\hsize = 6.5 true in %THE FIRST 14 LINES ARE TYPESETTING CODE. \input fontmac %delete to download, except on Univ. Mich. (MTS) equipment. \setpointsize{12}{9}{8} \parskip=3pt \baselineskip=14 pt \mathsurround=1pt \headline = {\ifnum\pageno=1 \hfil \else {\ifodd\pageno\righthead   \else\lefthead\fi}\fi} \def\righthead{\sl\hfil SOLSTICE } \def\lefthead{\sl Summer, 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 \centerline{\big SOLSTICE:} \vskip.5cm \centerline{\bf  AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS} \vskip5cm \centerline{\bf SUMMER, 1990} \vskip12cm \centerline{\bf Volume I, Number 1} \smallskip \centerline{\bf Institute of Mathematical Geography} \vskip.1cm \centerline{\bf Ann Arbor, Michigan} \vfill\eject \hrule \smallskip \centerline{\bf SOLSTICE} \line{Founding Editor--in--Chief:  {\bf Sandra Lach Arlinghaus}. \hfil} \smallskip \centerline{\bf EDITORIAL BOARD} \smallskip \line{{\bf Geography} \hfil} \line{{\bf Michael Goodchild}, University of California, Santa Barbara. \hfil} \line{{\bf Daniel A. Griffith}, Syracuse University. \hfil} \line{{\bf Jonathan D. Mayer}, University of Washington; joint appointment  in School of Medicine.\hfil} \line{{\bf John D. Nystuen}, University of Michigan (College of  Architecture and Urban Planning).} \smallskip \line{{\bf Mathematics} \hfil} \line{{\bf William C. Arlinghaus}, Lawrence Technological University. \hfil} \line{{\bf Neal Brand}, University of North Texas. \hfil} \line{{\bf Kenneth H. Rosen}, A. T. \& T. Information Systems Laboratory.        \hfil} \smallskip \line{{\bf Business} \hfil} \line{{\bf Robert F. Austin}, Director, Automated Mapping and Facilities Management, CDI. \hfil} \smallskip \hrule \smallskip        The  purpose of {\sl Solstice\/} is to promote  interaction   between geography and mathematics.   Articles in which  elements  of   one  discipline  are used to shed light on  the  other  are   particularly sought.   Also welcome,  are original contributions   that are purely geographical or purely mathematical.   These may   be  prefaced  (by editor or author) with  commentary  suggesting   directions  that  might  lead toward  the  desired  interaction.    Individuals  wishing to submit articles,  either short or full--  length,  as well as contributions for regular  features,  should   send  them,  in triplicate,  directly to the  Editor--in--Chief.    Contributed  articles  will  be refereed by  geographers  and/or   mathematicians.   Invited articles will be screened by  suitable   members of the editorial board.  IMaGe is open to having authors   suggest, and furnish material for, new regular features. \vskip2in \noindent {\bf Send all correspondence to:} \vskip.1cm \centerline{\bf Institute of Mathematical Geography} \centerline{\bf 2790 Briarcliff} \centerline{\bf Ann Arbor, MI 48105-1429} \vskip.1cm \centerline{\bf (313) 761-1231} \centerline{\bf IMaGe@UMICHUM} \vfill\eject        This  document is produced using the typesetting  program,  {\TeX},  of Donald Knuth and the American Mathematical  Society.   Notation  in  the electronic file is in accordance with that  of  Knuth's   {\sl The {\TeX}book}.   The program is downloaded  for  hard copy for on The University of Michigan's Xerox 9700 laser-- printing  Xerox machine,  using IMaGe's commercial account  with  that University.  Unless otherwise noted, all regular features are written by the Editor--in--Chief. \smallskip       {\nn  Upon final acceptance,  authors will work with IMaGe  to    get  manuscripts   into  a  format  well--suited  to   the  requirements   of {\sl Solstice\/}.  Typically,  this would mean  that  authors    would  submit    a  clean  ASCII  file  of  the  manuscript,  as well as   hard copy,  figures,  and so forth (in  camera--ready form).     Depending on the nature of the document  and   on   the  changing    technology  used  to  produce   {\sl  Solstice\/},   there  may  be  other    requirements  as   well.   Currently,  the  text  is typeset using   {\TeX};  in that  way,  mathematical formul{\ae} can be transmitted   as ASCII files and  downloaded   faithfully   and   printed   out.    The     reader  inexperienced  in the use of {\TeX} should note that  this    is  not  a what--you--see--is--what--you--get"  display;  however,    we  hope  that  such readers find {\TeX} easier to  learn  after    exposure to {\sl Solstice\/}'s e-files written using {\TeX}!}        {\nn  Copyright  will  be taken out in  the  name  of  the  Institute of Mathematical Geography, and authors are required to  transfer  copyright  to  IMaGe as a  condition  of  publication.   There  are no page charges;  authors will be given permission to  make reprints from the electronic file,  or to have IMaGe make a  single master reprint for a nominal fee dependent on  manuscript  length.   Hard  copy of {\sl Solstice\/}  will be sold  (contact  IMaGe  for  price--{\sl Solstice\/}  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, August, 1990, Institute of Mathematical Geography. All rights reserved. \vskip1cm ISBN: 1-877751-51-0 \vfill\eject \centerline{\bf SUMMARY OF CONTENT} \smallskip Note:  in this first issue, there is one of each type of article--this need not be the case in the future. \vskip.2cm 1. REPRINT. \smallskip William Kingdon Clifford, {\sl Postulates of the Science of Space\/}.       This  reprint  of a portion of Clifford's lectures  to  the  Royal  Institution in the 1870's suggests many geographic topics  of concern in the last half of the twentieth century.   Look for  connections  to boundary issues,  to scale problems,  to  self-- similarity and fractals,  and to non--Euclidean geometries (from  those  based on denial of Euclid's parallel postulate  to  those  based on a sort of mechanical polishing").  What else did,  or  might, this classic essay foreshadow? \smallskip 2.  FULL--LENGTH ARTICLE. \smallskip Sandra L. Arlinghaus, {\sl Beyond the Fractal\/}. Figures are transmitted in this e-file only for the half of the article described in the first paragraph below.       An   original  article.    The  fractal  notion  of  self-- similarity  is useful for characterizing change  in  scale;  the  reason  fractals are effective in the geometry of central  place  theory  is  because  that geometry is  hierarchical  in  nature.   Thus, a natural place to look for other connections of this sort  is  to  other geographical concepts that are also  hierarchical.   Within this fractal context,  this article examines the case  of  spatial diffusion.       When  the idea of diffusion is extended to see  adopters"  of  an innovation as attractors" of new adopters,  a Julia set  is  introduced as a possible axis against which to  measure  one  class  of  geographic phenomena.   Beyond the  fractal  context,  fractal  concepts,  such as compression" and space--filling"  are  considered in a broader graph--theoretic context. \smallskip 3.  SHORT ARTICLE. \smallskip William C. Arlinghaus, {\sl Groups, graphs, and God\/}        An original article based on a talk given before a MIdwest  GrapH TheorY (MIGHTY) meeting.   The author,  an algebraic graph  theorist, ties his research interests to a broader philosophical  realm,  suggesting  the  breadth  of range  to  which  algebraic  structure might be applied.        The  fact that almost all graphs are rigid  (have  trivial  automorphism  groups)  is exploited to argue   probabilistically   for the existence of God.  This  is  presented  in  the  context   that  applications  of mathematics need  not   be   limited   to   scientific ones. \smallskip  Note:   In  this  first  issue,  there is one of each  type  of  article--this need not be the case in the future. \smallskip 4.  REGULAR FEATURES \smallskip \item{i.} {\bf Theorem Museum} ---     Desargues's Two Triangle Theorem of projective geometry. \item{ii.} {\bf Construction Zone} ---     a  centrally symmetric hexagon is derived from an  arbitrary      convex hexagon. \item{iii.} {\bf Reference Corner} ---      Point set theory and topology. \item{iv.} {\bf Games and other educational features} ---      Crossword puzzle focused on spices. \item{v.} {\bf Coming attractions} ---      Indication of topics for the REGULAR FEATURES" section      in forthcoming issues. \smallskip \item{vi.}{\bf Solution to puzzle} \smallskip 5.  SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE \smallskip      This section shows the exact set of commands that  work  to  download this file on The University of Michigan's  Xerox  9700.   Because different universities will have different installations  of {\TeX}, this is only a rough guideline which {\sl might\/} be  of use to the reader or to the reader's computing center. \vfill\eject \noindent 1.  REPRINT \smallskip \centerline{\bf THE POSTULATES OF THE SCIENCE OF SPACE} \vskip.2cm \centerline{William Kingdon Clifford} \smallskip From a set of lectures given before the Royal Institution, 1873 -- The Philosophy of the Pure Sciences."  Reprinted excerpt longer than this one appears in  {\sl The  World of Mathematics\/}, edited by James R. Newman, New York:   Simon and Schuster, 1956. \smallskip \noindent   In my first lecture I said that,  out of the pictures which are all that we can really see, we imagine a world of solid things;  and  that  this world is constructed so as to  fulfil  a certain  code  of rules,  some called  axioms,  and  some  called definitions, and some called postulates, and some assumed  in the course of demonstration, but all laid down in one form or another in Euclid's Elements of Geometry.   It is this code of rules that   we have to consider to--day.   I do not, however, propose to take this book that I have mentioned, and to examine one after another the  rules as Euclid has laid them down or unconsciously  assumed them; notwithstanding that many things might be said in favour of such  a  course.   This  book  has been  for  nearly  twenty--two centuries the encouragement and guide of that scientific  thought which  is  one thing with the progress of man from a worse  to  a better  state.   The  encouragement;  for it contained a body  of knowledge that was really known and could be relied on,  and that moreover was growing in extent and application.   For even at the time this book was written---shortly after the foundation of  the Alexandrian  Museum--Mathematic  was no longer the  merely  ideal science of the Platonic school,  but had started on her career of conquest over the whole world of Phenomena.   The guide;  for the aim of every scientific student of every subject was to bring his knowledge  of  that subject into a form as perfect as that  which geometry  had attained.   Far up on the great mountain of  Truth, which all the sciences hope to scale, the foremost of that sacred sisterhood was seen,  beckoning to the rest to follow  her.   And hence  she was called,  in the dialect of the Pythagoreans,  the purifier  of the reasonable soul.'  Being thus in itself at  once the  inspiration and the aspiration of scientific  thought,  this Book of Euclid's has had a history as chequered as that of  human progress itself.  [Deleted text.]  The geometer of to--day  knows nothing  about  the  nature  of actually  existing  space  at  an infinite distance;  he knows nothing about the properties of this present space in a past or a future eternity.   He knows, indeed, that the laws assumed by Euclid are true with an accuracy that no direct experiment can approach,  not only in this place where  we are,  but  in places at a distance from us that no astronomer has conceived; but he knows this as of Here and Now; beyond his range is a There and Then of which he knows nothing at present, but may ultimately  come to know more.   So,  you see,  there is  a  real parallel between the work of Copernicus and his successors on the one hand,  and the work of Lobatchewsky and his successors on the other.   In both of these the knowledge of Immensity and Eternity is replaced by knowledge of Here and Now.  And in virtue of these two revolutions the idea of the Universe, the Macrocosm, the All, as  subject of human knowledge,  and therefore of human interest, has fallen to pieces.       It will now,  I think, be clear to you why it will not do to take for our present consideration the postulates of geometry  as Euclid  has laid them down.   While they were all certainly true, there  might  be  substituted  for  them  some  other  group   of equivalent propositions;  and the choice of the particular set of  statements  that should be used as the groundwork of the  science was  to  a  certain  extent  arbitrary,   being  only  guided  by convenience  of exposition.   But from the moment that the actual truth  of  these  assumptions  becomes  doubtful,  they  fall  of themselves into a necessary order and classification; for we then begin  to  see  which of them may be true  independently  of  the others.   And  for  the purpose of criticizing the  evidence  for them,  it is essential that this natural order should  be  taken; for  I  think you will see presently that any other  order  would   bring hopeless confusion into the discussion.       Space  is divided into parts in many ways.   If we  consider any material thing,  space is at once divided into the part where that  thing is and the part where it is not.   The water in  this glass,  for example,  makes a distinction between the space where it is and the space where it is not.   Now,  in order to get from one  of these to the other you must cross the {\it surface\/}  of the  water;  this surface is the boundary of the space where  the water  is  which  separates it from the space where  it  is  not. Every {\it thing\/},  considered as occupying a portion of space, has  a  surface which separates that space where it is  from  the space where it is not.  But, again, a surface may be divided into parts  in  various ways.   Part of the surface of this  water  is against  the air,  and part is against the glass.   If you travel over the surface from one of these parts to the other,  you  have to cross the {\it line\/} which divides them; it is this circular edge where water,  air,  and glass meet.  Every part of a surface is separated from the other parts by a line which bounds it.  But now  suppose,  further,  that this glass had been so  constructed that  the  part towards you was blue and the part towards me  was white,  as it is now.   Then this line, dividing two parts of the surface  of the water,  would itself be divided into  two  parts; there would be a part where it was against the blue glass,  and a part  where  it was against the white glass.   If you  travel  in thought along that line, so as to get from one of these two parts to  the other,  you have to cross a {\it point\/} which separates them,  and is the boundary between them.  Every part of a line is separated from the other parts by points which bound it.   So  we may say altogether --- \vskip.1cm The boundary of a solid ({\it i.e.\/},  of a part of space) is a surface. \vskip.1cm The boundary of a part of a surface is a line. \vskip.1cm The boundaries of a part of a line are points.       And  we are only settling the meanings in which words are to be  used.   But here we may make an observation which is true  of all  space that we are acquainted with:   it is that the  process ends  here.   There are no parts of a point which  are  separated from  one another by the next link in the series.   This is  also indicated by the reverse process.       For  I shall now suppose this point --- the last thing  that we   got  to --- to move round the tumbler so as to trace out  the line,  or edge,  where air, water, and glass meet.  In this way I get  a series of points,  one after another;  a series of such  a nature that,  starting from any one of them, only two changes are possible  that  will  keep it within  the  series:   it  must  go forwards or it must go backwards,  and each of these if perfectly definite.   The  line  may  then be regarded as an  aggregate  of points.   Now let us imagine,  further, a change to take place in this  line,  which  is nearly a circle.   Let us  suppose  it  to   contract  towards  the  centre of the circle,  until  it  becomes indefinitely small,  and disappears.   In so doing it will  trace out the upper surface of the water, the part of the surface where it  is  in   contact with the air.   In this way we shall  get  a series of circles one after another --- a series of such a nature that,  starting  from  any  one of them,  only  two  changes  are possible that will keep it within the series:   it must expand or it must contract.   This series,  therefore,  of circles, is just similar to the series of points that make one circle; and just as the line is regarded as an aggregate of points,  so we may regard this surface as an aggregate of lines.   But this surface is also in another sense an aggregate of point,  in being an aggregate of aggregates of points.  But, starting from a point in the surface, more  than two changes are possible that will keep it within  the surface, for it may move in any direction.  The surface, then, is an  aggregate of points of a different kind from  the  line.   We speak  of  the  line  as a  point--aggregate  of  one  dimension, because,  starting  from one point,  there are only two  possible directions  of change;  so that the line can be traced out in one motion.   In the same way,  a surface is a line--aggregate of one dimension,  because  it  can be traced out by one motion  of  the line; but it is a point--aggregate of two dimensions, because, in order to build it up of points, we have first to aggregate points into  a line,  and then lines into a surface.   It  requires  two motions of a point to trace it out.        Lastly,  let  us suppose this upper surface of the water to move downwards,  remaining always horizontal till it becomes  the under  surface.   In so doing it will trace out the part of space occupied  by the water.   We shall thus get a series of  surfaces one  after another,  precisely analogous to the series of  points which make a line,  and the series of lines which make a surface. The  piece  of solid space is an aggregate of  surfaces,  and  an aggregate  of  the same kind as the line is of points;  it  is  a surface--aggregate of one dimension.   But at the same time it is a  line--aggregate of two dimensions,  and a point--aggregate  of three  dimensions.   For if you consider a particular line  which has gone to make this solid,  a circle partly contracted and part of the way down,  there are more than two opposite changes  which it  can  undergo.   For it can ascend or descend,  or  expand  or contract,  or do both together in any proportion.  It has just as great  a  variety of changes as a point in a  surface.   And  the piece  of space is called a point--aggregate of three dimensions, because  it takes three distinct motions to get it from a  point. We  must first aggregate points into a line,  then lines  into  a surface, then surfaces into a solid.       At this step it is clear,  again, that the process must stop in all the space we know of.  For it is not possible to move that piece  of  space  in such a way as to change every point  in  it. When  we moved our line or our surface,  the new line or  surface contained no point whatever that was in the old one;  we  started with one aggregate of points, and by moving it we got an entirely new aggregate, all the points of which were new.  