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 \headline = {\ifnum\pageno=1 \hfil \else {\ifodd\pageno\righthead
 \def\righthead{\sl\hfil SOLSTICE }
 \def\lefthead{\sl Summer, 1990 \hfil}
 \font\big = cmbx17
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 \centerline{\big SOLSTICE:}
 \centerline{\bf SUMMER, 1990}
 \centerline{\bf Volume I, Number 1}
 \centerline{\bf Institute of Mathematical Geography}
 \centerline{\bf Ann Arbor, Michigan}
 \centerline{\bf SOLSTICE}
 \line{Founding Editor--in--Chief:  {\bf Sandra Lach Arlinghaus}. \hfil}
 \centerline{\bf EDITORIAL BOARD}
 \line{{\bf Geography} \hfil}
 \line{{\bf Michael Goodchild}, University of California, Santa Barbara. 
 \line{{\bf Daniel A. Griffith}, Syracuse University. \hfil}
 \line{{\bf Jonathan D. Mayer}, University of Washington; joint appointment
  in School of Medicine.\hfil}
 \line{{\bf John D. Nystuen}, University of Michigan (College of
  Architecture and Urban Planning).}
 \line{{\bf Mathematics} \hfil}
 \line{{\bf William C. Arlinghaus}, Lawrence Technological University. \hfil}
 \line{{\bf Neal Brand}, University of North Texas. \hfil}
 \line{{\bf Kenneth H. Rosen}, A. T. \& T. Information Systems Laboratory.
 \line{{\bf Business} \hfil}
 \line{{\bf Robert F. Austin},
 Director, Automated Mapping and Facilities Management, CDI. \hfil}
       The  purpose of {\sl Solstice\/} is to promote  interaction   
between geography and mathematics.   Articles in which  elements 
 of   one  discipline  are used to shed light on  the  other  are  
 particularly sought.   Also welcome,  are original contributions  
 that are purely geographical or purely mathematical.   These may  
 be  prefaced  (by editor or author) with  commentary  suggesting  
 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.
 \noindent {\bf Send all correspondence to:}
 \centerline{\bf Institute of Mathematical Geography}
 \centerline{\bf 2790 Briarcliff}
 \centerline{\bf Ann Arbor, MI 48105-1429}
 \centerline{\bf (313) 761-1231}
 \centerline{\bf IMaGe@UMICHUM}
       This  document is produced using the typesetting  program, 
 {\TeX},  of Donald Knuth and the American Mathematical  Society.  
 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
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 Copyright, August, 1990, Institute of Mathematical Geography.
 All rights reserved.
 ISBN: 1-877751-51-0
 \centerline{\bf SUMMARY OF CONTENT}
 Note:  in this first issue, there is one of each type of article--this
 need not be the case in the future.
 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?
 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
 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.
  Note:   In  this  first  issue,  there is one of each  type  of 
 article--this need not be the case in the future.
 \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.
 \item{vi.}{\bf Solution to puzzle}
      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.
 \noindent 1.  REPRINT
 \centerline{William Kingdon Clifford}
 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.
 \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 ---
 The boundary of a solid ({\it i.e.\/},  of a part of space) is a
 The boundary of a part of a surface is a line.
 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
       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
       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
       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
      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 
      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.
 \noindent 2.  FULL--LENGTH ARTICLE
 \centerline{\bf BEYOND THE FRACTAL} \vskip.1cm
 \centerline{\sl Sandra Lach Arlinghaus}
 \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."}
 {\sl Emily Dickinson, ``Chartless."}
  \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 
      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. \topinsert

Figure 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.

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


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

\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.

 {\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}}&
 I&{\phantom{0}1.40}  &{\phantom{0}4.41}

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 the middle adopter being gradually pulled, to the North, from 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.
Figure 3B.

 {\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}}&
 E&  &{\phantom{0}0.96}
 &&&&&&&&&&\cr &{\phantom{0}6.40}

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 adopters. \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 enclosure that you have added 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.

 {\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}}&
  &  &{14.29}\cr
  &\phantom{0}28.58\cr J&{\phantom{0}0.96}

\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.

 \settabs\+\indent \quad&Figure 3.B:  middle cell 2 units north
   \qquad\quad&Number of cells\qquad&(out of 300) in&\cr
 \+&Figure number:     &Number of cells &Largest value   \cr
 \+&Position of three  &in interaction  &(out of 300) in \cr
 \+&original adopters. &zone.           &individual cell.\cr
 \+&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
      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}.

Figure 4.

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, $D$.

 \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}

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.


\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 connection. Viewed broadly, the diffusion example sees adopters 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.

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

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

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
      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,
 \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}.
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.

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: Addison--Wesley. \ref Lefschetz, S. 1949. {\sl Introduction to Topology\/}. Princeton: Princeton University Press. \ref Loeb, A. L. 1976. {\sl Space Structures: Their Harmony and Counterpoint\/}. Reading, MA: Addison--Wesley. \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.


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.


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}, Reading: Addison--Wesley, 1961. \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.

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\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
 \noindent ACROSS
 \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
 \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 
 \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
 \item{63.}  Herb used in many pickled cucumbers.
 \item{64.}  ``Spiced--up" multiplication  tables  might  be  called
 ``---" tables.
 \noindent DOWN
 \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
 \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
 \item{48.}  Fine German white wine made from grapes harvested after
 frost:  ---wein.
 \item{49.}  Oyster Research Institute of Michigan, might be abbreviated
 \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."
 \noindent{\bf Coming attractions} ---
 \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}  
 \noindent{\bf Crossword puzzle solution}
Crossword Puzzle Solution.

