister genre of sacred cow machine to  which  we  seem addicted.  \heading {Vertical Control or Scale Transforms.}       There  are  signs  of a reaction setting  in.   Ralph  Nader  effectively  pointed  out that automobiles are  unsafe  at  any  speed."  The solution called for is not crash proof cars.   It is  reduction  of  exposure by reducing passenger miles  traveled  by  private automobiles.   We can accomplish this in two very general  ways:    by  developing mass transit systems and by reducing  the  number  and  length  of trips taken.  The latter calls  for  re-- ordering  land use patterns or changing our life style by  giving  up  some  of  our triumphs over space.   Trends  in  the  Detroit  Metropolitan Area suggest otherwise.  We are still in the process  of  completing an expressway system.   The state  has  authorized  one--half  cent  of the nine cent gasoline tax to be  devoted  to  mass transit systems;  a significant step but hardly a major re-- allocation  of  priorities.    SEMTA,  the  state  transportation  authority for Southeast Michigan,  has recently released its mass  transportation  plan calling for a 1990 completion date.   If the   experience  of systems such as the San Francisco Bay Area's  BART  can  be  taken  as an example,  significant  delays  due  to  the  operation  of political processes will set that date further into  the future, if indeed, the system is ever built. [As of 1990,  the Southeastern Michigan Transportation  Authority  (SEMTA)  is defunct.  Their mass transit plan,  released in 1975,  called for a 1990 completion date  (Figure 2).   All that came of  this  plan  was  the  elevated  downtown  Detroit  People  Mover,  delayed,  over  budget,  and out--of--control as the rest of  the  mass  transportation plan was never implemented and doomed to  go  out of business.  Too massive to tear down without great expense,  it  will  remain a bizarre monument to  inadequate  planning  and  fragmented  action.   On the other hand,  the Detroit  expressway  system is largely completed.  A final link in the circumferential  network,  I-696,  opened in 1989, twenty--five years after it was  proposed.   This  stretch  of expressway was met with  determined  opposition  from an upper--middle  class,  politically  effective  neighborhood.   The final links were modified to lessen impact on  adjacent   residents.    Neighborhoods  near  downtown  locations  succumbed  to  the  huge  concrete  corridors  years  ago.    The  expressways created huge barriers and the livable spaces  between  them proved too fragmented to sustain and are now abandoned.] \topinsert \vskip20cm \noindent {\bf Figure 2.}  Map from 1974 suggests a network that  was never built (as of 1990).
 \endinsert  Multi--million dollar transportation projects greatly affect land  use patterns and are once--and--for--all investments.   They come  infrequently  and permanently affect the geography of the region.   The massive water and interceptor plan of the Detroit Water Board  is  a similar large scale project with more benign  consequences.   This  brought water from Lake Huron via tunnel and aqueduct to  a  large  portion  of  the metropolitan  region.   [It  was  also  a  planning error.  In retrospect we see it was overbuilt due to the  decline in heavy industry in the city and the exodus of people to  the suburbs.]  Decisions  associated with large scale projects are  examples  of  factors  which  are out of the hands of the ordinary  citizen  or  even  the  large  land developers working in  the  region.   They  impose important constraints on land use possibilities.  They are  decisions  made by strangers and represent a loss of  private  or  small  community  freedom  of choice.   Many gross forms  in  the  Detroit metropolitan region are the consequence of decisions made  many decades ago.  Some individuals and communities try to resist  the pressures of single large scale commitments.   In the case of  water procurement,  this can be done by using local ground  water  wells and septic tanks or small municipal sewage plants.   At low  population  densities  these  local devices may work fine  and  a  decentralized  system  is  probably  best.   At  high  densities,  however,  local  environmental capacities  are  exceeded.   Other  public agencies,  such as the County Health Departments, may then  operate  to pressure communities into the larger system.   It  is  this  hierarchical ordering of systems that removes local control  from  one aspect after another of urban life.  When  the  problem  condition   in  the  environment  enlarges  previously   separate  problems begin to merge,  the best institutional response we have  yet  devised  is  to establish a  hierarchically  ordered  social  process to address the larger problem.   This change in scale may  result  in  qualitatively  different  situations.    Institutions  operating  at metropolitan levels may appear very inflexible  and  arbitrary    from  the  point  of  view  of  a  local  authority,  municipality,    or   private   home   owner.    The   need   for  standardization and routinization is absolutely crucial for  such  organizations.   Alienation  may develop between parties who view  things at different scales without anyone being at fault.  Politically, a metropolitan region is hierarchically organized by  spatial  jurisdictions.   Local problems are  most  appropriately  dealt  with by local authority and regional problems by  regional  authorities.   We  have  yet  to  devise a  means  of  graciously  transferring jurisdiction up or down the hierarchy to  correspond  to  changes in scale in the nature of the problems.   Our greatly  increased  capacity to overcome transportation and  communication  costs has led to changes in  population density and locations  of  jobs which have often exacerbated local problems and called forth  a scale transfer.  The local community, no longer able to perform  the  service,  loses  jurisdiction  over the problem   to  higher  authorities.   At a higher level, much of the loss of state power  to  the federal government has been a change of  this  sort.  [To  some  extent  deregulation efforts of recent years prior to  1990  have shifted responsibility back to local authorities, especially  from  Federal to State levels.   Hierarchies need to be  designed  that   set limits or levels of acceptable performance but  remain  tolerant  of variation in local actions.   State rules  regarding  equalization   of   county  property  taxes  and   local   school  performance are examples.]  \heading {Horizontal Control.}  Some  institutions  and corporations are cross--threaded  in  the  fabric  of  society.   Their interests and actions are  uncoupled  from the local community because they are interested in a  single  category  of  phenomena  and  not  in  the  mix  of  all  spatial  categories  at  one  location.   The  decision  makers  in  these  organizations  are very likely to be outlanders;  people who live  in  entirely  different communities or even  other  nations,  yet  whose decisions may be controlling factors in a  local situation.   The ability of multi--plant firms to make long distance decisions  is  closely tied to the effectiveness of channels of control  via  communication  and transportation facilities.   As  communication  improves  the management has the option to  centralize   decision  making,  thereby reducing the autonomy of each plant manager.  In  times   of   poorer  communication  major   decisions   regarding  enlargement  or  closing  of plants would have been made  at  the  headquarters of the central management.  A local community  finds  its fortunes very much in the hands of outlanders.   Three subtle  and disturbing aspects may characterize such a relationship.   In  the  first  place  the  central management may  act  in  what  it  believes  to  be  rational and moral purposes  in  closing  least  profitable  facilities  in  favor of expansion  in   areas  which  promise higher returns.  The overall result may be pernicious.  A  supermarket  chain operating under such rules may end up  closing  all  its  stores in the inner city in favor of  suburban  stores.   The  internal  firm reasons may make complete  sense;  close  the  oldest  facilities  on lots too small to accommodate  the  latest  technologies,  in  neighborhoods which have declining populations  and which do not yield high returns because of general low income  levels.   Inner city neighborhoods with older retired people  and  poverty  stricken  ethnic  groups,  losing  population  to  urban  renewal  or  expressway  construction end up losing  their  local  supermarket.   They are the least able to afford the loss.    The  decision  may be made in another city by outlanders  unresponsive  to  the  local  peoples' problems and with no  court  of  appeals  available.  A second difficulty for the local community with a plant owned by  an  international corporation is the policy of the corporation to  keep its young and most talented management moving from place  to  place in order that they can learn the business and eventually be  able to assume roles higher up in the corporate hierarchy.  It is  a   perfectly reasonable policy with respect to the internal firm  requirements.   The consequence,  however, is a cadre of talented  nomads who show little or no interest in the local welfare of the  community in which they are temporarily located.   Nor would  the  community  want  to commit political resources to such people  if  they expressed an interest.   