But this cannot be  done with the solid;  so that the process is at an  end.   We   arrive,  then,  at  the  result  that  {\it  space  is  of  three dimensions\/}.        Is  this,  then,  one of the postulates of the  science  of space?   No;  it  is not.   The science of space,  as we have it, deals  with relations of distance existing in a certain space  of three  dimensions,  but it does not at all require us  to  assume that  no relations of distance are possible in aggregates of more than  three  dimensions.   The  fact that there  are  only  three dimensions does regulate the number of books that we  write,  and the  parts of the subject that we study:   but it is not itself a postulate  of  the science.   We investigate a certain  space  of three  dimensions,   on  the  hypothesis  that  it  has   certain elementary  properties;  and  it  is  the  assumptions  of  these elementary properties that are the real postulates of the science of space.  To these I now proceed.       The first of them is concerned with {\it points\/}, and with the  relation  of  space  to them.   We spoke of  a  line  as  an aggregate  of  points.   Now there are two kinds  of  aggregates, which  are called respectively continuous and discrete.   If  you consider  this line,  the boundary of part of the surface of  the water,  you  will  find yourself believing that between  any  two points of it you can put more points of division, and between any two of these more again,  and so on; and you do not believe there can be any end to the process.  We may express that by saying you believe  that  between  any two points of the line  there  is  an infinite number of other points.  But now here is an aggregate of marbles,  which, regarded as an aggregate, has many characters of resemblance  with  the aggregate of points.   It is a  series  of marbles,  one  after  another;  and if we take into  account  the relations  of  nextness or contiguity which  they  possess,  then there are only two changes possible from one of them as we travel along  the series:   we must go to the next in front,  or to  the next behind.  But yet it is not true that between any two of them here  is an infinite number of other marbles;  between these two, for example, there are only three.  There, then, is a distinction at  once  between  the two kinds of  aggregates.   But  there  is another,  which  was pointed out by Aristotle in his Physics  and made the basis of a definition of continuity.   I have here a row of  two different kinds of marbles,  some white and  some  black. This aggregate is divided into two parts, as we formerly supposed the line to be.  In the case of the line the boundary between the two parts is a point which is the element of which the line is an aggregate.   In this case before us, a marble is the element; but here  we cannot say that the boundary between the two parts is  a  marble.   The boundary of the white parts is a white marble,  and the  boundary  of the black parts is a black  marble;  these  two adjacent parts have different boundaries.   Similarly, if instead of  arranging  my  marbles in a series,  I spread them out  on  a surface,  I may have this aggregate divided into two portions --- a  white  portion and a black portion;  but the boundary  of  the white portion is a row of white marbles,  and the boundary of the black portion is a row of black marbles.  And lastly, if I made a heap of white marbles, and put black marbles on the top of  them,   I  should have a discrete aggregate of three  dimensions  divided into two parts:   the boundary of the white part would be a layer of  white marbles,  and the boundary of the black part would be a layer  of  black  marbles.    In  all  these  cases  of  discrete aggregates,  when  they  are  divided into  two  parts,  the  two adjacent  parts have different boundaries.   But if you  come  to consider an aggregate that you believe to be continuous, you will see  that  you  think of two adjacent parts as  having  the  {\it same\/}  boundary.   What  is the boundary between water and  air here?   Is  it water?   No;  for there would still have to  be  a boundary to divide that water from the  air.  For the same reason it  cannot be air.   I do not want you at present to think of the actual  physical facts by the aid of any  molecular  theories;  I want  you  only  to  think of what appears to  be,  in  order  to understand  clearly a conception that we all have.   Suppose  the things actual in contact.   If,  however much we magnified  them, they  still  appeared  to be thoroughly  homogeneous,  the  water filling up a certain space,  the air an adjacent space;  if  this held   good  indefinitely  through  all  degrees  of  conceivable magnifying,  then we could not say that the surface of the  water was  a layer of water and the surface of air a layer of  air;  we should  have to say that the same surface was the surface of both of  them,  and was itself neither one nor the  other---that  this surface occupied {\it no\/} space at all.  Accordingly, Aristotle defined  the continuous as that of which two adjacent parts  have the  same boundary;  and the discontinuous or discrete as that of which two adjacent parts have direct boundaries.        Now  the  first postulate of the science of space  is  that space  in  a continuous aggregate of points,  and not a  discrete aggregate.  And this postulate---which I shall call the postulate of  continuity---is  really involved in those three  of  the  six postulates  of  Euclid for which Robert Simson has  retained  the name of postulate.   You will see, on a little reflection, that a discrete  aggregate  of points could not be so arranged that  any two  of  them  should be relatively situated to  one  another  in exactly the same manner,  so that any two points might be  joined by  a  straight line which should always bear the  same  definite relation  to them.   And the same difficulty occurs in regard  to the other two postulates.  But perhaps the most conclusive way of showing  that  this postulate is really assumed by Euclid  is  to adduce the proposition he probes, that every finite straight line may be bisected.   Now this could not be the case if it consisted of  an  odd   number of separate points.   As the  first  of  the postulates  of  the science of space,  the,  we must reckon  this postulate of Continuity; according to which two adjacent portions of  space,  or of a surface,  or of a line,  have the {\it same\/} boundary,  {\it viz\/}.--- a surface,  a line,  or a  point;  and between every two points on a line there is an infinite number of intermediate points.        The  next postulate is that of  Elementary  Flatness.   You know  that  if  you  get hold of a small piece of  a  very  large circle, it seems to you nearly straight.  So, if you were to take any  curved  line,  and magnify it  very   much,  confining  your  attention  to  a  small  piece  of  it,  that  piece  would  seem straighter to you than the curve did before it was magnified.  At least,  you can easily conceive a curve possessing this property, that  the more you magnify it,  the straighter it gets.   Such  a curve would possess the property of elementary flatness.   In the same  way,  if  you perceive a portion of the surface of  a  very large  sphere,  such as the earth,  it appears to you to be flat. If,  then,  you take a sphere of say a foot diameter, and magnify it more and more,  you will find that the more you magnify it the flatter  it gets.   And you may easily suppose that this  process would  go on indefinitely;  that the curvature would become  less and less the more the surface was magnified.   Any curved surface which  is such that the more you magnify it the flatter it  gets, is said to possess the property of elementary flatness.   But  if every  succeeding power of our imaginary microscope disclosed new wrinkles  and inequalities without end,  then we should say  that the surface did not possess the property of elementary flatness.        But  how  am  I to explain how solid space  can  have  this property  of  elementary flatness?   Shall I leave it as  a  mere analogy,  and say that it is the same kind of property as this of the curve and surface, only in three dimensions instead of one or two?   I think I can get a little nearer to it than that;  at all events I will try.        If we start to go out from a point on a surface, there is a certain  choice  of  directions  in  which  we  may  go.    These directions make certain angles with one another.   We may suppose a certain direction to start with, and  then gradually alter that by  turning it round the point:   we find thus a single series of directions in which we may start from the point.   According   to our  first  postulate,  it is a continuous series of  directions. Now  when  I  speak  of a direction from  the  point,  I  mean  a direction of starting;  I say nothing about the subsequent  path. Two  different paths may have the same direction at starting;  in this case they will touch at the point;  and there is an  obvious difference between two paths which touch and two paths which meet and form an angle.   Here,  then,  is an aggregate of directions, and they can be changed into one another.   Moreover, the changes by  which  they  pass  into  one  another  have  magnitude,  they constitute   distance--relations;   and  the  amount  of   change necessary  to  turn one of them into another is called the  angle between  them.   It  is involved in this postulate  that  we  are considering,   that   angles  can  be  compared  in  respect   of magnitude.   But  this  is  not all.   If we  go  on  changing  a direction of start,  it will,  after a certain amount of turning, come   round  into itself again,  and be the same  direction.   On every surface which has the property of elementary flatness,  the amount  of turning necessary to take a direction all  round  into its first position is the same for all points of the surface.   I will now show you a surface which at one point of it has not this property.   I  take this circle of paper from which a sector  has been cut out,  and bend it round so as to join the edges; in this way I form a surface which is called a {\it cone\/}.   Now on all points  of this surface but one,  the law of elementary  flatness  holds good.   At the vertex of the cone, however, notwithstanding that  there is an aggregate of directions in which you may start, such  that  by continuously changing one of them you may  get  it round into its original position,  yet the whole amount of change necessary  to effect this is not the same at the vertex as it  is at any other point of the surface.   And this you can see at once when  I unroll it;  for only part of the directions in the  plane have been included in the cone.  At this point of the cone, then, it does not possess the property of elementary flatness;  and  no amount  of  magnifying  would ever make a cone seem flat  at  its vertex.        To apply this to solid space, we must notice that here also there is a choice of directions in which you may go out from  any point;  but it is a much greater choice than a surface gives you. Whereas  in a surface the aggregate of directions is only of  one dimension, in solid space it is of two dimensions.  But here also there  are distance--relations,  and the aggregate of  directions may be divided into parts which have quantity.   For example, the directions  which start from the vertex of this cone are  divided into those which go  inside the cone,  and those which go outside the cone.   The part of the aggregate which is inside the cone is called  a solid angle.   Now in those spaces of three  dimensions which have the property of elementary flatness,  the whole amount of solid angle round one point is equal to the whole amount round another point.  Although the space need not be exactly similar to itself  in all parts,  yet the aggregate of directions round  one point  is  exactly similar to the aggregate of  directions  round another  point,  if  the  space has the  property  of  elementary flatness.        How   does  Euclid  assume  this  postulate  of  Elementary Flatness?   In his fourth postulate he has expressed it so simply and clearly that you will wonder how anybody could make all  this fuss.  He says, All right angles are equal.'        Why  could  I not have adopted this at once,  and  saved  a great  deal  of trouble?   Because it assumes the knowledge of  a surface  possessing the  property of elementary flatness  in  all its  points.   Unless such a surface is first made out to  exist, and  the definition of a right angle is restricted to lines drawn upon it---for there is no necessity for the word {\it straight\/} in that definition---the postulate in Euclid's form is  obviously not true.   I can make two lines cross at the vertex of a cone so that the four adjacent angles shall be equal,  and yet not one of them equal to a right angle.       I  pass on to the third postulate of the science of space--- the  postulate of Superposition.   According to this postulate  a body  can  be moved about in space without altering its  size  or shape.   This  seems  obvious enough,  but it is worth  while  to examine  a little more closely into the meaning of it.   We  must define  what we mean by size and by shape.   When we say  that  a body  can be moved about without altering its size,  we mean that it  can  be so moved as to keep unaltered the length of  all  the  lines in it.  This postulate therefore involves that lines can be compared  in  respect of magnitude,  or that they have  a  length independent of position; precisely as the former one involved the comparison  of angular magnitudes.   And when we say that a  body can  be moved about without altering its shape,  we mean that  it can be so moved as to keep unaltered all the angles in it.  It is not necessary  to make mention of the motion of a body,  although that  is  the easiest way of expressing and  of  conceiving  this postulate;  but we may,  if we like, express it entirely in terms which  belong  to  space,  and  that we should do  in  this  way. Suppose  a  figure to have been constructed in  some  portion  of space;  say  that  a triangle has been drawn whose sides are  the shortest  distances between its angular points.   Then if in  any other  portion  of  space two points  are  taken  whose  shortest distance  is equal to a side of the triangle,  and at one of them an  angle  is made equal to one of the  angles adjacent  to  that side,  and  a  line  of  shortest distance  drawn  equal  to  the corresponding  side of the original triangle,  the distance  from the  extremity  of this to the other of the two  points  will  be equal  to  the third side of the original triangle,  and the  two will be equal in all respects; or generally, if a figure has been constructed anywhere,  another figure, with all its lines and all its  angles  equal to the corresponding lines and angles  of  the first,  can  be constructed anywhere else.   Now this is  exactly what   is  meant by the principle of  superposition  employed  by Euclid to prove the proposition that I have just mentioned.   And we may state it again in this short form---All parts of space are exactly alike.       But   this  postulate  carries  with  it  a  most  important consequence.   In enables  us to make a pair of most  fundamental definitions---those  of the plane and of the straight  line.   In order to explain how these come out of it when it is granted, and how  they cannot be made when it is not granted,  I must here say something  more about the nature of the postulate  itself,  which might otherwise have been left until we come to criticize it.        We  have stated the postulate as referring to solid  space. But  a  similar  property  may  exist  in  surfaces.   Here,  for instance,  is  part of the surface of a sphere.   If I  draw  any figure  I like upon this,  I can suppose it to be moved about  in any way upon the sphere, without alteration of its size or shape. If  a  figure  has  been drawn on any part of the  surface  of  a sphere,  a figure equal to it in all respects may be drawn on any other part of the surface.   Now I say that this property belongs to the surface itself, is a part of its own internal economy, and does   not depend in any way upon its relation to space  of  three dimensions.   For  I can pull it about and bend it in all  manner of ways,  so as altogether to alter its relation to solid  space; and yet, if I do not stretch  it or tear it, I make no difference whatever  in the length of any lines upon it,  or in the size  of any angles upon it.   I do not in any way alter the figures drawn upon it,  or the possibility of drawing figures upon it,  {\it so far  as their relations with the surface itself are concerned\/}. This  property  of the surface,  then,  could be  ascertained  by  people who lived entirely in it,  and were absolutely ignorant of a third dimension.   As a point--aggregate of two dimensions,  it has  in itself properties determining the distance--relations  of the  points  upon it,  which  are absolutely independent  of  the existence of any points which are not upon it.       Now  here  is a surface which has not  that  property.   You observe that it is not of the same shape all over,  and that some parts  of  it are more curved than other parts.   If you  drew  a figure upon this surface,  and then tried to move it  about,  you would  find that it was impossible to do so without altering  the size  and  shape of the figure.   Some parts of it would have  to expand,  some to contract, the lengths of the lines could not all be  kept the same,  the angles would not hit off  together.   And this property of the surface---that its parts are different  from one another---is a property of the surface itself,  a part of its internal economy,  absolutely independent of any relations it may have with space outside of it.  For, as with the other one, I can pull  it  about in all sorts of ways,  and,  so long as I do  not stretch  it  or tear it,  I make no alteration in the  length  of lines drawn upon it or in the size of the angles.        Here,  then,  is an intrinsic difference between these  two surfaces,  as  surfaces.   They are both point--aggregates of two dimensions;  but  the  points in them have certain  relations  of distance (distance measured always {\it on\/} the  surface),  and these  relations of distance are not the same in one case as they are in the other.        The  supposed  people living in the surface and  having  no idea  of a third dimension might,  without suspecting that  third dimension  at  all,  make  a very accurate determination  of  the nature of their {\it locus in quo\/}.  If the people who lived on the  surface  of  the  sphere were to measure  the  angles  of  a triangle,  they  would find them to exceed two right angles by  a quantity proportional to the area of the triangle.   This  excess of  the angles above two right angles,  being divided by the area of the triangle, would be found to give exactly the same quotient at all parts of the sphere. That quotient is called the curvature of the surface;  and we say that a sphere is a surface of uniform curvature.   But  if the people living on this irregular  surface were  to do the same thing,  they would not find quite  the  same result.   