 \hsize = 6.5 true in
 \input fontmac
 \baselineskip=11 pt
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{1\phantom{0} \atop P}}& 
 {{2\phantom{0} \atop E}}&
 {{3\phantom{0} \atop P}}&
 {{4\phantom{0} \atop P}}&
 {{\phantom{0}\phantom{0} \atop E}}&
 {{\phantom{0}\phantom{0} \atop R}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{5 \phantom{0} \atop P}}&
 {{6 \phantom{0} \atop E}}&
 {{7 \phantom{0} \atop A}}&
 {{8 \phantom{0} \atop R}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{9 \phantom{0} \atop M}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{10 \atop \phantom{0}F}}&
 {{\phantom{0} \phantom{0} \atop I}}&
 {{11 \atop \phantom{0}L}}&
 {{12 \atop \phantom{0}E}}&\cr
 {{13 \atop \phantom{0}B}}&
 {{\phantom{0} \phantom{0} \atop A}}&
 {{\phantom{0} \phantom{0} \atop S}}& {{\phantom{0} \phantom{0} \atop I}}&
 {{\phantom{0} \phantom{0} \atop L}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{14 \atop \phantom{0}C}}&
 {{\phantom{0} \phantom{0} \atop A}}&
 {{\phantom{0} \phantom{0} \atop P}}&
 {{\phantom{0} \phantom{0} \atop E}}&
 {{\phantom{0} \phantom{0} \atop R}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{\phantom{0} \phantom{0} \atop N}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{\phantom{0} \phantom{0} \atop R}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{15 \atop \phantom{0}H}}&
 {{\phantom{0} \phantom{0} \atop R}}&\cr
 {{16 \atop \phantom{0}A}}&
 {{\phantom{0} \phantom{0} \atop R}}&
 {{\phantom{0} \phantom{0} \atop T}}&
 {{\phantom{0} \phantom{0} \atop S}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{17 \atop \phantom{0}M}}&
 {{\phantom{0} \phantom{0} \atop A}}&
 {{\phantom{0} \phantom{0} \atop P}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{18 \atop \phantom{0}O}}&
 {{\phantom{0} \phantom{0} \atop R}}&
 {{\phantom{0} \phantom{0} \atop E}}&
 {{19 \atop \phantom{0}G}}&
 {{\phantom{0} \phantom{0} \atop O}}&
 {{\phantom{0} \phantom{0} \atop N}}&\cr
 {{20 \atop \phantom{0}U}}&
 {{\phantom{0} \phantom{0} \atop S}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{21 \atop \phantom{0}A}}&
 {{22 \atop \phantom{0}H}}&
 {{23 \atop \phantom{0}T}}&
 {{\phantom{0} \phantom{0} \atop E}}&
 {{\phantom{0} \phantom{0} \atop N}}&
 {{\phantom{0} \phantom{0} \atop R}}&
 {{24 \atop \phantom{0}O}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{25 \atop \phantom{0}T}}&
 {{\phantom{0} \phantom{0} \atop A}}&
 {{\phantom{0} \phantom{0} \atop R}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{26 \atop \phantom{0}E}}&
 {{\phantom{0} \phantom{0} \atop U}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{\phantom{0} \phantom{0} \atop A}}&\cr
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\phantom{0} 
\phantom{0} \atop L}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{27 \atop \phantom{0}O}}&
 {{\phantom{0} \phantom{0} \atop A}}&
 {{\phantom{0} \phantom{0} \atop T}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{28 \atop \phantom{0}I}}&
 {{\phantom{0} \phantom{0} \atop N}}&
 {{29 \atop \phantom{0}C}}&
 {{\phantom{0} \phantom{0} \atop H}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{30 \atop \phantom{0}I}}&
 {{31 \atop \phantom{0}T}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{32 \atop \phantom{0}R}}&
 {{33 \atop \phantom{0}U}}&
 {{\phantom{0} \phantom{0} \atop E}}&\cr
 {{34 \atop \phantom{0}F}}&
 {{\phantom{0} \phantom{0} \atop E}}&
 {{35 \atop \phantom{0}N}}&
 {{\phantom{0} \phantom{0} \atop U}}&
 {{\phantom{0} \phantom{0} \atop G}}&
 {{\phantom{0} \phantom{0} \atop R}}&
 {{\phantom{0} \phantom{0} \atop E}}&
 {{36 \atop \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 \atop \bullet}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{\phantom{0} \phantom{0} \atop S}}& {{\bullet \atop \bullet}{\bullet 
\atop \bullet}}&\cr
 {{\phantom{0} \phantom{0} \atop N}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{44 \atop \phantom{0}A}}&
 {{45 \atop \phantom{0}A}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{46 \atop \phantom{0}A}}&
 {{\phantom{0} \phantom{0} \atop R}}&
 {{\phantom{0} \phantom{0} \atop S}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{47 \atop \phantom{0}O}}&
 {{\phantom{0} \phantom{0} \atop R}}&
 {{48 \atop \phantom{0}E}}&
 {{\phantom{0} \phantom{0} \atop G}}&
 {{\phantom{0} \phantom{0} \atop A}}&
 {{\phantom{0} \phantom{0} \atop N}}&
 {{49 \atop \phantom{0}O}}&
 {{\bullet \atop \bullet}{\bullet \atop \bullet}}&
 {{50 \atop \phantom{0}E}}&
 {{51 \atop \phantom{0}F}}&\cr
 {{52 \atop \phantom{0}N}}&
 {{53 \atop \phantom{0}U}}&
 {{\phantom{0} \phantom{0} \atop T}}&
 {{\phantom{0} \phantom{0} \atop M}}&
 {{\phantom{0} \phantom{0} \atop E}}&
 {{\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

\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 \noindent 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. 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