They are simply removed from making  a  local  community  contribution which they  might  easily  have  pursued  had they been permanently in the  community.   The  only  loyalty  that  makes sense to them is  company  loyalty.   Higher  corporate management is certainly not going to  discourage this.  A  third  tendency  of  horizontal  cross--community  control  in  society  is  the homogeneity of facilities  and  company  policy.  Hierarchies   work  best  under  standard  operating  procedures.  Economies  of scale are possible,  substitution of  material  and  personnel  from  one locality to another are facilitated  if  the   installations  are all the same.   If disciplined standardization  and  routinization  has  been enforced top  management  can  make  broad,  basic  decisions secure in the knowledge  that  countless  local  exceptions  will  not  subvert  their  intent  during  the  implementation  phase.   But  what happens when accommodation  to  local  situations  is required.   You may get a  machine  answer,  that  request  will  not compute!'' or more  likely  the  local  manager  will say,  I sure would like to help you but my  hands  are  tied  by company policy." He may not be telling  the  truth.   The  impersonal  corporate  presence is an easy way  to  solve  a  problem by defining oneself out of any concern or responsibility.   Of  course,  he may be telling the truth but be as  powerless  to  change corporate policy as the outsider seeking accommodation.  \heading {We Are the Enemy}  Pogo said, We have met the enemy, and he is us'' [Kelly, 1972].   All   metropolitan areas are complex.   The Detroit region is  no  exception.  There  is no one to blame for the mess.   We are  the  enemy;  we are the city of strangers.   There is no single leader  or group,  either evil or benign to blame.   The land use pattern  grows from our decentralized decision processes.   The  decisions  which  actually affect local land use extend over time and  space  well beyond the here and now.  It is true the channels of control  could  be in the hands of evil doers and we could improve our lot  by exposing and removing them.  But I  think we are not generally  in  the hands of the unscrupulous;  not even in the hands of  the  stupid and insensitive.  It just appears that way.  Each decision  or  action  is  contingent upon conditions that  are  beyond  the  control  of the individual or group making a  particular  choice.   There  is  rarely  an instance where these  constraints  are  not  present.   The  outcome  often  seems  stupid  or  callous.  Most  deleterious  outcomes  are  probably  unanticipated.    They  are  indirect effects not thought of by the decision makers.   We need  to  understand our urban processes well enough to take action  to  avoid  effects  which  cause discomfort or  inequity  to  others.   Constraints on decisions may be classed into three groups.  There  are  institutional  and legal policies.  There are  physical  and  natural  environmental limitations which have to do with laws  of  nature  and  the  technological  capacities  with  which  we  may  accommodate to those laws.  And finally, there are limitations to  our aspirations and goals,  the imagined conditions that motivate  our   actions.    These  aspirations  are  not  hampered  by  any  finiteness of imagination in any single pursuit,  for we all know  flights  of  imagination  are  boundless.  Rather  limits  appear  because we harbor multiple needs which are often in conflict.  We  choose  to  restrain our objectives in one pursuit  in  order  to  achieve goals in other pursuits.   For example we find it hard to  have large lots and big lawns which provide us with seclusion and  status  and  at  the  same  time have  many  close  and  friendly  neighbors  which make available to us the pleasures and  security  of sharing a close community.   Under most circumstances to  gain  one value is to lose the other. \heading {Scale Attributes of Value Systems}  A  definition of values is that they are an individual's feelings  about   and  identification  with  things  and  people   in   his  environment.   Values have scale attributes.   Another three fold  classification is convenient.  There are {\it individual/familial  identification\/},  a  commitment to proxemic space --- the space  within  which one touches,  tastes and smells  things.   Secondly  there   is  {\it  community  identification\/},   embracing   the  individual's  feelings and concern for those with whom he or  she  lives and interacts,  not in the same house,  but in the vicinity  or  neighborhood.  This is local space generally recognizable  by  sight  and  smell.  Finally  there  is  {\it  political--cultural  identification\/}  which refers to ideals and concerns  extending  beyond  the people and community with which the person has  daily  contact.   This  realm must be dealt with abstractly and  through  instruments,  either  mechanical or institutional for it  is  too  large to be perceived by the  senses directly.   This is national  or global space.   Machine space and control by outlanders may be  viewed  as  intrusions into our community space by  organizations  and facilities of this  larger domain.   How they look,  sound or  smell  has  not  been taken into account in the  design  of  such  facilities.   Examples include Edison power stations,  the  Lodge  and Ford expressways, and Detroit  Metropolitan Airport.  We give  up local community values for the benefits of the global mobility  and interaction.   Metropolitan life pushes us to scale extremes.   We   value   individual  rights  and  perogatives  and   mainline  connections  with the global culture over familial and  community  concerns.  Intermediate  spatial  scale  values  suffer  and  the  community declines along with them.   The consequences are visual  blight,   noise  pollution,   reduced  security,  and  injustice.   Community values include concern for our fellow man,  a sense  of  equity  and humaneness.  The mechanisms for enforcing a community  code  of  ethics are ostracism,  social pressure and the  use  of  sense  of  humor  to keep people responding to  others  as  human  beings.   These   mechanisms  do  not  work well  in  a  city  of  strangers   and  are  not  followed.    They   are   particularly  ineffective in those large impersonal machine spaces, the streets  and  expressways,   bus  stations,  terminals and  warehouse  and  factory  districts.  The urban code of ethics carefully preserves  the  privacy of individuals and tolerates eccentrics.   A  person  has  functional but fragmented value and is valued  for  specific  tasks he or she can do.   A major problem with the dehumanization  and  anonymity  of urban life is that the unscrupulous are  freed  from social control along with the rest of us.   We have distinct  evidence  that  we are being ripped off" at both  ends  of  the  spatial  scale  of involvement.  Corporations manipulate  markets  through advertisements thereby creating artificial shortages  and  rapid obsolescence of their products without fear of being called  to account.   Radical monopolies in the words of Ivan Illich.  At  the  other extreme individuals,  free of local  control,  satisfy  their  wants  by committing violent criminal acts against  others  and  then  disappearing  into the  crowd.  Ostracism  and  social  pressure  work  between friends.   They are  meaningless  to  the  corporate manipulator and street criminal.  We  are  in  a crisis of conflicting values when  we  attempt  to  reform the structure of society to eliminate these problems.   We  tend  to throw the baby out with the bath water.  Action  against  crime in the streets and the home is moving  toward hardening our  shelters,  walling up windows,  barring doors,  hiring guards and  guard dogs,  and restricting access.   Security guards in Detroit  are  big business.   Even entering the Federal District Court  in  downtown  Detroit now requires a personal search.   These actions  are destructive of community spirit.   They are a falling back to  greater individual isolation.   Burglar proof apartments are more  effective against neighbors than against burglars (Figure 3). \topinsert \vskip22cm \noindent {\bf Figure 3.} Photographs of Detroit scenes by John D. Nystuen, c. 1974.
 \endinsert   We  have barely recognized the assault on our well being  through  manipulation by national corporations,  let alone  having devised  counter  measures.   The  major instruments of global  firms  are  standardization  and routinization.   And Detroit is a symbol  of  giant  multinational corporations and the  Henry  Ford--perfected  assembly line.  A defensive action of sorts is uncoupling part of  one's  life  from the national distribution system.   Making  and  using homemade products are countermeasures.   The great rise  in  home  crafts,  community garden projects,  potters'  guilds,  art  fairs and galleries and counter--culture craft shops provide some  vehicles  for humanizing city space and reestablishing a sense of  community.     College  youth are showing the way.   Wearing  old  work clothes everywhere, worn and patched (whether needed or not)  is  a  symbol of a society moving beyond  mass  consumption.   Of  course, as soon as old work clothes become {\it de rigueur\/} the  agents  of  mass production can reassert  themselves  by  selling  pre--patched garments.   Community values benefit most by seeking  simple  handmade  products.    The  craft shop and  modern  craft  guilds  should  be valued for their local  community  effect  and  should be supported because of their community value (Table 1). \midinsert \smallskip \hrule \smallskip