The  sum of the angles would,  indeed,  differ from two right angles,  but sometimes in excess,  and sometimes in defect, according  to  the  part of the surface  where  they  were.   And though  for small triangles in any on neighbourhood the excess or defect  would be nearly proportional to the area of the  triangle, yet  the  quotient obtained by dividing this excess or defect  by the area of the triangle would vary from one part of the  surface to another.  In other words, the curvature of this surface varies from  point  to  point;   it  is  sometimes  positive,  sometimes negative, sometimes nothing at all.       But now comes the important difference.   When I speak of a triangle, what do I suppose the sides of that triangle to be?      If  I take two points near enough together upon  a  surface, and  stretch a string between them,  that string will take  up  a certain  definite position upon the surface,  marking the line of shortest distance from one point to  the other.   Such a line  is called a geodesic line.  It is a line determined by the intrinsic properties of the surface, and not by its relations with external space.   The  line would still be the shortest line,  however the surface  were  pulled about without  stretching  or  tearing.   A geodesic  line  may be {\it produced\/},  when a piece of  it  is given; for we may take one of the points, and, keeping the string stretched,  make  it go round in a sort of circle until the other end has turned through two right angles.   The new position  will then be a prolongation of the same geodesic line.        In speaking of a triangle,  then,  I meant a triangle whose sides  are  geodesic  lines.   But  in the case  of  a  spherical surface---or,   more  generally,   of   a  surface  of   constant curvature---these  geodesic lines have another and most important property.   They are {\it straight\/},  so far as the surface  is concerned.   On  this surface a figure may be moved about without altering its  size or shape.   It is possible, therefore, to draw a  line  which shall be of the same shape all along and  on  both sides.  That is to say, if you take a piece of the surface on one side of such a line,  you may slide it all along the line and  it will  fit;  and  you may turn it round and apply it to the  other side,  and it will fit there also.  This is Leibniz's  definition of a straight line, and, you see, it has no meaning except in the case of a  surface of constant curvature,  a surface all parts of which are alike.       Now let us consider the corresponding things in solid space. In this also we may have geodesic lines;  namely, lines formed by stretching  a string between two points.   But we may  also  have geodesic surfaces; and they are produced in this manner.  Suppose we  have  a point on a surface,  and this surface  possesses  the property of elementary flatness.   Then among all the  directions of  starting from the point,  there are some which start {\it  in the surface\/},  and do not make an angle with it.  Let all these be  prolonged  into geodesics;  then we may imagine one of  these geodesics  to  travel round and coincide with all the  others  in turn.   In so doing it will trace out a surface which is called a geodesic  surface.   Now in the particular case where a space  of three  dimensions has the property of superpositoin,  or  is  all over alike,  these geodesic surfaces are {\it planes\/}.  That is to  say,  since the space is all over alike,  these surfaces  are also   of  the  same shape all over and on both  sides;  which  is Leibniz's  definition of a plane.   If you take a piece of  space on one side of such a plane, partly bounded by the plane, you may slide it all over the plane, and it will fit; and you may turn it round and apply it to the other side, and it will fit there also. Now  it is clear that this definition will have no meaning unless the  third  postulate be granted.   So we may say that  when  the postulate  of Superposition is true,  then there are  planes  and straight  lines;  and they are defined as being of the same shape  throughout and on both sides.        It  is  found that the whole geometry of a space  of  three dimensions is known when we know the curvature of three  geodesic surfaces  at every point.  The third postulate requires that  the curvature  of all geodesic surfaces should be everywhere equal to the same quantity.       I  pass to the fourth postulate,  which I call the postulate of Similarity.   According to this postulate,  any figure may  be magnified or diminished in any degree without altering its shape. If  any figure has been constructed in one part of space,  it may be  reconstructed  to any scale whatever in any   other  part  of space,  so that no one of the angles shall be altered through all the lengths of lines will of course be altered.  This seems to be a sufficiently obvious induction from experience; for we have all frequently seen different sixes of the same shape; and it has the advantage of embodying the fifth and sixth of Euclid's postulates in a single principle, which bears a great resemblance in form to that of Superposition, and may be used in the same manner.  It is easy to show that it involves the two postulates of Euclid:  Two straight  lines cannot enclose a space,' and Lines in one  plane which never meet make equal angles with every other  line.'       This  fourth postulate is equivalent to the assumption  that the  constant curvature of the geodesic surfaces is zero;  or the third and fourth may be put together,  and we shall then say that the  three  curvatures  of space are all of them  zero  at  every point.       The  supposition  made by Lobatchewsky was,  that the  three first  postulates  were true,  but not the fourth.   Of  the  two Euclidean  postulates included in this,  he  admitted  one,  {\it viz\/}.,  that two straight lines cannot enclose a space, or that two  lines which once diverge go on diverging for ever.   But  he left  out the postulate about parallels,  which may be stated  in this  form.   If through a point outside of a straight line there be drawn another, indefinitely produced both ways; and if we turn this  second one round so as  to make the point  of  intersection travel  along the first line,  then at the very instant that this point  of intersection disappears at one end it will reappear  at the other,  and there is only one position in which the lines  do not intersect.  Lobatchewsky supposed, instead, that there was  a finite  angle through which the second line must be turned  after the  point of intersection had disappeared at one end,  before it reappeared  at the other.   For all positions of the second  line  within  this angle there is then no intersection.    In  the  two limiting positions,  when the lines have just done meeting at one end,  and when they are just going to meet at the other, they are called  parallel;  so that two lines can be drawn through a fixed point parallel to a given straight line.  The angle between these two depends in a  certain way upon the distance of the point from the line.   The sum of the  angles of a triangle is less than two right  angles  by  a quantity proportional to  the  area  of  the triangle.   The whole of this geometry is worked out in the style  of  Euclid,  and the most interesting conclusions are arrived at; particularly  in the theory of solid space,  in which  a  surface turns up which is not plane relatively to that space,  but which, for  purposes of drawing figures upon it,  is identical with  the Euclidean plane.        It was Riemann, however, who first accomplished the task of analysing  all the assumptions of geometry,  and showing which of them were independent.  This very disentangling and separation of them  is  sufficient to deprive them for the  geometer  of  their exactness and necessity;  for the process by which it is effected consists   in  showing  the  possibility  of  conceiving    these suppositions one by one to be untrue;  whereby it is clearly made out  how  much is supposed.   But it may be worth while to  state formally the case for and against them.       When  it is maintained that we know these postulates  to  be universally  true,  in  virtue  of certain  deliverances  of  our consciousness,  it  is implied that these deliverances could  not exist,  except upon the supposition that the postulates are true. If it can be shown,  then, from experience that our consciousness would  tell us exactly the same things if the postulates are  not true,  the ground of their validity will be taken away.  But this is a very easy thing to show.        That  same faculty which tells you that space is continuous tells  you  that this water is continuous,  and that  the  motion perceived  in  a wheel of life is continuous.   Now we happen  to know  that  if we could magnify this water as much again  as  the best microscopes can magnify it,  we should perceive its granular structure.   And what happens in a wheel of life is discovered by stopping the machine.   Even apart,  then,  from our knowledge of the way nerves act in carrying messages,  it appears that we have no means of knowing anything more about an aggregate than that it is too fine--grained for us to perceive its discontinuity,  if it has any.        Nor can we,  in general,  receive a conception as  positive knowledge which is itself founded merely upon inaction.   For the conception of a continuous thing is of that which looks just  the same however much you magnify it.  We may conceive the magnifying to  go  on to a certain extent without change,  and then,  as  it were,  leave  it going on,  without taking the  trouble to  doubt about the changes that may ensue.         In regard to the second postulate, we have merely to point to  the  example of polished surfaces.  The smoothest surface that can be made is the one  most completely covered with the minutest ruts and furrows.  Yet geometrical constructions can be made with extreme accuracy upon such a surface,  on the supposition that it is an exact plane.   If,  therefore, the sharp points, edges, and furrows of space are only small enough,  there will be nothing to hinder  our conviction of its elementary flatness.   It has  even been  remarked  by  Riemann that we must  not  shrink  from  this supposition   if  it  is  found  useful  in  explaining  physical  phenomena.       The  first  two postulates may therefore be doubted  on  the side  of  the   very small.   We may put  the  third  and  fourth together,  and doubt them on the side of the very great.   For if the  property  of elementary flatness exist on the  average,  the deviations from it being,  as we have supposed,  too small to  be perceived,  then,  whatever  were  the true nature of  space,  we should have exactly the conceptions of it which we now  have,  if only  the regions we can get at were small in comparison with the areas  of curvature.   If we suppose the curvature to vary in  an irregular manner,  the effect of it might be very considerable in a triangle formed by the nearest fixed stars;  but if we  suppose it  approximately  uniform to the limit of telescopic  reach,  it will  be  restricted  to very much  narrower  limits.   I  cannot perhaps do better than conclude by describing to you as well as I can  what  is the nature of things on the  supposition  that  the curvature of all space is nearly uniform and positive.        In this case the Universe,  as known, becomes again a valid conception;  for  the extent of space is a finite number of cubic miles.   And this comes about in a curious way.   If you were  to start in any direction whatever,  and move in that direction in a perfect  straight  line according to the definition  of  Leibniz; after  travelling  a  most  prodigious  distance,  to  which  the parallactic  unit---200,000  times the diameter  of  the  earth's orbit---would  be  only a few steps,  you would arrive  at---this place.   Only,  if you had started upwards, you would appear from below.   Now,  one of two things would be true.  Either, when you had  got half--way on your journey,  you came to a place that  is opposite to this,  and which you must have gone through, whatever direction you started in;  or else all paths you could have taken diverge  entirely  from each other till they meet again  at  this place.   In the former case,  every two straight lines in a plane meet in two points,  in the latter they meet only in  one.   Upon this  supposition of a positive curvature,  the whole of geometry is far more complete and interesting;  the principle of  duality, instead  of half breaking down over metric relations,  applies to all  propositions  without exception.   In fact,  I  do  no  mind confessing  that  I personally have often found relief  from  the dreary infinities of homaloidal space in the consoling hope that, after all, this other may be the true state of things. \vfill\eject \noindent 2.  FULL--LENGTH ARTICLE \vskip.5cm \centerline{\bf BEYOND THE FRACTAL} \vskip.1cm \centerline{\sl Sandra Lach Arlinghaus} \vskip.5cm \centerline{I never saw a moor,} \centerline{I never saw the sea;} \centerline{Yet know I how the heather looks,} \centerline{And what a wave must be."} \vskip.1cm {\sl Emily Dickinson, Chartless."} \vskip.5cm  \centerline{\bf Abstract.}        {\nn  The   fractal   notion   of   self--similarity    is    useful   for characterizing change in scale; the reason fractals  are   effective  in  the  geometry of central  place  theory  is  because   that  geometry is hierarchical  in  nature.   Thus,  a   natural   place  to  look  for other connections of this sort is  to  other   geographical   concepts  that  are  hierarchical  in  nature.    Within  this  fractal  context, this chapter examines   the  case of spatial diffusion.        When the idea of diffusion is extended to see  adopters"   of  an innovation as attractors" of new  adopters,   a   Julia   set   is introduced as a possible axis against which to  measure   one   class  of  geographic phenomena.    Beyond   the   fractal   context,  fractal concepts, such as compression" and space-- filling" are considered in a broader graph--theoretic context.}  \centerline{\bf Introduction.}        Because  a  fractal  may  be  considered  as  a   randomly   generated statistical image  (Mandelbrot,   1983),   one   place   to   look   for  geometric fractals tailored to  fit  geographic  concepts   is   within  the  set   of   ideas   behind   spatial   configurations   traditionally characterized using   randomness.    The   spatial   diffusion  of  an innovation is one  such  case;  H\"agerstrand   characterized it using probabilistic  simulation   techniques   (H\"agerstrand,   1967).    This  chapter    builds   directly   on  H\"agerstrand's  work  in  order to  demonstrate,  in some detail, how fractals might arise  in  spatial diffusion.   From  there,  and  with   a   view   of   an   adopter   of   an  innovation  as  an  attractor"   of   other   adopters,    the  connected Julia set $z = z^2-1$ is examined,  only broadly,  for  its  potential to serve as an axis from which to measure spatial  attraction."       More generally,  it is not necessary to consider  fractal-- like  concepts  such  as  attraction,"  space--filling,"  or  compression"  relative  to any metric,  as  in  the  diffusion  example,   or   relative  to any axis,  as in  the   Julia   set   case.    These   broad   fractal notions are examined,  in  some  detail,  in a graph--theoretic  realm,  free  from   metric/axis   encumbrance,  as  one  step  beyond  the fractal.  An effort has  been made to explain key geographical  and mathematical concepts  so that much of the material, and  the  flow of ideas, is self--  contained and accessible to readers from various disciplines.  \centerline{\bf A fractal connection to spatial diffusion}        The  diffusion   of  the  knowledge   of   an   innovation   across  geographic  space  may be simulated  numerically   using   Monte    Carlo   techniques   based   in   probability    theory  (H\"agerstrand,  1967).   A simple example illustrates the basic   mechanics of H\"agerstrand's procedure.         Consider  a geographic region and cover  it  with  a  grid   of  uniform  cell  size suited to the scale  of  the   available   empirical information about the innovation.   Enter  the  number   of  initial adopters of the innovation in the grid:  an entry of  $1$"  means  one  person   (household,   or   other   set   of   people)  knows   of   the innovation.   Over time,  this  person  will tell others.  Assume that the spread of the news, from this  person to  others,   decays  with distance.   To  simulate  this   spread,   probabilities   of   the likelihood of contact will be  assigned  to  each  cell  surrounding each initial  adopter.   A   table   of   random  numbers  is  used  in conjunction with  the  probabilities, as follows.        Given  a gridded geographic region and a  distribution  of  three  initial adopters of an innovation  (Figure  1).    Assume   that  an initial telling occurs no  more  than  two  cells  away   from   the initial adopters' cells.   This assumption creates  a  five--by--five  grid in which interchange can occur  between  an  initial  adopter   in  the  central  cell  and  others.   Assign  probabilities  of  contact  to each of these twenty--five  cells  as  a  percentage likelihood that a randomly chosen  four  digit  number falls within a given interval of numbers assigned to each  cell  (Figure  2).    Because   the   intervals  in   Figure   2   partition   the  set  of  four  digit  numbers,   the percentage  probabilities assigned to each cell add to 100\%.   Pick up  the  five--by--five  grid  and center it on the original  adopter  in  cell H3 (Figure 1).   Choose the first number, 6248, in the list  of  random  numbers  (Figure 2).   It falls in the  interval  of  numbers in the central cell.  Enter a $+1$" in the  associated  cell,  H3,  to represent this new adopter.   Move the five--by-- five grid across the distribution of original adopters, stopping  it and repeating  this procedure with the next random number  in  the  list  each  time a  new original  adopter  is  encountered.   Center the five--by--five grid on H4;  the next random number is  0925  which  falls  in  the interval  in  the  cell  immediately  northwest  of  center (Figure 2).   Enter a $+1$" in  cell  G3  (Figure 1),  the  cell  immediately  northwest  of  H4. Finally,  center the moving grid on H5.   The  next  random  number, 4997,  falls in the center cell; therefore, enter a $+1$" in cell H5.  Once  this procedure has been applied to all original  adopters,  the   population    of   adopters  doubles   and    a    first   generation"  of adopters,   comprising  original  adopters   and    newer   adopters represented as $+1$'s", emerges  (Figure  1).    Any   number   of additional generations  of  adopters  of   the   innovation may be simulated by iteration of this procedure. \topinsertFigure 1.
\noindent{\bf Figure 1}.
\smallskip Three original adopters, represented as 1's. Positions are simulated for three new adopters, represented as $+1$'s. The two sets taken together form a first generation of adopters of an innovation (grid after H\"agerstrand). \smallskip North at the top.
TYPESETTING, USING TeX, FOR THIS IMAGE APPEARS BELOW