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

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].

\ref Abbot,   Edwin A.   (1956)  Flatland."  reprinted in  {\sl
The World  of  Mathematics\/},  James  R.  Newman,  editor.   New
York:   Simon and  Schuster.

\ref Barratt,  Krome  (1980)  {\sl Logic and Design:   The Syntax
of   Art,   Science,    and   Mathematics\/}.    Westfield,   NJ:
Eastview Editions, 1980.

\ref  Copi,   Irving M.   (1986)  {\sl Introduction to  Logic\/}.
Seventh  Edition.    New  York:   Macmillan  Publishing  Company,
(first edition, 1953).

\ref  Engel,   S.   Morris (1982)  {\sl With  Good  Reason:    An
Introduction  to Informal  Fallacies\/}.    Second Edition.   New
York:   St.  Martins  Press.

\ref  Kelley,    John  L.    (1963)   {\sl  General  Topology\/}.
Princeton:  D. Van Nostrand.

\ref Lach,  Alma  S.  (1974)   {\sl The Hows and Whys  of  French
Cooking\/},  Chicago:  The University of Chicago Press.

\ref  Mandelbrot,   Benoit  (1983)  {\sl The Fractal Geometry  of
Nature\/}.   San  Francisco:  Freeman.

\ref  Nicod,     Jean    (1969)   {\sl Geometry  and   Induction:
Containing  Geometry  in  the Sensible World' and  The  Logical
Problem  of   Induction'  with Prefaces by Roy Harrod,   Bertrand
Russell,   and  Andre   Lalande\/}.     London:    Routledge  and
Kegan   Paul,   New  translation.

\ref  Nystuen,   John   D.   (1963)   Identification  of   some
fundamental   spatial   concepts."    {\sl Papers   of   Michigan
Academy  of  Letters,  Sciences, and Arts\/}.  48: 373-384.

\ref  Robinson,   W.   (1950)  Ecological  correlations  and  the
behavior of  individuals,  {\sl American Sociological  Review\/}.
15: 351-357.

\ref  Rockwell,    John    (1990)     Fractals:    A    Mystery
Lingers."   Review/Music,  {\sl The New York Times\/},  Thursday,  April 26.  \ref Thompson, D'Arcy Wentworth (1956)  On Magnitude."  In {\sl  The World  of Mathematics\/},   James R.   Newman,   Editor.  New  York:  Simon and  Schuster.  \ref  Tufte,  Edward  (1989)  {\sl  Envisioning  Information\/}.   Cheshire, CT.  \ref  Upton,   Graham J.   G.  (1990) Information from Regional  Data,"  in   {\sl  Spatial  Statistics:     Past,   Present,  and  Future\/},  edited by Daniel  A.    Griffith.   IMaGe  Monograph,   \#12.   Ann  Arbor:   Michigan  Document Services.  \ref  {\sl Webster's  Seventh New Collegiate Dictionary\/} (1965)   Springfield,  MA:  G. and C. Merriam Company. \vfill\eject \centerline{\bf PARALLELS BETWEEN PARALLELS} \smallskip \centerline{\sl Sandra Lach Arlinghaus} \smallskip \smallskip \centerline{\it I have a little shadow that goes in and out with me,} \centerline{\it And what can be the use of him is more than I can see."} \smallskip \centerline{\sl Robert Louis Stevenson } \centerline{My Shadow" in {\sl A Child's Garden of Verses}}  {\narrower\smallskip{\bf Abstract}:       The earth's sun introduces a symmetry in  the  perception  of its trajectory in the sky that naturally  partitions  the  earth's surface into zones of affine and hyperbolic geometry.  The  affine zones, with single geometric  parallels,  are  located  north  and south of the geographic tropical parallels.  The hyperbolic  zone, with  multiple  geometric  parallels,  is  located   between   the geographic  tropical  parallels.   Evidence  of   this   geometric partition is  suggested in  the  geographic  environment---in  the design of houses and of gameboards.\smallskip}  \heading 1. Introduction.       Subtle influences shape our perceptions of  the  world.   The breadth of a world--view is a function not only of  real"--world experience, but also of the abstract"--world context within which that experience can be structured.  As  William  Kingdon  Clifford asked in his Postulates of the Science of Space [3], how  can  one recognize flatness when  magnification  of  the  landscape  merely reveals new wrinkles to traverse?       Geometry is a source of form" not only in mathematics [10], but also in the real" world [2].  Street patterns are geometric; architectural designs are geometric; and, diffusion  patterns  are geometric.  In this study, the geometric notion of parallelism  is examined in relation to the manner in which the  sun's  trajectory in  the  earth's  sky  is  observed  by  inhabitants  at   various latitudinal positions:  from north and south  of  the  tropics  to between  the  tropical  parallels  of  latitude.   A   fundamental geometrical notion is thus aligned with  fundamental  geographical and astronomical relationships; this alignment is  interpreted  in cultural contexts ranging from the  design  of  rooflines  to  the design of board games.  \heading 2.  Basic Geometric Background.        To  understand how geometry  might  guide  the   perception   of  form,  it  is  therefore   important   to   understand   what  geometry"  might be.   Projective geometry is totally symmetric  and  possesses  a completely dual" vocabulary:   points"  and  lines,"  collinear"  and concurrent," and a host of others,  are interchangeable terms  [6].   Indeed, a  Principle of Duality  serves as a linguistic axis,   or mirror, halving the  difficulty   of  proving  theorems.    Thus,  because two points determine a  line" is true, it follows, dually,  that  two  lines  determine   a  point"   is   also   true.   The  corresponding situation does  not hold in the Euclidean plane:   two  lines do not  necessarily  determine  a  point because parallel  lines  do not  determine  a  point [6].       Coxeter classifies other  geometries  as  specializations  of projective geometry based on the notion of parallelism,  depending on whether a geometry admits zero, one, or  more  than  one  lines parallel to a given line, through a point not on  the  given  line [6]. In the elliptic" geometry of Riemann, there are no parallel lines, much as there are none in the geometry of the  sphere  that includes great circles  as  the  only  lines,  any  two  of  which intersect at antipodal points.  In  affine"  geometry,  there is exactly one line parallel to a given line, through a point not  on that line.  Affine geometry is further subdivided  into  Euclidean and Minkowskian geometries. Finally, in the hyperbolic" geometry of Lobachevsky, there are at least two lines parallel to  a  given line through a point not on that line.        To  visualize,  intuitively,  the possibility of more  than   one   line parallel to a given line it is helpful  to  bend   the   lines,   sacrificing straightness" in order to retain the non-- intersecting  character of parallel lines.   Thus,  two  upward-- bending  lines $m$ and $m'$ passing through a point $P$ not on  a  given  line $\ell$ never intersect $\ell$;  they are  divergently  parallel to $\ell$ (Figure 1.a).  Or, one might imagine lines $m$  and $m'$ that are asymptotically parallel to $\ell$ (Figure  1.b)  [8]. \topinsert \vskip15cm

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

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

\topinsert \vskip15cm
{\bf Figure 2.}  The Poincar\'e Disk Model of the hyperbolic plane.