$$\matrix{ {\phantom{+0} \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {1 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {2 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {3 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {4 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {5 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {6 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {7 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {8 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}&\cr A&&&&&&&&\cr B&&&&&&&&\cr C&&&&&&&&\cr D&&&&&&&&\cr E&&&&&&&&\cr F&&&&&&&&\cr G&&&{{+1} \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}&&&&&\cr H&&&{{+1} \atop \phantom{+0}}{\phantom{0} \atop 1}& {\phantom{+0} \atop \phantom{+0}}{\phantom{0} \atop 1}& {{+1} \atop \phantom{+0}}{\phantom{0} \atop 1}&&&\cr I&&&&&&&&\cr J&&&&&&&&\cr K&&&&&&&&\cr }$$
\endinsert
\topinsert

\noindent {\bf Figure 2}.
\smallskip
Five--by--five grid overlay.  Numerical entries in  cells
show the percentage of four digit  numbers  associated  with  each
cell.  The given listing of cells shows which cell  is  associated
with which range of four digit numbers.
\smallskip
North at the top.
Figure 2.

TYPESETTING USING TeX OF THE SCANNED IMAGE, ABOVE, APPEARS BELOW

$$\matrix{ {\phantom{0} \atop \phantom{0}}& {\phantom{0}1\phantom{.00} \atop \phantom{00.00}}& {\phantom{0}2 \atop \phantom{00.00}}& {\phantom{0}3 \atop \phantom{00.00}}& {\phantom{0}4 \atop \phantom{00.00}}& {\phantom{0}5 \atop \phantom{00.00}}&\cr 1&{\phantom{0}0.96 \atop \phantom{00.00}} &{\phantom{0}1.40 \atop \phantom{00.00}} &{\phantom{0}1.68 \atop \phantom{00.00}} &{\phantom{0}1.40 \atop \phantom{00.00}} &{\phantom{0}0.96 \atop \phantom{00.00}}\cr 2&{\phantom{0}1.40 \atop \phantom{00.00}} &{\phantom{0}3.01 \atop \phantom{00.00}} &{\phantom{0}5.47 \atop \phantom{00.00}} &{\phantom{0}3.01 \atop \phantom{00.00}} &{\phantom{0}1.40 \atop \phantom{00.00}}\cr 3&{\phantom{0}1.68 \atop \phantom{00.00}} &{\phantom{0}5.47 \atop \phantom{00.00}} &{44.31 \atop \phantom{00.00}} &{\phantom{0}5.47 \atop \phantom{00.00}} &{\phantom{0}1.68 \atop \phantom{00.00}}\cr 4&{\phantom{0}1.40 \atop \phantom{00.00}} &{\phantom{0}3.01 \atop \phantom{00.00}} &{\phantom{0}5.47 \atop \phantom{00.00}} &{\phantom{0}3.01 \atop \phantom{00.00}} &{\phantom{0}1.40 \atop \phantom{00.00}}\cr 5&{\phantom{0}0.96 \atop \phantom{00.00}} &{\phantom{0}1.40 \atop \phantom{00.00}} &{\phantom{0}1.68 \atop \phantom{00.00}} &{\phantom{0}1.40 \atop \phantom{00.00}} &{\phantom{0}0.97 \atop \phantom{00.00}}\cr }$$

\smallskip
A random set of numbers (source:
{\sl CRC Handbook of Standard Mathematical Tables\/}):
\smallskip
\line{6248, 0925, 4997, 9024, 7754 \hfil}
\smallskip
\line{7617, 2854, 2077, 9262, 2841 \hfil}
\smallskip
\line{9904, 9647, \hfil}
\smallskip
\noindent and so forth.
\vskip.5cm
Random number assignment to matrix cells, with cell number given
as an ordered pair whose first entry  refers  to  the  reference
number on the left of the matrix in this figure and whose second
entry refers to the reference number at the top of that matrix.
\vskip.2cm
\line{(1,1):  0000-0095; (1,2):  0096-0235; (1,3):  0236-0403 \hfil}
\line{(1,4):  0404-0543; (1,5):  0544-0639 \hfil}
\smallskip
\line{(2,1):  0640-0779; (2,2):  0780-1080; (2,3):  1081-1627 \hfil}
\line{(2,4):  1628-1928; (2,5):  1929-2068 \hfil}
\smallskip
\line{(3,1):  2069-2236; (3,2):  2237-2783; (3,3):  2784-7214 \hfil}
\line{(3,4):  7215-7761; (3,5):  7762-7929 \hfil}
\smallskip
\line{(4,1):  7930-8069; (4,2):  8070-8370; (4,3):  8371-8917 \hfil}
\line{(4,4):  8918-9218; (4,5):  9219-9358 \hfil}
\smallskip
\line{(5,1):  9359-9454; (5,2):  9455-9594; (5,3):  9595-9762 \hfil}
\line{(5,4):  9763-9902; (5,5):  9903-9999 \hfil}
\endinsert

There are numerous side issues,  which are important,  that
may  complicate  this  basic  procedure  (H\"agerstrand,   1967;
Haggett {\it et al.\/},  1977).  How are the percentages for the
five--by--five  grid chosen?   Indeed,  how is the dimension  of
five"  chosen for a side of this grid?   Should the choices of
percentages  and of  dimension be based on  empirical  data,  on
other abstract  considerations,   or on a mix of the two?   What
sorts  of criteria should  there  be  in judging suitability  of
empirical  data?   What  if  a  random  entry falls outside  the
given  grid;   what  sorts  of  boundary/barrier considerations,
both  in terms of  the  position  of  new  adopters relative  to
the   regional  boundary   and   of   the   symmetry   of    the
probabilities  within the five--by--five grid,  should be  taken
into account?

Independent  of  how many generations are calculated  using
this procedure,  the pattern of filling in" of new adopters is
heavily influenced by the shape of the set of original adopters.
Indeed,  over time,  knowledge of the innovation diffuses slowly
initially,  picks up in speed of transmission,  tapers off,  and
eventually  the population becomes saturated with the knowledge.
Typically   this  is characterized as  a  continuous  phenomenon
using   a  differential equation of inhibited growth that has as
an  initial  supposition that the population may not exceed $M$,
an  upper  bound,  and that $P(t)$,  the population $P$ at  time
$t$,  grows  at  a rate proportional to the size of  itself  and
proportional  to  the  fraction left to grow  (Haggett  {\it  et
al.\/}, 1977; Boyce and DiPrima, 1977).  An equation such as
$${dP(t) \over dt} = k\, P(t)(1- (P(t)/M))$$
serves  as a mathematical model for this sort of growth in which
$k >0$ is a growth constant and the fraction $(1-(P(t)/M)$  acts
as   a damper on the rate of growth (Boyce and  DiPrima,  1977).
The  graph  of  the  equation  is   an   $S$--shaped   (sigmoid)
logistic   curve  with  horizontal  asymptote  at  $P(t)=M$  and
inflection point at $P(t)=M/2$.  When $dP/dt > 0$ the population
shows growth;  when $d^2 P/dt^2 > 0$ (below $P(t)=M/2$) the rate
of growth is increasing;  when $d^2P/dt^2<0$ (above  $P(t)=M/2$)
the rate of growth is decreasing.

The    differential    equation    model    thus    yields
information   concerning  the  rate  of  change  of  the   total
population  and   in  the rate of change in growth of the  total
population.    It   does   not show how to  determine  $M$;  the
choice  of  $M$  is  given {\it a  priori\/}.

Iteration of the H\"agerstrand procedure gives a  position
for $M$  once  the procedure has been run for all  the  generations
desired.

For,  it  is  a relatively easy matter to  accumulate  the
distributions  of  adopters and stack them next to  each  other,
creating  an  empirical  sigmoid logistic  curve  based  on  the
simulation (Haggett {\it et al.\/}, 1977).  Finding the position
for  the  asymptote (or for {\it an\/} upper bound close to  the
asymptotic position) is then straightforward.

Neither  the  H\"agerstrand  procedure  nor  the  inhibited
growth  model   provides  an  estimate   of   saturation   level
(horizontal asymptote position) (Haggett,  {\it et al.\/}, 1977)
that  can be  calculated early in the measurement of the growth.
The  fractal  approach suggested below offers a means for making
such  a calculation  when self--similar  hierarchical  data  are
involved;   allometry  is  a  special  case  of  this  procedure
(Mandelbrot,  1983).    The  reasons for wanting to make such  a
calculation  might  be  to determine where to  position  adopter
seeds"  in  order  to produce  various  levels  of  innovation
saturation.

As  is  well--known,  not all innovations  diffuse   in   a
uniform manner;  Paris fashions readily available in major U. S.
cities   up  and  down each coast might  seldom   be   seen   in
rural  midwestern towns.   To determine how the ideas of fractal
space--filling"  and this sort of diffusion--related space--
filling" might be aligned, consider the following example.

Given   a  distribution  of   three   original    adopters
occupying cells H3,  H4, and H5  in  a  linear  pattern  (Figure
3.A).    The  probabilities for positions for new adopters   are
encoded   within  each  cell surrounding  each  of   these   (as
determined  from  the five--by--five grid of Figure  2).   Thus,
for  example,  when  the  grid of Figure 2 is  superimposed  and
centered on the  original  adopter in cell H3,  a probability of
3.01\% is assigned to  the likelihood for contact from H3 to G4;
when it is superimposed and centered on the original adopter  in
H4,  there  is  a  5.47\% likelihood  for contact from H4 to G4;
and,  when  it  is superimposed  and  centered on  the  original
adopter in H5,  there is  a  3.01\% likelihood  for contact from
H5  to  G4.   Therefore,  the  percentage likelihood of   a  new
first--generation  adopter  in  cell  G4,   given this   initial
configuration  of  adopters,  is  the sum  of  the   percentages
divided by the number  of  initial  adopters,  or  11.49/3.  For
ease  in inserting fractions into the grid,  only the numerator,
11.49,  is shown as the entry (Figure 3.A).  It would be useful,
for  purposes  of comparison of this distribution to those  with
sets  of  initial adopters of sizes other than 3,  to divide  by
the  number of initial adopters in order to derive a  percentage
that   is  independent  of the size of the  initial distribution
({\it i.e.\/}, to normalize  the numerical entries).
\topinsert
\noindent{\bf Figure 3.A}.
\smallskip

The   simulation  is  run  on  three  original   adopters   with
positions given below.   Numerical entries show the  likelihood,
out  of  300,  that a new adopter will fall into a  given  cell.
Zones   of interaction between overlapping five--by--five  grids
are outlined by a heavy line (begin at the upper left--hand
corner (ulhc) of cell F2; move horizontally to the upper right--hand
corner (urhc) of cell F6; vertically to lower right--hand corner
(lrhc) of cell J6; horizontally to lower left--hand corner (llhc)
of cell J2; vertically to ulhc of F2 --- should be a rectangular
enclosure that you have added to this figure).
{\bf Original adopters are in cells H3, H4, H5.}
\smallskip
North at the top.LINES DESCRIBED ABOVE WERE ADDED TO THE SCANNED IMAGE IN COREL PHOTO-PAINT.
Figure 3A.

TYPESETTING USING TeX OF THE FIGURE. $$\matrix{ {\phantom{+0} \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {1 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {2 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {3 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {4 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {5 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {6 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {7 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {8 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {Totals}\cr A&&&&&&&&&\cr B&&&&&&&&&\cr C&&&&&&&&&\cr D&&&&&&&&&\cr E&&&&&&&&&\cr F&{\phantom{0}0.96} &{\phantom{0}2.36} &{\phantom{0}4.04} &{\phantom{0}4.48} &{\phantom{0}4.04} &{\phantom{0}2.36} &{\phantom{0}0.96} & &\phantom{0}19.20\cr G&{\phantom{0}1.40} &{\phantom{0}4.41} &{\phantom{0}9.88} &{11.49} &{\phantom{0}9.88} &{\phantom{0}4.41} &{\phantom{0}1.40} & &\phantom{0}42.87\cr H&{\phantom{0}1.68} &{\phantom{0}7.15} &{51.46} &{55.25} &{51.46} &{\phantom{0}7.15} &{\phantom{0}1.68} & &175.83\cr I&{\phantom{0}1.40} &{\phantom{0}4.41} &{\phantom{0}9.88} &{11.49} &{\phantom{0}9.88} &{\phantom{0}4.41} &{\phantom{0}1.40} & &\phantom{0}42.87\cr J&{\phantom{0}0.96} &{\phantom{0}2.36} &{\phantom{0}4.04} &{\phantom{0}4.48} &{\phantom{0}4.04} &{\phantom{0}2.36} &{\phantom{0}0.96} & &\phantom{0}19.20\cr K&&&&&&&&&\cr &&&&&&&&&&\cr &{\phantom{0}6.40} &{20.69} &{79.30} &{87.19} &{79.31} &{20.70} &{\phantom{0}6.41} & &{300\phantom{.00}}\cr }$$ \endinsert
It  is  easy to see that the values in the cells of  Figure
3.A must add to a total of 300 if one views them as derived from
each  of  three five--by--five grids centered on  each  original
adopter.   A zone of interaction" of entries from two or  more
five--by--five  grids is outlined by a heavy line;  25 cells are
enclosed in it  in Figure 3.A.   The pattern of numbers exhibits
bilateral   symmetry,  insofar as  is  possible  (allowing   for
the   appendix"   of   .01  required  to  make  the  numerical
partition  associated  with Figure 2 complete) with  respect  to
both   North--South and East--West axes (with  the   origin   in
cell  H4).    Sum  and  column  totals  are calculated;   as the
shape  of  the  distribution  of initial   adopters  is  altered
(below),    these   totals   will   tag   sets   of   cells   to
demonstrate   how  changes  in  the  zone  of  interaction   are
occurring.

Next    consider   a   distribution   of   three   initial
adopters  derived  from  the linear one  by  moving  the  middle
adopter  one  unit to the North (Figure 3.B).   When interaction
values   are  calculated  as  they  were   for    the    initial
distribution   in   Figure  3.A,   a comparable,  but  different
numerical pattern emerges (Figure  3.B). Here, the column totals
are  the same as those in Figure  3.A,   but the row totals  are
different.   The  zone  of interaction contains  23  cells;  the
highest individual cell value of  50.33  is  less  than that  of
the highest cell value,  55.25,  in  Figure  3.A.   Because both
sets  of values are partitions of the number 300,  and   because
there  are  more cells with potential for contact in Figure  3.B
than in Figure 3.A,  the concentration of entries in Figure  3.B
is  not as compressed as in Figure 3.A.  This  is  reflected  in
the   row  totals;  a  visual device useful  for  tracking  this
compression is to think of the five--by--five grid  centered  on
under   the  set  of entries in Figure 3.A.   In Figure 3.B  the
top  of  this  middle  grid slips out  from  under,  failing  to
intersect the bottom row,   J,  of the grid.  With this view, it
is  easy  to understand why  only  the row totals,  and not  the
column totals, change.
\topinsert
\noindent{\bf Figure 3.B}.
\smallskip
The  simulation  is  run  on  three  original   adopters    with
positions  given below.   Numerical entries show the likelihood,
out  of  300,  that a new adopter will fall into a  given  cell.
Zones   of interaction between overlapping five--by--five  grids
are outlined by a heavy line
(begin
ulhc of F2;
horizontally to urhc of F6;
vertically to lrhc of I6;
horizontally to lrhc of I5;
vertically to lrhc of J5;
horizontally to llhc of J3;
vertically to llhc of I3;
horizontally to llhc of I2;
vertically to ulhc of F2
--- should be a fat" T--shaped enclosure that you have added
to this figure).
{\bf Original adopters are in cells H3, G4, H5.}
\smallskip
North at the top. 
(HERE, AND IN SUBSEQUENT RELATED FIGURES,
DOTTED LINE TO INDICATE T-SHAPED POLYGON ADDED IN SCANNED IMAGE USING COREL PHOTO-PAINT, 5.0.)
Figure 3B.

TYPESETTING THAT PRODUCED FIGURE ABOVE $$\matrix{ {\phantom{+0} \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {1 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {2 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {3 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {4 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {5 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {6 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {7 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {8 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {Totals}\cr A&&&&&&&&&\cr B&&&&&&&&&\cr C&&&&&&&&&\cr D&&&&&&&&&\cr E& &{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}1.68} &{\phantom{0}1.40} &{\phantom{0}0.96} & & &{\phantom{0}6.40}\cr F&{\phantom{0}0.96} &{\phantom{0}2.80} &{\phantom{0}5.65} &{\phantom{0}8.27} &{\phantom{0}5.65} &{\phantom{0}2.80} &{\phantom{0}0.96} & &\phantom{0}27.09\cr G&{\phantom{0}1.40} &{\phantom{0}4.69} &{12.34} &{50.33} &{12.34} &{\phantom{0}4.69} &{\phantom{0}1.40} & &\phantom{0}87.19\cr H&{\phantom{0}1.68} &{\phantom{0}6.87} &{49.00} &{16.41} &{49.00} &{\phantom{0}6.87} &{\phantom{0}1.68} & &131.51\cr I&{\phantom{0}1.40} &{\phantom{0}3.97} &{\phantom{0}8.27} &{\phantom{0}7.70} &{\phantom{0}8.27} &{\phantom{0}3.98} &{\phantom{0}1.40} & &\phantom{0}34.99\cr J&{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}2.64} &{\phantom{0}2.80} &{\phantom{0}2.65} &{\phantom{0}1.40} &{\phantom{0}0.97} & &\phantom{0}12.82\cr K&&&&&&&&&\cr &&&&&&&&&&\cr &{\phantom{0}6.40} &{20.69} &{79.30} &{87.19} &{79.31} &{20.70} &{\phantom{0}6.41} & &{300\phantom{.00}}\cr }$$ \endinsert
Naturally,   as   the   middle    initial    adopter    is
pulled   successively  one   unit   to   the   north   in    the
configuration  of original adopters,  the middle  five--by--five
grid  is  also pulled one unit to the north (Figures  3.C,  3.D,
3.E,  and 3.F).  The numerical consequence is to reduce the size
of  the  zone of interaction among the initial adopters  and  to
spread  the  range  of cells  over  which the value  of  300  is
partitioned.  This implies less  concentration near the original
adopters  and less filling in"  around  them as  one  proceeds
from Figure 3.A to Figure 3.F.   Thus,  in  Figure  3.C the zone
of   interaction   shrinks   to   21  cells   with   a   largest
individual  cell entry of 47.39.   At the stage shown in  Figure
3.D,  the  largest  cell  entry  is  45.99;  because  the  cells
associated with this value are not overlapped by the  five--by--
five  grid  centered on the middle adopter,  this largest  value
will  not  change  as  the middle adopter is pulled more to  the
north.    Table  1  shows  the sizes of the zones of interaction
of  the largest  individual  cell entry for each of Figures  3.A
to 3.F.  No new  information  arises from moving the middle cell
to  the north beyond  the  position  in Figure 3.F;  the  five--
by--five   grid  is  revealed  and no longer intersects the  two
overlapping   grids  associated  with  the  other  two   initial
\topinsert
\noindent{\bf Figure 3.C}.
\smallskip

The  simulation  is  run  on  three  original   adopters    with
positions  given below.   Numerical entries show the likelihood,
out  of  300,  that a new adopter will fall into a  given  cell.
Zones   of interaction between overlapping five--by--five  grids
are outlined by a heavy line
(begin
ulhc of F2;
horizontally to urhc of F6;
vertically to lrhc of H6;
horizontally to lrhc of H5;
vertically to lrhc of J5;
horizontally to llhc of J3;
vertically to llhc of H3;
horizontally to llhc of H2;
vertically to ulhc of F2 --- should be a less--fat" T--shaped
to this figure).
{\bf Original  adopters are in cells H3, F4, H5.}
\smallskip
North at the top.
Figure 3C.

TYPESETTING THAT PRODUCED THIS FIGURE $$\matrix{ {\phantom{+0} \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {1 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {2 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {3 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {4 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {5 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {6 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {7 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {8 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {Totals}\cr A&&&&&&&&&\cr B&&&&&&&&&\cr C&&&&&&&&&\cr D& &{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}1.68} &{\phantom{0}1.40} &{\phantom{0}0.96} & & &{\phantom{0}6.40}\cr E& &{\phantom{0}1.40} &{\phantom{0}3.01} &{\phantom{0}5.47} &{\phantom{0}3.01} &{\phantom{0}1.40} & & &{14.29}\cr F&{\phantom{0}0.96} &{\phantom{0}3.08} &{\phantom{0}8.11} &{47.11} &{\phantom{0}8.11} &{\phantom{0}3.08} &{\phantom{0}0.96} & &\phantom{0}71.41\cr G&{\phantom{0}1.40} &{\phantom{0}4.41} &{\phantom{0}9.88} &{11.49} &{\phantom{0}9.88} &{\phantom{0}4.41} &{\phantom{0}1.40} & &\phantom{0}42.87\cr H&{\phantom{0}1.68} &{\phantom{0}6.43} &{47.39} &{12.62} &{47.39} &{\phantom{0}6.44} &{\phantom{0}1.68} & &123.63\cr I&{\phantom{0}1.40} &{\phantom{0}3.01} &{\phantom{0}6.87} &{\phantom{0}6.02} &{\phantom{0}6.87} &{\phantom{0}3.01} &{\phantom{0}1.40} & &\phantom{0}28.58\cr J&{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}2.64} &{\phantom{0}2.80} &{\phantom{0}2.65} &{\phantom{0}1.40} &{\phantom{0}0.97} & &\phantom{0}12.82\cr K&&&&&&&&&\cr &&&&&&&&&&\cr &{\phantom{0}6.40} &{20.69} &{79.30} &{87.19} &{79.31} &{20.70} &{\phantom{0}6.41} & &{300\phantom{.00}}\cr }$$ \endinsert

\topinsert
\noindent{\bf Figure 3.D}.
\smallskip

The   simulation  is  run  on  three  original   adopters   with
positions given below.   Numerical entries show the  likelihood,
out  of  300,  that a new adopter will fall into a  given  cell.
Zones   of interaction between overlapping five--by--five  grids
are outlined by a heavy line
(begin
ulhc of F2;
horizontally to urhc of F6;
vertically to lrhc of G6; horizontally to lrhc of G5;
vertically to lrhc of J5;
horizontally to llhc of J3;
vertically to llhc of G3;
horizontally to llhc of G2;
vertically to ulhc of F2
--- should be a less--fat" T--shaped enclosure that you have added
to this figure).
{\bf Original adopters are in  cells  H3, E4, H5.}
\smallskip
North at the top.
Figure 3D.

TYPESETTING, USING TeX, THAT PRODUCED THIS FIGURE. $$\matrix{ {\phantom{+0} \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {1 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {2 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {3 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {4 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {5 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {6 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {7 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {8 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {Totals}\cr A&&&&&&&&&\cr B&&&&&&&&&\cr C& &{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}1.68} &{\phantom{0}1.40} &{\phantom{0}0.96} & & &{\phantom{0}6.40}\cr D& &{\phantom{0}1.40} &{\phantom{0}3.01} &{\phantom{0}5.47} &{\phantom{0}3.01} &{\phantom{0}1.40} & & &{14.29}\cr E& &{\phantom{0}1.68} &{\phantom{0}5.47} &{44.31} &{\phantom{0}5.47} &{\phantom{0}1.68} & & &{58.61}\cr F&{\phantom{0}0.96} &{\phantom{0}2.80} &{\phantom{0}5.65} &{\phantom{0}8.27} &{\phantom{0}5.65} &{\phantom{0}2.80} &{\phantom{0}0.96} & &\phantom{0}27.09\cr G&{\phantom{0}1.40} &{\phantom{0}3.97} &{\phantom{0}8.27} &{\phantom{0}7.70} &{\phantom{0}8.27} &{\phantom{0}3.98} &{\phantom{0}1.40} & &\phantom{0}34.99\cr H&{\phantom{0}1.68} &{\phantom{0}5.47} &{45.99} &{10.94} &{45.99} &{\phantom{0}5.47} &{\phantom{0}1.68} & &117.22\cr I&{\phantom{0}1.40} &{\phantom{0}3.01} &{\phantom{0}6.87} &{\phantom{0}6.02} &{\phantom{0}6.87} &{\phantom{0}3.01} &{\phantom{0}1.40} & &\phantom{0}28.58\cr J&{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}2.64} &{\phantom{0}2.80} &{\phantom{0}2.65} &{\phantom{0}1.40} &{\phantom{0}0.97} & &\phantom{0}12.82\cr K&&&&&&&&&\cr &&&&&&&&&&\cr &{\phantom{0}6.40} &{20.69} &{79.30} &{87.19} &{79.31} &{20.70} &{\phantom{0}6.41} & &{300\phantom{.00}}\cr }$$ \endinsert

\topinsert
\noindent{\bf Figure 3.E}.
\smallskip

The   simulation  is  run  on  three  original   adopters   with
positions given below.   Numerical entries show the  likelihood,
out  of  300,  that a new adopter will fall into a  given  cell.
Zones   of interaction between overlapping five--by--five  grids
are outlined by a heavy line
(begin
ulhc of F2;
horizontally to urhc of F6;
vertically to lrhc of F6;
horizontally to lrhc of F5;
vertically to lrhc of J5;
horizontally to llhc of J3;
vertically to llhc of F3;
horizontally to llhc of F2;
vertically to ulhc of F2
--- should be a less--fat" T--shaped enclosure that you have added
to this figure).
{\bf Original adopters  are  in  cells  H3,  D4,  H5.}
\smallskip
North at the top.
Figure 3E.

TYPESETTING, USING TeX, THAT PRODUCED THIS FIGURE. $$\matrix{ {\phantom{+0} \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {1 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {2 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {3 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {4 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {5 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {6 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {7 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {8 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {Totals}\cr A&&&&&&&&&\cr B& &{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}1.68} &{\phantom{0}1.40} &{\phantom{0}0.96} & & &{\phantom{0}6.40}\cr C& &{\phantom{0}1.40} &{\phantom{0}3.01} &{\phantom{0}5.47} &{\phantom{0}3.01} &{\phantom{0}1.40} & & &{14.29}\cr D& &{\phantom{0}1.68} &{\phantom{0}5.47} &{44.31} &{\phantom{0}5.47} &{\phantom{0}1.68} & & &{58.61}\cr E& &{\phantom{0}1.40} &{\phantom{0}3.01} &{\phantom{0}5.47} &{\phantom{0}3.01} &{\phantom{0}1.40} & & &{14.29}\cr F&{\phantom{0}0.96} &{\phantom{0}2.36} &{\phantom{0}4.04} &{\phantom{0}4.48} &{\phantom{0}4.04} &{\phantom{0}2.37} &{\phantom{0}0.96} & &\phantom{0}19.21\cr G&{\phantom{0}1.40} &{\phantom{0}3.01} &{\phantom{0}6.87} &{\phantom{0}6.02} &{\phantom{0}6.87} &{\phantom{0}3.01} &{\phantom{0}1.40} & &\phantom{0}28.58\cr H&{\phantom{0}1.68} &{\phantom{0}5.47} &{45.99} &{10.94} &{45.99} &{\phantom{0}5.47} &{\phantom{0}1.68} & &117.22\cr I&{\phantom{0}1.40} &{\phantom{0}3.01} &{\phantom{0}6.87} &{\phantom{0}6.02} &{\phantom{0}6.87} &{\phantom{0}3.01} &{\phantom{0}1.40} & &\phantom{0}28.58\cr J&{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}2.64} &{\phantom{0}2.80} &{\phantom{0}2.65} &{\phantom{0}1.40} &{\phantom{0}0.97} & &\phantom{0}12.82\cr K&&&&&&&&&\cr &&&&&&&&&&\cr &{\phantom{0}6.40} &{20.69} &{79.30} &{87.19} &{79.31} &{20.70} &{\phantom{0}6.41} & &{300\phantom{.00}}\cr }$$ \endinsert

\topinsert
\noindent{\bf Figure 3.F}.
\smallskip

The   simulation  is  run  on  three  original   adopters   with
positions given below.   Numerical entries show the  likelihood,
out  of  300,  that a new adopter will fall into a  given  cell.
Zones   of interaction between overlapping five--by--five  grids
are outlined by a heavy line
(begin at ulhc of F3;
horizontally to urhc of F5;
vertically to lrhc of J5;
horizontally to llhc of J3;
vertically to ulhc of F3
--- should be a rectangular enclosure that you have added to this figure).
{\bf Original adopters are in cells H3, C4, H5.}
\smallskip
North at the top.
Figure 3F.

TYPESETTING, USING TeX, THAT PRODUCED THIS FIGURE. $$\matrix{ {\phantom{+0} \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {1 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {2 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {3 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {4 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {5 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {6 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {7 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {8 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {Totals}\cr A& &{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}1.68} &{\phantom{0}1.40} &{\phantom{0}0.96} & & &{\phantom{0}6.40}\cr B& &{\phantom{0}1.40} &{\phantom{0}3.01} &{\phantom{0}5.47} &{\phantom{0}3.01} &{\phantom{0}1.40} & & &{14.29}\cr C& &{\phantom{0}1.68} &{\phantom{0}5.47} &{44.31} &{\phantom{0}5.47} &{\phantom{0}1.68} & & &{58.61}\cr D& &{\phantom{0}1.40} &{\phantom{0}3.01} &{\phantom{0}5.47} &{\phantom{0}3.01} &{\phantom{0}1.40} & & &{14.29}\cr E& &{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}1.68} &{\phantom{0}1.40} &{\phantom{0}0.97} & & &{\phantom{0}6.41}\cr F&{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}2.64} &{\phantom{0}2.80} &{\phantom{0}2.64} &{\phantom{0}1.40} &{\phantom{0}0.96} & &\phantom{0}12.80\cr G&{\phantom{0}1.40} &{\phantom{0}3.01} &{\phantom{0}6.87} &{\phantom{0}6.02} &{\phantom{0}6.87} &{\phantom{0}3.01} &{\phantom{0}1.40} & &\phantom{0}28.58\cr H&{\phantom{0}1.68} &{\phantom{0}5.47} &{45.99} &{10.94} &{45.99} &{\phantom{0}5.47} &{\phantom{0}1.68} & &117.22\cr I&{\phantom{0}1.40} &{\phantom{0}3.01} &{\phantom{0}6.87} &{\phantom{0}6.02} &{\phantom{0}6.87} &{\phantom{0}3.01} &{\phantom{0}1.40} & &\phantom{0}28.58\cr J&{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}2.64} &{\phantom{0}2.80} &{\phantom{0}2.65} &{\phantom{0}1.40} &{\phantom{0}0.97} & &\phantom{0}12.82\cr K&&&&&&&&&\cr &&&&&&&&&&\cr &{\phantom{0}6.40} &{20.69} &{79.30} &{87.19} &{79.31} &{20.70} &{\phantom{0}6.41} & &{300\phantom{.00}}\cr }$$ \endinsert

\topinsert
\hrule
\smallskip
\centerline{TABLE 1}
\vskip.2cm
\noindent   Sizes  of  zones  of  interaction  and  of   largest
individual  cell value for each of the distributions of  initial
adopters in Figure 3. \vskip.2cm
Table 1.

TYPESETTING, USING TeX, THAT PRODUCED TABLE 1. \settabs\+\indent \quad&Figure 3.B:  middle cell 2 units north   \qquad\quad&Number of cells\qquad&(out of 300) in&\cr \smallskip \+&Figure number:     &Number of cells &Largest value   \cr \+&Position of three  &in interaction  &(out of 300) in \cr \+&original adopters. &zone.           &individual cell.\cr \smallskip \+&Figure 3.A: linear arrangement         &25  &55.25\cr \+&Figure 3.B: middle cell 1 unit north   &23  &50.33\cr \+&Figure 3.C: middle cell 2 units north  &21  &47.39\cr \+&Figure 3.D: middle cell 3 units north  &19  &45.99\cr \+&Figure 3.E: middle cell 4 units north  &17  &45.99\cr \+&Figure 3.F: middle cell 5 units north  &15  &45.99\cr \smallskip \hrule \endinsert       The example depicted in Figure 3 shows that even as early  as the first generation, the pattern of the positions of the  initial adopters affects significantly  the  configuration  of  the  later adopters.  Figure 3.A with the heaviest possible filling of  space using three initial adopters represents  a  most  saturated  case, which,  taken  together  with  an  underlying  symmetry  that   is bilateral relative to mutually perpendicular axes,  suggests  that an associated space--filling curve should have dimension 2, should have a  rectilinear  appearance,  and  should  be  formed  from  a generator whose shape is related to the pattern  of  placement  of the original adopters.  One space--filling curve that  meets these requirements  is  the  rectilinear  curve  of  Figure  4.A.    The generator is composed of three nodes hooked together by two  edges in a straight path.  This is scaled--down, by a factor of 1/2, and hooked to the endpoints of the original generator.   Iteration  of this procedure leads  to  a  rectilinear  tree  with  the  desired properties.  The approach of looking for a geometric form to fit a given set of conditions is like the calculus approach  of  looking for a differential equation to fit a given set of conditions.  The difference here is that the  shape  of  the  generator  and  other information from early stages may be used to estimate the relative saturation or space--filling level.        The  spatial position of the original adopters in   Figure   3.B  suggests  a fractal generator in  the  shape  of   a   V"   with  an interbranch angle, ${\theta}$, of 90 degrees, while the  V  in  Figure  3.C suggests a  generator  with  $\theta \approx 53^{\circ}$,  that  of  Figure  3.D  one  with  $\theta \approx 37^{\circ}$,  that  of  Figure  3.E  one  with  $\theta \approx 28^{\circ}$,  and  that of  Figure  3.F one with $\theta \approx 23^{\circ}$.   Figures 4.B,  4.C,  4.D,  4.E,  and  4.F  suggest  trees that can be generated using these values for $\theta$.
 \topinsert \vskip 5in \noindent {\bf Figure 4}. \smallskip THIS FIGURE DOES NOT TRANSMIT---AVAILABLE ONLY AS HARD COPY
(THUS, SCANNED IMAGE ONLY IS AVAILABLE HERE...NO CORRESPONDING TeX CODE.)


Figure 4.

\smallskip
Fractal  trees  derived  from  the  diffusion  grids  of  Figure  3;  labels  A  through F correspond in the two   Figures.    The  position of the distribution of  original  adopters  in   Figure   3  determines the positions for generators for  fractal   trees.    The interbranch angle,  $\theta$,  is constant  within  a  tree;   values  of  $\theta$ decrease from A.  to F. as does the fractal  dimension, $D$.
Table.