 \item{a.} The diameter,  $\ell$,  is a Poincar\'e line of the model, as are  arcs  $m$  and $m'$ which are orthogonal to the   boundary   $C$.    The   Poincar\'e  lines  $\ell$  and $m$  are  parallel  (do  not  intersect);  the  lines  $\ell$  and $m'$ are  not  parallel  (do  intersect). \item{b.} The sum of the angles of $\Delta OPQ$ is less than $180^{\circ}$.   The  triangle   is  formed  by  sides  $\ell$,   $m$,   $n$;  the  Poincar\'e  lines  $\ell$  and  $m$  are   diameters,   and   the  Poincar\'e line $n$ is  an  arc  of  a  circle  orthogonal to C. \item{c.} A  Lambert  quadrilateral with three right angles and  one  acute  angle $(PRQ)$.  Pairs of opposite sides are parallel. \endinsert        The  hyperbolic  plane  is represented as  the  disk,  $D$,   interior   to  an Euclidean circle  $C$.   Because  the  bounding  circle, $C$, is not included, the notion of infinity is suggested  by  choosing  points of $D$ closer and closer to this unreachable   boundary.    Points  in the hyperbolic plane correspond to points   in   $D$.     Lines   in   the  hyperbolic  plane  correspond  to   diameters   of  $D$  or  to  arcs  of circles orthogonal to  $C$.   These  arcs and diameters are referred to as Poincar\'e" lines.   Because  $C$ is not included in the model,  the endpoints of  the  Poincar\'e lines are not included,  suggesting  the notion of two  points  at  infinity.   Two Poincar\'e lines $\ell$ and  $m$  are  parallel  if and only if they have no common  point.   Thus,  the  disk diameter $\ell$ and the circular arc, $m$, orthogonal to $C$  are  parallel because they do not intersect;  however,  the  disk   diameter $\ell$ and the circular arc, $m'$, orthogonal to $C$ are  not parallel because they do intersect (Figure 2a).  \heading 5.  Hyperbolic Triangles and Quadrilaterals.       Any triangle in the hyperbolic plane is such that the sum of   its angles is less than $180^{\circ}$.   When a triangle is drawn  in  the   Poincar\'e model this becomes  quite  believable;  draw  Poincar\'e  lines   $\ell$  and $m$ as disk  diameters  and  draw  Poincar\'e  line  $n$ as an arc of a  circle  orthogonal  to  the   disk  boundary  (Figure  2b)  [8].   The  triangle formed in this  manner  has  one side  that  has caved--in"  suggesting how  it  happens that the angle sum can be less than $180^{\circ}$   (note  that three diameters cannot intersect in a triangle because   all   diameters   are   concurrent   at  the  center   of   the  disk).    Triangles  formed  from more than one Poincar\'e line that is  an  arc of a circle would become even more concave.      Because   all   triangles   have   angle   sum   less   than  $180^{\circ}$,  there  can be no rectangles (quadrilaterals  with  four   right  angles)  in the hyperbolic plane.  The  idea   that   corresponds   to   that  of  a rectangle is a quadrilateral  with  three  right  angles,   one  acute angle,  and pairs of  opposite  sides  parallel  (in  the  hyperbolic sense).   The sides,  $OP$,  $OQ$,  $PR$,  and  $RQ$,  of  this  quadrilateral  are  drawn  on  Poincar\'e lines that are segments of disk  diameters  or arcs of  circles  orthogonal  to  the outer circle  (Figure  2c;  $OQ$  is  parallel   to  $PR$  and  $RQ$  is  parallel  to   $PO$).    This  quadrilateral  is  called a Lambert  quadrilateral  after  Johann  Heinrich  Lambert  [8],  creator of  the   Lambert"   azimuthal   equal   area   map projection (among others) [12].   When such  a  quadrilateral  is   drawn  in  the Poincar\'e  model,  the  acute  angle  at  $R$  can  be  drawn to  suggest  that  its  sides  are  divergent,  asymptotic,  or intersecting.  Here, these sides have  been drawn to intersect (Figure 2c)  and  to  evidently  compress  the  angle at $R$ as a suggestion of the angular compression [12]  present  in  azimuthal  map  projections  (including   those   of  Lambert) around the projection center.  \heading 6.  Tiling the Hyperbolic Plane.        If one views a map grid as a tiling by  quadrilaterals   of   a  portion of the Euclidean plane, then it might  be  instructive   to  consider  a  tiling of the map"  of  the   Poincar\'e  disk  model  by  Lambert  and   other   quadrilaterals   [5].    Gluing   quadrilaterals together along Poincar\'e lines produces a variety  of  quadrilaterals (Figure 3).   All have pairs of opposite sides   parallel;  Poincar\'e  lines represented as arcs are   orthogonal   to   the   outer   circle.    Naturally,  the  tiling  can  never  completely  cover  the disk,  because the disk  boundary  is  not  included.   Thus,  tilings  of  this map have  quadrilaterals  of  shrinking  dimensions   as   the  outer  circle   is  approached.   This  permits  hyperbolic  tilings"  to  suggest the  infinite;  indeed,   they  have  served  as  artistic  inspiration  for  the  limitless" art of M. C. Escher [7]. \midinsert \vskip11cm