TYPESETTING THAT PRODUCED THE ABOVE TEXT
 \line{A. $\theta = 180^{\circ}$,      $D = 2$. \hfil} \line{B. $\theta = 90^{\circ}$,      $D \approx 0.72$. \hfil} \line{C. $\theta \approx 53.13^{\circ}$, $D \approx 0.47$. \hfil} \line{D. $\theta \approx 36.87^{\circ}$, $D \approx 0.38$. \hfil} \line{E. $\theta \approx 28.07^{\circ}$, $D \approx 0.33$. \hfil} \line{F. $\theta \approx 22.62^{\circ}$, $D \approx 0.30$. \hfil} \endinsert


A  rough measure of how much space each  one  fills"  may
be calculated using Mandelbrot's formula for fractal  dimension,
D, as,
$$D = {{\hbox{ln}\,N} \over {\hbox{ln}\,(1/r)}}$$
where  $N$  represents  the number of sides in  the  generator,
which  in all cases here is the value 2,  and where $r$ is  some
sort  of  scaling value that remains  constant  independent   of
scale   (Mandelbrot,  1977).   The  difficulty in  the  case  of
trees,    deriving    from   the  complication  of  intersecting
branches,  is  to  select  a  suitable description for $r$.  One
angle,  $\phi$, that remains constant throughout the iteration,
and that produces the desired effect for  the  case in which the
diffusion  is  the  most saturated,  is the base  angle  of  the
isoceles triangle with apex angle $\theta /2$ whose equal  sides
have  the  length of the equal sides of the two branches of  the
generator (Figure 5).   When $r$ is taken as the cosine of $\phi$,  then  $D=2$  in  the case of Figure 4.A  and  it   decreases
dramatically   as  the  trees generated by the  distribution  of
original adopters fill less space (Table 2).

\topinsert \vskip 3.5in
\noindent{\bf Figure 5}.
The construction of the angle $\phi$ used in the calculation of
the fractal dimension, $D$, of the trees in Figure 4.

Figure 5.
\endinsert

\topinsert
\hrule
\smallskip
\centerline{TABLE 2}
\smallskip
\noindent $D$--values, which suggest extent of space--filling, for
the trees (Figure 4) representing the patterns of initial adopters
in Figure 3.
Table 2.

TYPESETTING, USING TeX, THAT PRODUCED TABLE 2. \smallskip \settabs\+\noindent &Figure 3.C: middle cell 2 units north\quad &Figure 4.C: $\theta \approx 53.13^{\circ}$\quad &$=(180-(\theta /2))/2$ \quad &$(\hbox{ln}\,(1/\hbox{cos}\,\phi))$ &\cr \smallskip \+&Figure number:     &Size of interbranch  &Size     &$D$--value:\cr \+&Position of three  &angle, $\theta$, in &of $\phi$     &$D=(\hbox{ln}\, 2)/$\cr \+&original adopters. &associated tree.     &$=(180-(\theta /2))$     &$(\hbox{ln}\, (1/\hbox{cos}\,\phi))$\cr \smallskip \+&Figure 3.A: linear arrangement       &Figure 4.A:                    $\theta = 180^{\circ}$    &$45^{\circ}$  &2       \cr \+&Figure 3.B: middle cell 1 unit north &Figure 4.B:                    $\theta = 90^{\circ}$    &$67.5^{\circ}$&0.721617\cr \+&Figure 3.C: middle cell 2 units north&Figure 4.C:                    $\theta \approx 53.13^{\circ}$ &76.78         &0.471288\cr \+&Figure 3.D: middle cell 3 units north&Figure 4.D:                    $\theta \approx 36.87^{\circ}$ &80.78         &0.378471\cr \+&Figure 3.E: middle cell 4 units north&Figure 4.E:                    $\theta \approx 28.07^{\circ}$ &82.98         &0.32971 \cr \+&Figure 3.F: middle cell 5 units north&Figure 4.F:                    $\theta \approx 22.62^{\circ}$ &84.35         &0.299116\cr \smallskip \hrule \endinsert
This  decreasing sequence of $D$--values  corresponds  only
loosely  to  Mandelbrot's  measurements  of  fractal  dimensions
of  trees (Mandelbrot,  1983);  here,  however,  when $D=1$  the
corresponding   tree  is  one  with  an  interbranch  angle   of
$120^{\circ}$.   This has some appeal if one notes that then the
tree  associated with $D=1$ might therefore represent a  Steiner
network   (tree   of  shortest  total   length   under   certain
circumstances)  or   part  of  a   central   place   net.    The
numerical unit $D$--value would thus correspond to optimal forms
for  transport   networks   or  for  urban    arrangements    in
abstract  geographic space (in which  H\"agerstrand's  diffusion
procedure also exists).

One  use   for   these  $D$--values,   which  measure  the
relative  space--filling   by   trees,   might   be   as   units
fundamental   to  developing   an   algebraic   structure    for
planning  the  eventual saturation level to arise in communities
into  which  an  innovation is introduced to selected  adopters.
By  choosing  judiciously  the pattern  of   initial   adopters,
the   relative   space--filling   of associated trees  might  be
guided  by  local municipal authorities so as  not  to  conflict
with,  or to interfere  with,   other  issues  of local concern.
The $D$--values associated with  triads of original adopters (as
in  Table  2)  might serve as  irreducible   elements   of  this
algebra,  into  which larger sets could be decomposed (much   as
positive  integers  ($> 1$) can be decomposed into a product  of
powers   of   prime  numbers).    The  manner   in   which   the
decomposition   is  to take place would likely be  an  issue  of
considerable  algebraic difficulty,  no doubt requiring the  use
of   geographic  constraints to  limit   it.     (For,    unlike
the    parallel   with   integer decomposition,  this one  would
seem  not  to  be unique.)  An  initial direction  for  such   a
diffusion--algebra  might  therefore be  to exploit the parallel
with the Fundamental Theorem of Arithmetic.

Another use might involve a self--study by the National Center
for Geographic  Information  and  Analysis  (NCGIA)  in  order  to
monitor the  diffusion  of  Geographic  Information  System  (GIS)
technology through the various programs designed to  promote  this
technology in the academic arena.  University test--sites  for the
materials of the NCGIA, for example, might be selected as seeds"
with deliberate plans for using a  diffusion  structure  based  on
these seeds to bring later adopters up to date.

Another use might involve  the  determination  of  sites  for
locally unwanted land uses such as waste sites,  prisons,  and  so
forth.  Regions expected  to  experience  high  concentrations  of
population  coming  from  the  totality  of  innovations   already
introduced, or to be introduced, might be overburdened by  such  a
landuse.  When relative fractal saturation estimates are run on  a
computer in conjunction with a GIS,  local  municipal  authorities
might examine issues such as this for themselves.

\centerline{\bf Attraction:  the Julia set $z = z^2 - 1$}

A different way to view the space--filling characteristics of
the diffusion example is to consider each initial  adopter  as  an
attractor" of other adopters, once  again  suggesting  a fractal
attracted to points within  an  abstract  geographic  space.   The
fractal connection is to describe  space--filling  rather  than to
describe the pattern or the  direction  of  the  attraction.   The
material below suggests a means of  viewing  the  broad  class  of
spiral geographic phenomena as repelled  away  from  a  Julia  set
toward points of  attraction  within  and  beyond  the fractal":
hence, pattern and direction of attraction.

The  familiar   Mandelbrot   set,   comprising   a   large
central  cardioid  and circles tangent to the  cardioid,   along
with   points  interior  and  exterior  to  this  boundary,   is
associated with $z = z^2 +c$, where $z$" is a complex variable
and $c$" is a complex constant (Mandelbrot,  1977; Peitgen and
Saupe,  1988).   When constant  values for $c$ are chosen, Julia
sets   fall  out  of  the  Mandelbrot  set (Peitgen  and  Saupe,
1988).

When  $c=0$,  the  corresponding  Julia set is   the   unit
circle centered at the origin.  The boundary itself is fixed, as
a whole, under the transformation $z \mapsto z^2$, although only
the  individual point $(1,0)$ is itself fixed.   Points interior
to the boundary are attracted  to  the   origin:     for   them,
iteration   of   the transformation leads eventually to a  value
of  0.     Points   outside  the  circle  are  attracted  toward
infinity;    the   boundary   repels points not on it   (Peitgen
and   Saupe,   1988).    Various  natural associations might  be  made
between   this   simple   Julia   set   and  astronomical
phenomena such as orbits or compression within  black holes.

When $c = -1$,  the corresponding Julia set is described  by
$z= z^2 -1$ (Figure 6).   The attractive fixed points are 0, $- 1$,  and infinity.   The repulsive fixed points  on  the   Julia
set,  found using the quadratic" formula on $z^2-z-1 = 0$, are
at   distances  of $(1+\sqrt 5 )/2$ and $(1-\sqrt 5 )/2$  units
from  the  origin along the  real axis (distinguished on  Figure
6).   Points within the Julia set are attracted alternately to 0
and to  $-1$  as  attractive  two--cycle" fixed points; points
outside it are attracted to infinity.   To see the two--cycle"
effect,  iterate  the  transformation using $z =1.59$  (located
within the Julia set) as the initial value.
equation

TYPESETTING, USING TeX, THAT PRODUCED THE EQUATION ABOVE. \eqalign{ 1.59 & \mapsto 1.5281 \mapsto 1.3350896 \mapsto 0.7824643 \mapsto -0.3877497 \cr & \mapsto -0.849650 \mapsto -0.2780946 \mapsto -0.9226634 \mapsto -0.1486922 \cr & \mapsto -0.9778906 \mapsto -0.0437299 \mapsto -0.9980877 \mapsto -0.003821 \cr & \mapsto -0.9999854 \mapsto -0.0000292 \mapsto -1 \mapsto -0.00000000016 \cr & \mapsto -1 \mapsto 0. \cr }

This value of $z$ is attracted to $-1$ faster than it is to  0.
In this case, iter\-a\-tion strings close down on points of at\-
trac\-tion;  this  is  not the case for all Ju\-lia  sets.   The
choice  of  the   value of $c$ determines whether  or  not  such
strings  can escape (Peitgen and Saupe, 1988).

\topinsert \vskip 5in
\noindent{\bf Figure 6}.
\smallskip
THIS FIGURE DOES NOT TRANSMIT---AVAILABLE ONLY AS HARD COPY
(THUS, ONLY AS A SCANNED IMAGE HERE WITH NO ACCOMPANYING TeX.)

Figure 6.
\smallskip
The   Julia  set $z = z^2 - 1$.    Fixed   points  $((1 \pm \sqrt 5 )/2, 0)$ are distinguished on the boundary.
\endinsert

The  movement of an initial point  toward   an   attractor,
and away from a fixed boundary (as above), suggests  a  view  of
this  Julia  set as an axis:   lines from which the movement  of
points  are measured are axes."  Indeed,  the repulsive  fixed
points  on this set,  located at $((1+ \sqrt 5 )/2,0)$ and $((1- \sqrt 5 )/2,0)$,  might serve  as units."  They are the  non--
zero  terms  of the coefficients in the generating function  for
the Fibonacci numbers  (thanks to  W.  Arlinghaus for suggesting
this   connection   to   the    Fibonacci  generating  function;
Rosen,  1988).   For,  the $n$th Fibonacci number,  $a_n =a_{n- 1}+a_{n-2}, \quad a_0=0, \quad a_1=1$, is generated by
equation.

TYPESETTING FOR THE EQUATION ABOVE $$a_n = {1 \over \sqrt 5} ((1+\sqrt 5)/2)^n -{1 \over \sqrt 5} ((1-\sqrt 5)/2)^n.$$

Because  the  Fibonacci  sequence  can  be  expressed  using   the logarithmic spiral, this particular Julia set with these values as units" might therefore serve as an axis  from  which  to measure spiral phenomena at various scales ranging from the global to  the local:   from,   for   example,   the   climatological   to    the meteorological.       The mechanics of using this curve as an axis might involve an approach different from  that  customarily  employed.   The  curve might, for  example,  be  mounted  as  an  equator  on  the  globe partitioning the earth into two pieces in much the way that a seam serves as an equatorial line to partition the hide on a  baseball. In this circumstance, there would be freedom  to  choose  how  the equator partitions the earth's landmass.  It might be  located  in such a way that exactly half of the earth's water and half of  the earth's land lie on either side of the Julia set  (using  theorems from algebraic topology (Lefschetz, 1949; Dugundji, 1966; Spanier, 1966)).  \centerline{\bf Beyond  the  fractal:  a graph theoretic connection.}       The notions of attraction" and repulsion"  have also been expressed in the physical world, using graph theory (Harary, 1969; Uhlenbeck, 1960).  Fractals rely on distance, angle, or some other quantifier; graphs do not, and in that respect, are  more  general than are fractals.  Fractal--like concepts, such as space--filling and the associated image  compression  (Barnsley,  1988),  may  be characterized using graphs, as below (Arlinghaus, 1977; 1985).       This strategy will be expressed in terms of cubic trees  (all nodes are of degree three, unless they are at the tip of a branch) of shortest total length (Steiner trees) of maximal branching.  It could be expressed in terms of graphs of various linkage patterns; what is important is to begin with  some  systematic  process  for forming graphs.       Given  a geographic region  whose  periphery  is   outlined   by landmark positions at $P_1$,  $P_2$,  $P_3$, $P_4$, and $P_5$  (Figure  7.A).    View the landmarks as the nodes of a graph and  the  peripheral outline as the edges linking these nodes (Figure  7.A).   A  global" network within the entire pentagonal region  might  lie along   lines of a Steiner (shortest total  distance)  tree  (Figure  7.A)  (Arlinghaus,  1977;  1985) attached to  the  pentagonal hull joining neighboring branch tips  (Balaban,  {\it  et al.\/}, 1970).
 \topinsert \vskip 6in \noindent{\bf Figure 7}. \smallskip THIS FIGURE DOES NOT TRANSMIT---AVAILABLE ONLY AS HARD COPY.
(THUS SHOWN HERE AS A SCANNED IMAGE ONLY, WITHOUT THE ACCOMPANYING TeX.)
Figure 7.


\smallskip
Network  location  within geographic regions.   Points   of  the
pentagonal hull have P" as a notational base;  Steiner  points
have S" as a notational base.
A.  A  Steiner  (shortest  total distance) tree linking five locations.
B.  Partition  into  three distinct, contiguous geographic regions.
C.  Steiner networks  in each geographic region;
boundaries separating regions are removed.
D.  Steiner  networks  in  two  quadrangular  circuits;
circuit boundaries removed.
E.  Process   repeated   on   remaining quadrangular cell;
the  result  is  a  tree  with  local  Steiner
characteristics that provides global linkage following  the  basic
pattern of the global Steiner tree (Figure 7.A)).
\endinsert

Figure  7.B will be used as an initial figure  from   which
to  produce  a  network that  penetrates  triangular  geographic
subregions  (introducing  edges  $P_2P_5$   and  $P_2P_4$)  more
deeply than does the global network of Figure 7.A,  yet  retains
the   Steiner   characteristic  locally within  each  geographic
subregion.   An  iterative   process using Steiner trees  (as  a
Steiner  transformation")  will  be  applied  to  Figure   7.B
(Arlinghaus, 1977; 1983), as follows.

Introduce   Steiner  networks  into  each  of  the   three
triangular regions and remove the edges $P_2P_5$ and $P_2P_4$ so
that  a  new  network,  containing two  quadrangular  cells,  is
hooked   into   the  pentagon  $P_1P_2P_3P_4P_5$  (Figure  7.C).
Repeat this procedure in the network of Figure 7.C,  introducing
Steiner  networks into all circuits that do not have an edge  in
common  with  the pentagon  $P_1P_2P_3P_4P_5$.   Thus,  the  two
four--sided circuits,  $P_5S_1P_2S_2$; $P_2S_2P_4S_3$, in Figure
7.C  are replaced with the lines of  the   network,   $P_5S_1'$,
$S_1S_1'$,    $S_1'S_2'$,   $P_2S_2'$,   $S_2'S_2$;   $S_2S_3'$,
$P_2S_3'$,  $S_3'S_4'$,  $S_4'P_4$, $S_4'S_3$, shown  in  Figure
7.D.   Repeat  this  process in Figure 7.D,   using  a   Steiner
tree,   $S_2S_1''$,   $S_2'S_1''$,   $S_1''S_2''$,   $S_2''P_2$,
$S_2''S_3'$,   to   replace   the   single   four--sided   cell,
$P_2S_2'S_2S_3'$,  not  sharing an edge with  $P_1P_2P_3P_4P_5$.
The  result,  shown  in Figure 7.E,  is a tree which  cannot  be
further  reduced   using   the   Steiner   transformation.    It
satisfies  the initial conditions of  generating  a  tree   more
local   than   the  Steiner  network  of  maximal  branching  on
$P_1P_2P_3P_4P_5$  (but  with  local  Steiner  characteristics),
while retaining the global structure  of a graph--theoretic tree
hooked  into  $P_1P_2P_3P_4P_5$ in a pattern similar to that  of
the  global Steiner tree (with only local  variation   as  along
the edge $S_2S_1''$).   This process attempts to integrate local
with  global  concerns.   In this case,  the process  terminates
after a  finite  number  of  steps;   were  it   to    continue,
greater  space--filling  of the geographic region by  lines   of
the network would occur (Arlinghaus, 1977; 1985).

A  natural  question  to ask is   whether   or   not   this
process  necessarily terminates;   do  successive   applications
generate  a  finite  reduction sequence   of   the   cellular"
structure  into  a tree" structure  within  $P_1P_2P_3P_4P_5$?
Or,   is   it  possible  that   this   transformation,   applied
iteratively,  might  fill   enough  space  to choke  the  entire
region  with  an  infinite regeneration of cells  and  of  lines
bounding those cells (Arlinghaus, 1977; 1985)?

In this vein, take  Figure  7.B  and  add  one  edge  to  it,
creating  four triangular geographic regions (Figure  8.A).  Apply
the same  process to it as above, producing the networks  shown in
Figures 8.B  and 8.C.   Clearly,  further  iteration  would simply
produce a greater  number of polygonal  cells,  tightly compressed
around  the  node  $P_2$.   Discovering  a  means to calculate the
dimension of this  compression  is  an  open  issue.   It  is  not
difficult, however,  to  understand  under  what  conditions  this
sequence might,  or  might  not,  terminate  (Comments  (based  on
material in Arlinghaus, 1977; 1985) below).
\vskip.1cm
\noindent Definition (Harary, 1969; Tutte, 1966),
\vskip.1cm

A wheel $W_n$ of order $n$, $n>3$, is a graph obtained from
an $n$--gon by inserting one new vertex, the hub, and by joining
the  hub  to at least two of the vertices of the $n$--gon  by  a
finite  sequence of edges ($P_2$ is the hub of a wheel formed in
Figure 8.A).

\topinsert \vskip 6in
\noindent{\bf Figure 8}.
\smallskip
THIS FIGURE DOES NOT TRANSMIT---AVAILABLE ONLY AS HARD COPY.
(THUS, IT IS PRESENTED HERE AS A SCANNED IMAGE ONLY, WITHOUT ACCOMPANYING TeX.) \smallskip

Figure 8.

A  modification of Figure 7.    An extra edge is added to Figure
7.A,   creating   a   graph--theoretic   wheel."    When   the
procedure  displayed   in   Figure  7   is   applied   to   this
initial configuration,  cells are added within the hull (B.  and
C.), rather than removed.
\endinsert

\noindent Comment 1
\vskip.1cm

Hubs of wheels are invariant,  as  hubs  of  wheels,   under   a
sequence    of   successive   applications   of   the    Steiner
transformation described above.
\vskip.1cm
\noindent Comment 2
\vskip.1cm

Suppose   that   there  exists  a  finite  set  of    contiguous
triangles,   $T$.   If  $T$  contains  a  wheel, then a sequence
of successive applications of the Steiner transformation to  $T$
fails  to produce an irreducible tree.   The sequence fails   to
terminate  (as  long  as the Steiner trees  produced   at   each
stage  are  not degenerate).
\vskip.1cm
\noindent Comment 3
\vskip.1cm

Suppose   that  there  exists  a  finite  set  of    contiguous
triangles  $T = \{L_1 \ldots L_m \}$ with vertex set $V = \{P_1 \ldots P_n \}, \quad n>m$ (as in Figure 7.B, $m = 3$, $n = 5$).
Suppose that $T$ does not contain a wheel.   The number of steps
$M$,  in   the   sequence   of  successive applications  of  the
Steiner  transformation to $T$ required to reduce $T$ to a  tree
is
$$M = (\hbox{max} (\hbox{degree} (P_i ))) - 1.$$

Since $T$ does not contain a wheel, it follows from  Comment
2 that the reduction sequence is finite.  The actual size of $M$
might  be  found using mathematical induction on the  number  of
cells in $T$ and on the graph--theoretic degree of $P_i$.

The examples shown in Figures 7  and  8,   together   with
the  Comments  above,  suggest  that a  sequence  of  successive
applications  of  the  Steiner transformation  to  such  geo--
graphs" resolves scale problems in the same manner as  fractals.
A   natural  next  step beyond the fractal might be to note that
a  graph  is   a  simplicial complex of   dimension   0   or   1
(Harary,   1969).    Thus,   similar  strategy might be  applied
there:     the   triangles   of   Figure   7.B  might  represent
simplexes  of  arbitrary dimension in  a  simplicial complex  of
higher  dimension.    Theorems  from  algebraic  topology  might
then  be  turned back on the mapping of  geographic  information
using  a  computer.   This  notion  is  already   in   evidence:
because  point,"  line,"  and  area"  translate  into  the
topological notions of 0--cell," 1--cell," and 2--cell" in
a  Geographic  Information  System,   cells  in  the  underlying
computerized  sim--pixel"  complex  can  then  be  colored  as
inside" or outside"  a  given data set.  This  follows  from
the  Jordan  Curve  Theorem   (of   algebraic topology).

Independent of choice of theoretical  tool---from fractal  to
graph to simplicial complex---the resolution of scale  is achieved
by uniting  local  and  global  mathematical  structures:   within
fractal geometry as well as beyond it.
\vskip.1cm
{\narrower\noindent
In nature, parts  clearly  do  fit  together  into  real
structures,  and  the  parts  are   affected   by   their
environment.    The   problem   is   largely    one    of
understanding.  The mystery that remains lies largely  in
the nature of structural hierarchy, for  the  human  mind
can examine nature on many different scales sequentially,
but not simultaneously." \smallskip}
{\sl C. S. Smith, in Arthur L. Loeb, 1976\/}.
\vfill\eject

\centerline{\bf References.}

\ref Arlinghaus, S. L. 1985. {\sl Essays on Mathematical Geography\/}.
Institute of Mathematical Geography,  Monograph  \#3.   Ann  Arbor:
Michigan Document Services.
\ref Arlinghaus, S. L. 1977. On Geographic Network Location Theory."
Unpublished  Ph.D.  dissertation,  Department  of  Geography,  The
University of Michigan.

\ref Balaban, A. T.; Davies, R. O.; Harary, F.; Hill, A.; and Westwick,
R.  1970.  Cubic identity graphs and planar  graphs  derived  from
trees.  {\sl Journal\/}, Australian Mathematical Society 11:207-215.

\ref Barnsley, M. 1988. {\sl Fractals  Everywhere\/}. New  York: Academic
Press.

\ref Boyce, W. E. and DiPrima, R. C. 1977. {\sl Elementary  Differential
Equations\/}.  New York:  Wiley.

\ref Dugundji, J.  1966.  {\sl Topology \/}.  Boston:  Allyn and Bacon.

\ref H\"agerstrand, T. 1967.  {\sl Innovation Diffusion as a
Spatial Process\/}. Postscript and translation by Allan Pred.
Chicago:  University of Chicago Press.

\ref Haggett, P.;  Cliff,  A.  D.;  and  Frey,  A. 1977. {\sl Locational
Analysis in Human Geography\/}.  New York:  Wiley.

\ref Harary, F.  1969.  {\sl Graph Theory\/}.  Reading, MA:

\ref Lefschetz,  S.   1949. {\sl Introduction  to  Topology\/}.
Princeton:  Princeton University Press.

\ref Loeb,  A. L. 1976. {\sl Space  Structures:  Their Harmony and

\ref Mandelbrot, B.  1983.  {\sl The Fractal Geometry of Nature\/}.
New York:  W. H. Freeman.

\ref Peitgen, H.-O. and Saupe, D., editors.  1988. {\sl The Science of
Fractal Images\/}.  New York:  Springer.

\ref Rosen, K. H. 1988. {\sl Elementary Number Theory and its
Applications\/}.  Reading, MA:  Addison--Wesley.  \ref Spanier, E. H. 1966. {\sl Algebraic Topology\/}. New York: McGraw--Hill.  \ref Tutte, W. T. 1966. {\sl Connectivity in Graphs\/}. London: Oxford University Press.  \ref Uhlenbeck, G.  E.   1960.   Successive  approximation  methods  in classical  statistical  mechanics.   {\sl Physica \/} (Congress on Many Particle Problems, Utrecht), 26:17-27. \vfill\eject \noindent 3.  SHORT ARTICLE \smallskip \centerline{GROUPS, GRAPHS, AND GOD} \vskip.2cm \centerline{\sl William C. Arlinghaus} \vskip.5cm \centerline{\bf Abstract}       {\nn The fact that almost all graphs are rigid (have trivial automorphism groups) is exploited to argue  probabilistically  for the existence of God.  This  is  presented  in  the  context  that applications of mathematics need  not  be  limited  to  scientific ones.}       Recently I was teaching some elementary  graph  theory  to  a class studying finite mathematics when, inevitably, someone  asked the question, But what is all this good for?"  This  question is posed often, and the answer rarely satisfies either the  poser  or the responder.       Usually the responder is a little annoyed  at  the  question, for often a deeper look by  the  poser  would  have  yielded  some insight into the question.  But also the responder is irritated on account of inability to give a satisfactory answer.   Two  obvious choices present themselves: \vskip.1cm       1.  Most mathematicians find  the  process  of  discovery  in mathematics rewarding in itself.  An elegantly concocted proof  of a pleasingly stated theorem gives a sense of  satisfaction  and  a joy  in  the  appreciation  of  beauty   that   makes  real--world application unnecessary.  But the  questioner  usually  lacks  the mathematical maturity necessary to appreciate this answer. \vskip.1cm      2.  The most readily available sources of application  are  in the physical sciences, although there  is  an  increasing  use  of mathematics in the social sciences.  But often  the  mathematician lacks confidence in the extent of his knowledge of the appropriate science.  This makes response somewhat tentative,  and  again  the response fails to satisfy the questioner. \vskip.1cm       On this  occasion,  a  third  alternative  presented  itself. Being human, all people have some interest in philosophy,  varying from formal study to informal discussion.  What  better  place  to find a meeting ground to answer the above question?  {\bf  Definition  1}  Let $G$ be a  finite   graph.   Then  the  automorphism  group of $G$,  Aut $G$,  is the set of all  edge-- preserving 1--1 maps of the vertex set $V(G)$ onto itself,  with   composition  the  binary operation.  Informally,  one  can view the size of Aut $G$ as a measure   of   the amount of symmetry that $G$ possesses,  the structure of Aut  $G$ as a measure of the way in which the symmetry occurs.  {\bf  Definition  2}   Let $g(n)$ be the number  of  $n$--point  graphs which have non--identity  automorphism group,  $h(n)$ the  number of $n$--point graphs.  Define $f(n)=(g(n))/(h(n))$.  It is well--known [2, 3, 4, 6] that  $$\lim_{n\to \infty} f(n) = 0.$$ In  other  words,  almost all graphs have identity automorphism  group.        Viewed from a philosophical perspective,  this says   that   the  probability  of  symmetry existing in a complex  world   is   virtually  zero.   Yet symmetry  abounds  in  our  own   complex   world.    This provides plausibility for the view that the world  did   not   evolve randomly,  that some force  shaped  it;  {\it  i.e.\/}, it may be taken as a proof" for the existence of God.       One might point out at this point that many other proofs  for the existence of God rely on mathematical foundations.   Causality depends on the belief that  infinite  regress  through  successive causes must eventually reach an infinite  First  Cause.   Anselm's ontological argument involves the idea of being able  to  abstract the idea of perfection and then  posit  its  existence.   Pascal's view that one should behave as if  God  exists  on  the  basis  of expected value of reward if He  does  is  surely  a  probabilistic view.       Since there is a whole first--order class of logical sentences about graphs [1] each of which is either  almost  always  true  or almost never true, further examples of this nature should be  easy to find.  Indeed, to close with one, observe that [3] almost every tree has non--trivial automorphisms.  Thus even a random  tree  has some symmetry.  This might lead one  to  question  Joyce  Kilmer's statement that Only God can make a tree."  \centerline{\bf References.}  \ref 1.  Blass, A. and F. Harary, Properties of almost all graphs and complexes.  {\sl J. Graph Theory\/} 3 (1979) 225-240.  \ref 2.  Erdos, P. and A. Renyi, Asymmetric graphs. {\sl Acta Math. Acad. Sci. Hungar.\/} 14 (1963) 293-315.  \ref 3.  Ford, G. W. and G. E. Uhlenbeck, Combinatorial problems in the theory of graphs.   {\sl Proc. Nat. Acad. Sci. U.S.A.\/} 42 (1956) 122-128, 529-535; 43(1957) 163-167.  \ref 4.  Harary, F., {\sl Graph  Theory\/}.  Addison--Wesley, Reading, Mass. (1969).  \ref 5.  Harary, F. and E. M. Palmer, {\sl Graphical Enumeration\/}. Academic, New York (1973).  \ref 6.  Riddell, R. J., Contributions to the theory of condensation. Dissertation, Univ. of Michigan, Ann Arbor (1951). \vskip.5cm  The author is Associate Professor and Chairperson, Department of Mathematics and Computer Science, Lawrence Technological University, 21000 West  Ten Mile Road, Southfield, MI 48075. This material was presented as a paper to the MIchigan GrapH TheorY (MIGHTY) meeting, Saturday, October 29, 1988 at Oakland University, Rochester, Michigan. \vfill\eject \noindent 4.  REGULAR FEATURES \smallskip \noindent{\bf Theorem Museum} ---       One  purpose  of  a  museum is to  display  to  the  public  concepts  of  an  enduring character in some sort  of  hands--on  manner  that will promote grasp and retention of  that  concept.   When  the display also piques the interest of the  observer,  so  much the better.       This  particular  feature  is  motivated by  a  variety  of  sources.   About  ten  years ago,  William E.  Arlinghaus and  I  submitted  a proposal to {\sl The Mathematical  Intelligencer\/}  for  a  museum exhibit,  based on constructing a  giant  Rubik's  (trademarked  name)  Cube,  to teach  people elements  of  group  theory  by  carrying them physically (in Ferris  wheel  fashion)  through  group theoretic motions while riding inside  the  cube.   At the same time,  I also submitted another proposal to the same  journal  for another museum exhibit to be called The Garden of  Shadows."  This was to be an outdoor display based on using  the  sun  as  a  point source of light  at  infinite"  distance  to  physically  demonstrate  a  number of theorems  from  projective  geometry.       A  number of years later,  I came to know fine artist David  Barr   who  specializes  in  large  outdoor   sculpture.    Bill  Arlinghaus  and John Nystuen are continuing participants  at  my  IMaGe meetings; over the years others have joined us, and one of  the  most regular is David Barr.   Often,  we have,  as a group,  discussed various aspects of using outdoor sculpture to  educate  the  public as well as colleagues.   John Nystuen suggested that  we  build  an actual,  physical  Theorem  Museum,  dedicated  to  Theorems  that  could be portrayed in sculpture (similar to  the  {\sl Intelligencer} proposals).   Barr informs us that  interest  in  this sort of idea is well--established in the world of  Art:   Swiss  artist Max Bill,  and other Western European painters and  sculptors,  create  art determined by mathematical equations  of  various sorts.   Here, we are suggesting that it is the theorem,  itself,  that  is art.   This feature is therefore  the  written  groundwork  for such a museum.   If you have a favorite theorem,  and  can  suggest how to express it  physically  using  artistic  media,  you  might  want  to  consider  submitting  it  to  {\sl  Solstice} for this section.   Theorems that can be so envisioned  may  also be ones that  are easiest to mold to fit other  real-- world phenomena.        Projective  geometry is a highly general geometry that  is  perfectly  symmetric in its statements.   The reason for this is  that  parallel"  lines meet in ideal" points,  lying  on  an  ideal" line, at infinity. Thus, in the projective plane, as in  the Euclidean plane,  two points determine a line;  however,  in  the  projective plane a dual statement (that is NOT true in  the  Euclidean plane) that two lines determine a point is also  true.   Duality  in  language results in symmetry of form.    Here is  a  remarkable theorem from projective geometry  (see reference  for  proof). \smallskip \centerline{\bf Desargues's Two Triangle Theorem.}       Given two triangles,  $PQR$ and $P'Q'R'$ such that   $PP'$,  $QQ'$,  and  $RR'$ are concurrent at point O.   It  follows that  the  intersection  points  of corresponding  sides  of  the  two  triangles are collinear.   That is,  suppose that  corresponding  sides $PQ$ and $P'Q'$ intersect at point L, that $QR$ and $Q'R'$  intersect  at  point M,  and that $PR$ and $P'R'$  intersect  at  point N.   Then,  the points L,  M, and N all lie along a single  straight   line   (please  draw  your  own  figure  from   these  directions).