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

\item{a.} A   Saccheri   quadrilateral,   formed   from     two     Lambert  quadrilaterals.   It  has  two  right  angles   and   two   acute   angles.   Pairs of opposite sides are parallel,  as drawn in  the   Poincar\'e Disk Model. \item{b.} West  Sumatran  Minangkabau house.   Roofline is suggestive of  a  Saccheri quadrilateral.  Photograph by John D. Nystuen. \endinsert \vfill\eject       Games children play often  reveal  deeper  traditions  of  an entire society.  As the sun moves  through  its  entire  range  of possible positions, shadows dance across the full range of compass positions on Indonesian soil and come alive, as shadow puppets," in Indonesian theatrical productions.  Elegant cut--outs traced on goat skins and other hides are mounted on sticks and  dance  in  a plane of light between a single point--source and a screen, casting their filigreed, shadowy outlines high enough for all to see.  The motions of the Indonesian puppetteer are regulated by the world of projective geometry, with  shadows  stretching  out  diffuse  arms toward the infinite.       A  commonly played  Indonesian  board  game  is  Sodokan,"   a  variant of  checkers  [1].   Two  people  play  until  all  of   an   opponent's  ten  pieces,    arranged   initially   on    the   intersection  points of the last two lines of a $5\times 5$ board  (Figure 5a),  have been  captured.   Pieces move across the board  horizontally,  vertically,  or diagonally, one square at a  time.    What   is   unusual  is  the  method of  capture;   to  take   an   opponent's   marker   requires  a  surprise" attack  along  the  loops  outside  the  apparent natural grid of the gameboard. \topinsert \vskip11cm {\bf Figure 5.}

\item{a.} Sodokan  game  board in Euclidean space.   Markers  travel  along  lines   separating  regions  of  contrasting  color  and    along   circular loops at the corners. \endinsert       For example, with just two pieces remaining  (so  that  there are no intervening pieces), black may capture white  (Figure  5b). To do so, black must traverse at least one loop;  in  the  act  of capture, black can slide across as many open grid intersections as required to gain entry to a loop.  Then, still in the  same  turn, black slides around  the  loop,  re--enters  the  game  board, and continues to slide across grid intersections and  loops  until  an opponent's marker is reached, and therefore captured. \midinsert \vskip11cm {\bf Figure 5.}

Click here for Figure 5b. \item{b.} Sample of capture.  Black captures white---a single move. \endinsert      The  name,  Sodokan," means push out."   Its name  seems   to  apply  only loosely to the $5\times 5$ Euclidean  game  board  (Figure 5a) because the loops are   not,   themselves,   pushed   out" from  the natural  gameboard  grid.    If  they  were,   the   corners  of  the Euclidean grid would disappear.   However,  when  the  game  board  is drawn on a grid in the Poincar\'e disk model  of the hyperbolic plane (Figure 5c),  the loops appear  naturally   from   grid   intersections  outside the  circular  boundary.   A  marker   engaged   in  a   capture    on   this    non--Euclidean   (hyperbolic)   board   traverses   the  entire  hyperbolic  plane  (universe"),  passes  across  the infinite and  is  provided  a  natural avenue within the  system  for  return  to  the universe.   The  loops  are naturally pushed out" of the  underlying  grid,   tiled   partially   by   Lambert   quadrilaterals;   they   might  suggest  paths   along   which   gods   [11],   skipping   across   space,  interrupt  (sacrifice)  elements within  the  predictable  universe  of the life--space in the disk.   However,  independent   of  speculation  as  to what such paths  might  mean,  the   fact   remains   that  it  is within the hyperbolic geometric framework,  only,  that  this  game board emerges as a part of a natural grid  system.    Thus,   capture  is no longer a mysterious event  from   outside"  the system;   the change  in  theoretical  framework,   from   an  Euclidean   to   an hyperbolic viewpoint,  made  it  a  logical occurence. \topinsert \vskip20cm {\bf Figure 5.} \item{c.}