\topinsert \vskip7.5in
Figure to accompany Desargues's Two Triangle Theorem

Figure 9.
\endinsert

From  a  geographic  viewpoint,   this  says  that  if  a  rigid
tetrahedron were built of metal rods with apex at point O,  that
any two triangles that fit perfectly inside this structure would
have this property.   One triangle projects" from a point  (as
for example in gnomonic or stereographic  map projection) to the
other.    This  might  suggest  a  way  to  deform  cells  of  a
triangulation  of a region of the earth into one another in such
a  way  that  this Desargues's line serves as some  sort  of  an
invariant of the deformation.

This  observation might then make one wonder what sorts of
geometries exist that do not obey Desargues's  Theorem.    There
is  a  whole  class  of Combinatorial  geometries"  or  finite
projective planes that do not.
\smallskip
References
\smallskip

\ref Coxeter, H. S. M. {\sl Introduction to Geometry\/}, New York:
Wiley, 1961.

\ref Coxeter, H. S. M.  {\sl Projective Geometry\/}, Toronto:  University
of Toronto Press, 1974.
\vfill\eject
\noindent{\bf Construction Zone} ---

One  possible direction for application of Desargues's  Theorem
is to deform one tesselation of a region into  another,  leaving
something  invariant.   Another related issue with  tesselations
is  to  try to regularize a tesselation composed of  irregularly  shaped
cells.   The following construction shows how to derive a
centrally symmetric hexagon from an arbitrary convex hexagon.
Given an arbitrary convex hexagon,  $V_1V_2V_3V_4V_5V_6$.   Join
alternate  vertices to inscribe a six--pointed star within  this
hexagon---that  is,  draw lines  $V_1V_3$,  $V_2V_4$,  $V_3V_5$,
$V_4V_6$,  $V_5V_1$, $V_6V_2$ (it is suggested that you do so on
a separate sheet of paper, at this point).

\topinsert \vskip5.5in
Figure to accompany construction of centrally symmetric hexagon.

Figure 10.

\endinsert
This  produces  six  distinct triangles  (of  interest  here--of
course there are more):
$$\triangle V_1V_2V_3; \quad \triangle V_2V_3V_4; \quad \triangle V_3V_4V_5; \quad \triangle V_4V_5V_6; \quad \triangle V_5V_6V_1; \quad \triangle V_6V_1V_2.$$
To find the center of gravity of any triangle,  find the  point
at  which  the  medians are concurrent (the median is  the  line
joining  a vertex to the midpoint of the opposite  side).   This
point is the center of gravity.  Find the centers of gravity
$$G_1, \quad G_2, \quad G_3, \quad G_4, \quad G_5, \quad G_6$$
of  each  of  the triangles distinguished above  (in  the  order
suggested).  The hexagon determined by these centers of  gravity
will  be centrally symmetric.   That is,  opposite sides will be
equal in length and parallel to each other:
$$G_1G_2 \parallel G_4G_5; \quad |G_1G_2|=|G_4G_5|;$$
$$G_2G_3 \parallel G_5G_6; \quad |G_2G_3|=|G_5G_6|;$$
$$G_3G_4 \parallel G_6G_1; \quad |G_3G_4|=|G_6G_1|.$$
Another  way of visualizing the symmetry is to observe that  the
three lines joining $G_1G_4$,  $G_2G_5$, $G_3G_6$ are concurrent
at a single point (call it $O$).   In this way,  one might  also
determine  a  center" for this symmetric hexagon  which  might
then  serve  as  a  point to which a reference  value  might  be
attached  for the arbitrary hexagon from which it  was  derived.
This  centrally symmetric hexagon is called the Dirichlet region
of  the  arbitrary convex hexagon.   This  construction  can  be
proved  using Euclidean geometry (if requests come in,  I'll put  it in
a later issue).
\smallskip
This feature is based on discussions in
\smallskip
\ref Kasner, Edward, and Newman, James R. New names for old," in
{\sl The World of Mathematics\/}, edited by James R. Newman, Volume III,
1996-2010.  New York:  Simon and Schuster, 1956.

\ref Coxeter, H. S. M. {\sl Introduction to Geometry\/}, New York:
Wiley, 1961.
\vfill\eject
\noindent{\bf Reference Corner} ---
\vskip.5cm

Point set theory and topology.   A recent pleasant evening
spent  with  Hal  Moellering had him  questioning  me  and  Bill
Arlinghaus as to what might be reasonable, or useful, references
from which graduate students in geography could get some sort of
grasp  of the elements of point set topology.   A few references
are  listed  below;  send  in your favorites and  they  will  be
printed  next  time.   Hope  that   mathematicians  as  well  as
geographers  will do so.  Future topics to include graph  theory
and number theory as well as others suggested by reader input.
Thanks  Hal for the idea (generated by your questions) of  doing
this feature!

\noindent Some long--time favorites and classics:

\ref Dugundji, James.  {\sl Topology\/}.  Boston:  Allyn and Bacon,
1960.

\ref Hall, Dick Wick and Guilford L. Spencer II, {\sl Elementary
Topology\/}, New York:  Wiley, 1955.

\ref Halmos, Paul R., {\sl Na\"{\i}ve Set Theory\/}, Princeton: D. Van
Nostrand, 1960.

\ref Hausdorff, Felix, {sl Mengenlehre\/}, Berlin:  Walter de Gruyer,
1935.

\ref Hocking, John G. and Gail S. Young, {\sl Topology},

\ref Kelley, John L., {\sl General Topology\/}, Princeton:
D. Van Nostrand, 1955.

\ref Landau, Edmund, {\sl Grundlagen der Analysis\/}, New York:
Chelsea, 1946.  Third edition, 1960.

\ref Mansfield, Maynard J.  {\sl Introduction to Topology\/}, Princeton:
D. Van Nostrand, 1963.
\vfill\eject
\noindent{\bf Games and other educational features} ---
\smallskip
\centerline{\bf Crossword puzzle.}
The focus of this puzzle is on herbs and spices.  Spice trade
has helped to shape many geographic alignments and spices such  as
pepper, known from its preservative  characteristic,  helped  make
long voyages possible.  Puzzles should be fun;  they  should  also
stimulate thought and offer some sort of  educational  value.   If
you think that this puzzle might be of use  to  your  students  in
this capacity, feel free to copy it from this page.  Think of the
asterisks as the blank squares, or as tiles with letter on the other
side.  Each set of four bullets represents a black square.

Crossword Puzzle.

TYPESETTING, USING TeX, THAT PRODUCED THIS CROSSWORD PUZZLE.
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\setpointsize{12}{9}{8}
\parskip=3pt
\baselineskip=14 pt
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\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
 \noindent ACROSS \vskip.1cm \item{1.}   Plant of the Capsicum family, native to the Americas. Good source of vitamins A and C.  Some varieties are native to Tabasco in Mexico. \item{5.}   Fruit native to the Americas is the prickly ---. \item{10.}  Powder made from young sassafras leaves that is essential in making creole gumbo. \item{13.}  The royal" herb -- often the dominant herb in Pesto. \item{14.}  Bush--bud often seen in Tartare sauce or with an anchovy coiled around it. \item{15.}  Hour -- abbreviation. \item{16.}  College of Liberal ---. \item{17.}  A fundamental tool of the geographer and of the mathematician. \item{18.}  U.S. state -- remove one letter from the spice in 47 across to form an anagram of this state name. \item{20.}  United States -- abbreviation. \item{21.}  Jumble of letters in another." \item{25.}  Black, sticky substance. \item{26.}  Eastern Uganda -- abbreviation. \item{27.}  The bran of this grain is much in vogue. \item{28.}  Unit on a ruler. \item{30.}  He, she, ---. \item{32.}  Herb sometimes used in fruit cup.  Can cause severe  allergic reactions.  Also:  French for street. \item{34.}  Along with coriander or cumin, this is a dominant ingredient in many curries. \item{37.}  A  plant  extract  from  which  candies  can  be  made. \item{39.}  In humans, the color blue, for this, is a recessive genetic trait. \item{40.}  Senior -- abbreviation. \item{41.}  Spice with flavor close to nutmeg. \item{43.}  Chronological or mental ---. \item{44.}  Association of American Geographers:  ---G. \item{46.}  A poem should be palpable and  mute;  As  a  globed  fruit," from Archibald MacLeish's --- Poetica." \item{47.}  Often found in Italian sauces. \item{50.}  Fifth and sixth letters of the alphabet used in English. \item{52.}  Spice often ground freshly and sprinkled on eggnog. \item{54.}  Eau de ---. \item{55.}  Noise a lion might make. \item{57.}  First two letters of Spanish for United States. \item{58.}  Jumble of the letters in the name of an herb with a licorice flavor. \item{61.}  Word that might describe the flavor of a julep (adjectival form). \item{62.}  This broadleaf big onion" is a key ingredient in Vichyssoise. \item{63.}  Herb used in many pickled cucumbers. \item{64.}  Spiced--up" multiplication  tables  might  be  called ---" tables. \vskip.1cm \noindent DOWN \vskip.1cm \item{1.}   This herb supposedly has the power to destroy the scent of garlic and onion. \item{2.}   East, in French. \item{3.}   Italian city -- home to Fibonacci. \item{4.}   Postal letter (abbreviation) \item{5.}   Orangish powder often association with Hungarian dishes. \item{6.}   East Prussia (abbreviation). \item{7.}   Almost everywhere (mathematical term -- abbreviation). \item{8.}   Railroad (abbreviation). \item{9.}   First initial and last name of former Panamanian leader. \item{10.}  A complimentary copy is a --- one. \item{11.}  Left hand opponent (duplicate bridge term, abbreviation). \item{12.}  Jumble of the word neared." \item{17.}  ---s and bounds." \item{19.}  Spiritual guide in Hinduism. \item{22.}  Poland China is a variety of these. \item{23.}  This herb is often held in vinegar because its leaf veins stiffen when dried and do not resoften when cooked.  Estragon" in French. \item{24.}  --- A Clear Day" \item{25.}  Though" -- some newspapers spell that word in this way. \item{29.}  This herb loses most of its flavor when dried:   Pluches  de cerfeuil" refers to sprigs of this herb. \item{31.}  If/---":  typical manner in which a theorem is stated. \item{33.}  Removes from political office. \item{34.}  One variety of this herb, often used in conjunction with fat fish and lentils, is known as Florence ---. \item{35.}  Tidy. \item{36.}  Paramedic vans are often marked with these three letters. \item{38.}  Uncontrolled anger. \item{42.}  Company (abbreviation) \item{45.}  Running --- (Malay word).  To be in a violently frenzied state. \item{48.}  Fine German white wine made from grapes harvested after frost:  ---wein. \item{49.}  Oyster Research Institute of Michigan, might be abbreviated thus. \item{51.}  Popular description of wok cookery:  stir---. \item{53.}  Employ. \item{56.}  Identity element of the integers under multiplication. \item{58.}  Anno Domini (abbreviation) \item{59.}  National income (abbreviation) \item{60.}  Elevated train (abbreviation) -- forms Loop" in Chicago. \item{61.}  Prefix meaning muscle." \vfill\eject \noindent{\bf Coming attractions} --- \vskip.5cm \line{Feigenbaum's number \hfil} \line{Pascal's theorem from projective geometry \hfil} \line{Braikenridge--MacLaurin construction for a conic in the projective      plane. \hfil}   \vfill\eject \noindent{\bf Crossword puzzle solution}
Crossword Puzzle Solution.

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\phantom{0}E}}& {{\phantom{0} \phantom{0} \atop K}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{37 \atop \phantom{0}H}}& {{\phantom{0} \phantom{0} \atop O}}& {{38 \atop \phantom{0}R}}& {{\phantom{0} \phantom{0} \atop E}}& {{\phantom{0} \phantom{0} \atop H}}& {{\phantom{0} \phantom{0} \atop O}}& {{\phantom{0} \phantom{0} \atop U}}& {{\phantom{0} \phantom{0} \atop N}}& {{\phantom{0} \phantom{0} \atop D}}&\cr {{39 \atop \phantom{0}E}}& {{\phantom{0} \phantom{0} \atop Y}}& {{\phantom{0} \phantom{0} \atop E}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{40 \atop \phantom{0}S}}& {{\phantom{0} \phantom{0} \atop R}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{41 \atop \phantom{0}M}}& {{\phantom{0} \phantom{0} \atop A}}& {{42 \atop \phantom{0}C}}& {{\phantom{0} \phantom{0} \atop E}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{43 \atop \phantom{0}A}}& {{\phantom{0} \phantom{0} \atop G}}& {{\phantom{0} \phantom{0} \atop E}}& {{\bullet \atop \bullet}{\bullet 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{{\phantom{0} \phantom{0} \atop G}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{54 \atop \phantom{0}V}}& {{\phantom{0} \phantom{0} \atop I}}& {{\phantom{0} \phantom{0} \atop E}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{55 \atop \phantom{0}R}}& {{56 \atop \phantom{0}O}}& {{\phantom{0} \phantom{0} \atop A}}& {{\phantom{0} \phantom{0} \atop R}}&\cr {{57 \atop \phantom{0}E}}& {{\phantom{0} \phantom{0} \atop S}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\phantom{0} \phantom{0} \atop O}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\phantom{0} \phantom{0} \atop O}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{58 \atop \phantom{0}A}}& {{59 \atop \phantom{0}N}}& {{60 \atop \phantom{0}E}}& {{\phantom{0} \phantom{0} \atop I}}& {{\phantom{0} \phantom{0} \atop S}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{61 \atop \phantom{0}M}}& {{\phantom{0} \phantom{0} \atop I}}& {{\phantom{0} \phantom{0} \atop N}}& {{\phantom{0} \phantom{0} \atop T}}& {{\phantom{0} \phantom{0} \atop Y}}&\cr {{62 \atop \phantom{0}L}}& {{\phantom{0} \phantom{0} \atop E}}& {{\phantom{0} \phantom{0} \atop E}}& {{\phantom{0} \phantom{0} \atop K}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\phantom{0} \phantom{0} \atop N}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{63 \atop \phantom{0}D}}& {{\phantom{0} \phantom{0} \atop I}}& {{\phantom{0} \phantom{0} \atop L}}& {{\phantom{0} \phantom{0} \atop L}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{64 \atop \phantom{0}T}}& {{\phantom{0} \phantom{0} \atop H}}& {{\phantom{0} \phantom{0} \atop Y}}& {{\phantom{0} \phantom{0} \atop M}}& {{\phantom{0} \phantom{0} \atop E}}& {{\phantom{0} \phantom{0} \atop S}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}&\cr }$$ \endinsert \vfill\eject

\hsize = 6.5 true in %THE FIRST 14 LINES ARE TYPESETTING CODE.
\setpointsize{12}{9}{8}
\parskip=3pt
\baselineskip=14 pt
\mathsurround=1pt
\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
\smallskip
This section shows the exact set of commands that  work  to
Because different universities will have different installations
of {\TeX}, this is only a rough guideline which {\sl might\/} be

This document prints out to be about 50 pages; on UM equipment,
there are varying rates at varying times of day.  At the minimum
rate, the cost to print this out, using  {\TeX} , is  about  six
dollars.

ASSUME YOU HAVE SIGNED ON AND ARE AT THE SYSTEM PROMPT, \#.
\# create $-$t.tex

\# percent--sign t from pc c:backslash words backslash
solstice.tex to mts $-$t.tex char notab

[this command sends my file, solstice.tex, which I did as
a WordStar (subdirectory, words") ASCII file to the
mainframe]

\# run *tex par=$-$t.tex

\# run *dvixer par=$-$t.dvi

\# control *print* onesided

\# run *pagepr scards=$-$t.xer, par=paper=plain
\bye