Sodokan  game  board  drawn on the Poincar\'e Disk Model  of  the  hyperbolic  plane.    The  four   central   quadrilaterals    are   Lambert    quadrilaterals---the    intersecting    versions    of  quadrilateral  $(OPQR)$  in  Figure  3.   When  their  sides  are  extended,  the   gameboard   loops are formed naturally by  these  grid lines  and  their  intersection points. \endinsert       A change in the underlying symmetry  introduced  order.   The meta" earth--sun system, when viewed as that  which introduces a symmetric  partition  of  the  earth   according   to   bands   of sun--delivered affine and hyperbolic  geometry,  offered  order in understanding roofline and gameboard shape  where  none  had  been apparent.       Sources of evidence for  other  similar  interpretations  are plentiful:  from Indonesian calendars based on a nested  hierarchy of cycles, to the loops within loops creating the syncopated forms characteristic of Indonesian gamelan music.   Perhaps  Indonesians and  other  between--the--parallels  dwellers   have  escaped  the asymmetric confines of Euclidean thought, enabling them to include a comfortable  vision  of  infinity  as  part  of  the  underlying symmetry of their daily circle of life. \vfill\eject \heading 8.  References.  \ref 1.  R. C. Bell, {\sl The Boardgame Book\/}  Open Court, New  York, 1983.  \ref 2.   William Wheeler Bunge,  {\sl Theoretical  Geography\/}  Lund Studies in Geography, ser. C, no. 1, Lund, 1966.  \ref  3.   William  Kingdon Clifford,   The postulates  of   the   science   of  space,  1873.   Reprinted  in {\sl  The  World  of   Mathematics\/} ed.  J.  R.  Newman, 552-567, Simon and Schuster,  New York,  1956.   [Portions also reprinted in {\sl Solstice\/},  Vol. I, No. 1, Summer, 1990.]  \ref  4.   Richard  Courant and Herbert Robbins,  {\sl  What  Is  Mathematics?\/} Oxford University Press, London, 1941.  \ref 5.   H.  S.  M.  Coxeter,  {\sl Introduction to Geometry\/}  Wiley, New York, 1961.  \ref  6.   H.  S.  M.  Coxeter,  {\sl Non--Euclidean Geometry\/}  University of Toronto Press, Toronto, 1965.  \ref 7.   Maurits C.  Escher, Circle Limit IV (Heaven and Hell),  woodcut, 1960.  \ref 8.  Marvin J. Greenberg, {\sl Euclidean and Non--Euclidean   Geometries:   Development  and History\/}  W.  H.  Freeman,  San  Francisco, 1974.  \ref  9.    R.  K.  Luneburg,  {\sl  Mathematical   Analysis  of  Binocular Vision\/} Princeton University Press, Princeton, 1947.  \ref  10.   Saunders  Mac  Lane,  {\sl  Mathematics:   Form  and  Function\/} Springer, New York, 1986.  \ref 11.  John D. Nystuen, Personal communication, 1989.  \ref 12.    J.   A.   Steers,    {\sl An  Introduction  to   the   Study  of  Map Projections\/} London University  Press,  London,  1962.  \ref  13.    Raymond  L.   Wilder,   {\sl Introduction  to   the   Foundations of Mathematics\/} New York:  Wiley, New York, 1961.  \heading Acknowledgment       The author wishes to thank John D. Nystuen for  his  kindness in sharing information, concerning various aspects  of  Indonesian culture,  gathered  in  field  work.   Nystuen  pointed  out   the connection between  West Sumatran,  Minangkabau  house--types  and Saccheri quadrilaterals, and taught the author and others to  play the board game he had learned of in Indonesia.  The photograph  of the West Sumatran house was taken by Nystuen and appears here with his permission.     She also wishes to thank  Istv\'an Hargittai of the  Hungarian Academy  of  Sciences  and  Arthur  Loeb of Harvard University for earlier efforts with this manuscript; this  paper  was  originally accepted by {\sl Symmetry\/}---Dr. Hargittai  was Editor  of  that journal  and  Professor Loeb  was  the  Board  member  of that now defunct journal who communicated this work to Hargittai. The paper appears here exactly as it was communicated to {\sl Symmetry\/}. \vfill\eject \centerline{\bf THE HEDETNIEMI MATRIX SUM:  A REAL--WORLD APPLICATION} \smallskip \centerline{\sl Sandra L. Arlinghaus, William C. Arlinghaus, John D. Nystuen.} \smallskip  In  a  recent paper,  we presented an algorithm for  finding  the  shortest distance between any two nodes in a network of $n$ nodes  when  given only distances between  adjacent  nodes  [Arlinghaus,  Arlinghaus,  Nystuen,  1990(b)].   In that previous research,  we  applied  the  algorithm  to the generalized  road  network  graph  surrounding  San  Francisco  Bay.   The  resulting  matrices  are  repeated here (Figure 1),  in order to examine consequent changes  in  matrix  entries when the underlying adjacency pattern of  the  road  network was altered by the 1989 earthquake that closed  the  San Francisco--Oakland Bay Bridge.   Thus,  we test the algorithm  against  a  changed  adjacency configuration  and  interpret  the  results  with  the  benefit of hindsight from  an  actual  event.  Figure 1 shows a graph, with edges weighted with time--distances,  representing  the  general  expressway  linkage  pattern  joining  selected  cities surrounding San Francisco Bay.   The matrix  $A$  displays  these  time--distances in  tabular  form;  an  asterisk  indicates  that there is no direct linkage between  corresponding  entries.   Thus,  an  asterisk  in entry $a_{13}$ indicates  that  there  is no single edge of the graph linking San  Francisco  and  San Jose (all paths have 2 or more edges).   Higher powers of the  matrix $A$ count numbers of paths of longer length---$A^2$ counts  paths of 2 edges as well as those of one edge.  Thus, one expects  in  $A^2$  to see a number measuring time--distance  between  San  Francisco and San Jose;  indeed, there are two such paths, one of  length  30+50=80,  and  one of length 30+25=55.   The  Hedetniemi  matrix operator always selects the shortest.   Readers wishing to  understand  the mechanics of this algorithm should refer  to  the  other  references  related to this topic in the list at  the  end  [Arlinghaus,  Arlinghaus,  and Nystuen;  W.  Arlinghaus].   It is  sufficieint here simply to understand generally how the procedure  works, as described above.       When  a recent earthquake caused a disastrous collapse of  a  span  on  the  San Francisco--Oakland  Bay  Bridge,  forcing  the  closing of the bridge, municipal authorities managed  to keep the  city  moving  using a well--balanced combination of  added  ferry  boats,  media  messages urging people to stay off the roads,  and  dispersal  of information concerning alternate route  strategies.  National  telecasts  showed a city on the  move,  albeit  slowly,  although  outside forecasters of doom were predicting  a  massive  grid--lock  that  never  occured.    What  would  the  Hedetniemi  algorithm have forecast in this situation?       To find out, we compare the matrices of Figure 1 to those of  Figure  2,  derived  from  the graph of Figure 1  with  the  link  between  San Francisco and Oakland removed;  that  is,  the  edge  linking  vertex 4 to vertex 1 is removed --- the results show  in  the matrix entries $a_{14}$ and $a_{41}$.  Thus in Figure 2,  the  adjacency  matrix $A$,  describing 1--step edge linkages  differs  from  that of Figure 1 only in the $a_{14}$ ($a_{41}$)  position.   The  value of * replaces the time--distance of 30 minutes in that  graph because the bridge connection was destroyed.   When 2--edge  paths are counted,  there is spread of increased  time--distances  across these paths, as well.  What used to take 30 minutes, under  conditions of normal traffic, to go from San Francisco to Oakland  now takes 70 minutes,  under conditions of normal traffic,  going  by way of San Mateo.  The trip from San Francisco to Walnut Creek  had  been  possible along a 2--edge path passing through  Oakland  (and taking a total of 60 minutes);  the asterisk in $A^2$ in the  $a_{15}$  entry indicates that that path no longer  exists.   The  journey  from San Francisco to Richmond,  along a  2--edge  path,  increased  in time--distance from 50 to 60 minutes---going around  the  longer" side of the rectangle.   Note that what  is  being  evaluated   here   is  change  in  trip--time   under   normal"  circumstances,  according  to  whether  or  not  routing  exists;  congestion  fluctuates  but actual road lengths do not  (once  in  place).  These values therefore  form a set of benchmarks against  which  to  measure  time--distance changes  resulting  from  more  variable quantities, such as increased congestion.       When  three--edged  paths are brought into  the  system,  in  $A^3$ (Figure 2), the trip from San Francisco to Walnut Creek now  becomes possible, but takes 100 rather than 60 minutes. Also, the  trip from San Francisco to Vallejo now becomes  possible (in both  pre-- and  post--earthquake systems) although it takes 10 minutes  longer with removal of the bridge.  When paths of length four are  introduced,  no  changes occur in these entries;  the  system  is  stable and the effects are confined to locations close'' to the  bridge that was removed.   The relatively small number of changes  in  the basic underlying route choices,  forced by the removal of  the  Bay Bridge,  suggest {\bf why} it was possible,  with  swift  action   by  municipal  authorities  and  citizens   to   control  congestion,  to  avert a situation that appeared destined to lead  to gridlock.       What if the Golden Gate Bridge had been removed rather  than  the  San Francisco--Oakland Bay Bridge?   Figure 3 shows that the  same  sort of clustered,  localized results  follow.   When  both  bridges  are removed (Figure 4),  the position of affected matrix  entries is identical to the union of the positions of entries  in  Figures  1 and 2,  but the magnitude of time--distances has  been  magnified by the combined removal.  With hindsight,  the test seems to be reasonable.  One  direction  for   a  larger  application  might  therefore  be  to   consider  historical  evidence  in which bridge bombing (or some such)  was  critical  to associated circulation patterns.   When  large  data  sets  are  entered  into a computer,  and manipulated  using  the  Hedetniemi  matrix  algorithm,  previously  unnoticed  historical  associations   might   emerge   and   maps   showing    alternate  possibilities could be produced.  In short, this might serve as a  tool useful in historical discovery.   Other important directions  for  application  of the Hedetniemi algorithm involve those in  a  discrete mathematical setting that focus on tracing actual  paths  [W. Arlinghaus, 1990---includes program for algorithm], and those  using  the  Hedetniemi algorithm in the computer architecture  of  parallel processing [Romeijn and Smith]. \vfill\eject


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


Click here for Figure 2, graph.

Click here for Figure 2, matrix.

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


Click here for Figure 3, graph.

Click here for Figure 3, matrix.

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


Click here for Figure 4, graph.

Click here for Figure 4, matrix.

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

\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}

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.

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?

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
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

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

\endinsert

If  both geometric environments are  then viewed as  composed  of
these  highly--crenulated  elements   (which do fit  together  to
cover the plane), then the hexagonal environment is the one  that
permits infinite iteration  without loss  of  all  pixel content.
This approach is akin to that  of Barnsley, which stores sets  of
transformations that are used   to  drive image production.  What
is suggested here  is  a  possible  way to vastly  improve  image
resolution corresponding, to some extent, to Barnsley's successful
strategy to improve data compression (Barnsley).

This  approach  is also similar,  in  general  strategy  to  that
employed  by  Hall  and  G\"okmen;  both  seek   transformations,
applied   in  an  electronic  environment,   under   which   some
properties   are   preserved.   Hall   and  G\"okmen   focus   on
transformations  linking  hexagonal and rectangular  pixel  space
whereas  the  transformations  employed  here  function  entirely
within  a single type of geometric environment (using one on  the
other appears to be of interest).   Additionally,  this  approach
offers  a systematic characterization,  in the infinite,  for the
aggregate   7--kernels  of  hexagons,   at  various   levels   of
aggregation,  suggested  only  as finite sequences in Gibson  and
Lucas.   Finally,  Tobler's  maps of Swiss migration patterns  at
three  levels  of  spatial resolution  suggest  a  methodological
handle  of an attractivity function to implement ideas  involving
spatial   resolution  in  an  electronic   environment.    Deeper
analysis, of the sort represented in the works mentioned here, is
beyond the scope of this particular short piece.

Table 1
shows a set of fractal dimensions for selected L\"oschian numbers.
\midinsert

Click here for Table 1. 
TYPESETTING THAT PRODUCED TABLE 1.
\smallskip \hrule \smallskip \centerline{ \bf Table 1} \centerline{(derived from a Table in Arlinghaus and Arlinghaus, 1989)} \settabs\+&$K=3,\,D=1.262$;\quad&$K=12,\,D=1.116$;\quad&$K=27,\,D=1.087$;\quad           &$K=49,\,D=1.074$&$\ldots$&\cr \+&K=3, D=1.262;&K=12, D=1.116;&K=27, D=1.087;&K=48, D=1.074;&$\ldots$\cr \+&K=7, D=1.129;&K=19, D=1.093;&K=37, D=1.078;&K=61, D=1.069;&$\ldots$\cr \+&K=4, D=1.585;&K=13, D=1.255;&K=28, D=1.168;&K=49, D=1.129;&$\ldots$\cr \smallskip \hrule \smallskip \endinsert
The line of L\"oschian numbers that begins with $K=4$, those that
are  organized according to an transportation"  principle,  are
the  ones   that   fill  two  dimensional  space   most  thickly.
Thus,    when   introducing   smaller   and   smaller   hexagonal
cells    to    improve   resolution  in  the  quality  of   curve
representation,   or   when  zooming  in,"  it   would   appear
appropriate  to  let  the  orientation of  successive  layers  of
smaller   and  smaller  cells correspond to the  $K=4$  type   of
hierarchy.   Clutter would not enter as fast as in the  Manhattan
environment, even in this densest arrangement.  Super,"  rather
than  high,"  definition of resolution   could  therefore  fall
naturally    from  an  underlying  hexagonal  pixel geometry with
measures  of  clutter  and information  content  determined using
fractal dimensions.

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.

\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:

\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 $y=x^2+c$.


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


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

\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.

\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