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e-mail
\centerline{\big SOLSTICE:}
\vskip.5cm
\centerline{\bf  AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS}
\vskip5cm
\centerline{\bf SUMMER, 1990}
\vskip12cm
\centerline{\bf Volume I, Number 1}
\smallskip
\centerline{\bf Institute of Mathematical Geography}
\vskip.1cm
\centerline{\bf Ann Arbor, Michigan}
\vfill\eject
\hrule
\smallskip
\centerline{\bf SOLSTICE}
\line{Founding Editor--in--Chief:  {\bf Sandra Lach Arlinghaus}. \hfil}
\smallskip
\centerline{\bf EDITORIAL BOARD}
\smallskip
\line{{\bf Geography} \hfil}
\line{{\bf Michael Goodchild}, University of California, Santa Barbara.
\hfil}
\line{{\bf Daniel A. Griffith}, Syracuse University. \hfil}
\line{{\bf Jonathan D. Mayer}, University of Washington; joint appointment
in School of Medicine.\hfil}
\line{{\bf John D. Nystuen}, University of Michigan (College of
Architecture and Urban Planning).}
\smallskip
\line{{\bf Mathematics} \hfil}
\line{{\bf William C. Arlinghaus}, Lawrence Technological University. \hfil}
\line{{\bf Neal Brand}, University of North Texas. \hfil}
\line{{\bf Kenneth H. Rosen}, A. T. \& T. Information Systems Laboratory.
\hfil}
\smallskip
\line{{\bf Robert F. Austin},
Director, Automated Mapping and Facilities Management, CDI. \hfil}
\smallskip
\hrule
\smallskip

The  purpose of {\sl Solstice\/} is to promote  interaction
between geography and mathematics.   Articles in which  elements
of   one  discipline  are used to shed light on  the  other  are
particularly sought.   Also welcome,  are original contributions
that are purely geographical or purely mathematical.   These may
be  prefaced  (by editor or author) with  commentary  suggesting
directions  that  might  lead toward  the  desired  interaction.
Individuals  wishing to submit articles,  either short or full--
length,  as well as contributions for regular  features,  should
send  them,  in triplicate,  directly to the  Editor--in--Chief.
Contributed  articles  will  be refereed by  geographers  and/or
mathematicians.   Invited articles will be screened by  suitable
members of the editorial board.  IMaGe is open to having authors
suggest, and furnish material for, new regular features.
\vskip2in
\noindent {\bf Send all correspondence to:}
\vskip.1cm
\centerline{\bf Institute of Mathematical Geography}
\centerline{\bf 2790 Briarcliff}
\centerline{\bf Ann Arbor, MI 48105-1429}
\vskip.1cm
\centerline{\bf (313) 761-1231}
\centerline{\bf IMaGe@UMICHUM}
\vfill\eject

This  document is produced using the typesetting  program,
{\TeX},  of Donald Knuth and the American Mathematical  Society.
Notation  in  the electronic file is in accordance with that  of
hard copy for on The University of Michigan's Xerox 9700 laser--
printing  Xerox machine,  using IMaGe's commercial account  with
that University.

Unless otherwise noted, all regular features are written by the
Editor--in--Chief.
\smallskip
{\nn  Upon final acceptance,  authors will work with IMaGe
to    get  manuscripts   into  a  format  well--suited  to   the
requirements   of {\sl Solstice\/}.  Typically,  this would mean
that  authors    would  submit    a  clean  ASCII  file  of  the
manuscript,  as well as   hard copy,  figures,  and so forth (in
camera--ready form).     Depending on the nature of the document
and   on   the  changing    technology  used  to  produce   {\sl
Solstice\/},   there  may  be  other    requirements  as   well.
Currently,  the  text  is typeset using   {\TeX};  in that  way,
mathematical formul{\ae} can be transmitted   as ASCII files and
inexperienced  in the use of {\TeX} should note that  this    is
not  a what--you--see--is--what--you--get"  display;  however,
we  hope  that  such readers find {\TeX} easier to  learn  after
exposure to {\sl Solstice\/}'s e-files written using {\TeX}!}

{\nn  Copyright  will  be taken out in  the  name  of  the
Institute of Mathematical Geography, and authors are required to
transfer  copyright  to  IMaGe as a  condition  of  publication.
There  are no page charges;  authors will be given permission to  make
reprints from the electronic file,  or to have IMaGe make a
single master reprint for a nominal fee dependent on  manuscript
length.   Hard  copy of {\sl Solstice\/}  will be sold  (contact
IMaGe  for  price--{\sl Solstice\/}  will  be  priced  to  cover
expenses  of journal production);  it is the desire of IMaGe  to
kind of academic newsstand at which one might browse,  prior  to
making purchasing decisions.  Whether or not it will be feasible
to  continue distributing complimentary electronic files remains
to be seen.}
\vskip.5cm
Copyright, August, 1990, Institute of Mathematical Geography.
\vskip1cm
ISBN: 1-877751-51-0
\vfill\eject
\centerline{\bf SUMMARY OF CONTENT}
\smallskip
Note:  in this first issue, there is one of each type of article--this
need not be the case in the future.
\vskip.2cm
1. REPRINT.
\smallskip
William Kingdon Clifford, {\sl Postulates of the Science of Space\/}.

This  reprint  of a portion of Clifford's lectures  to  the
Royal  Institution in the 1870's suggests many geographic topics
of concern in the last half of the twentieth century.   Look for
connections  to boundary issues,  to scale problems,  to  self--
similarity and fractals,  and to non--Euclidean geometries (from
those  based on denial of Euclid's parallel postulate  to  those
based on a sort of mechanical polishing").  What else did,  or
\smallskip
2.  FULL--LENGTH ARTICLE.
\smallskip
Sandra L. Arlinghaus, {\sl Beyond the Fractal\/}.
Figures are transmitted in this e-file only for the half of the
article described in the first paragraph below.

An   original  article.    The  fractal  notion  of  self--
similarity  is useful for characterizing change  in  scale;  the
reason  fractals are effective in the geometry of central  place
theory  is  because  that geometry is  hierarchical  in  nature.
Thus, a natural place to look for other connections of this sort
is  to  other geographical concepts that are also  hierarchical.
spatial diffusion.

When  the idea of diffusion is extended to see  adopters"
of  an innovation as attractors" of new adopters,  a Julia set
is  introduced as a possible axis against which to  measure  one
class  of  geographic phenomena.   Beyond the  fractal  context,
fractal  concepts,  such as compression" and space--filling"
are  considered in a broader graph--theoretic context. \smallskip
3.  SHORT ARTICLE.
\smallskip
William C. Arlinghaus, {\sl Groups, graphs, and God\/}

An original article based on a talk given before a MIdwest
GrapH TheorY (MIGHTY) meeting.   The author,  an algebraic graph
theorist, ties his research interests to a broader philosophical
realm,  suggesting  the  breadth  of range  to  which  algebraic
structure might be applied.

The  fact that almost all graphs are rigid  (have  trivial
automorphism  groups)  is exploited to argue   probabilistically
for the existence of God.  This  is  presented  in  the  context
that  applications  of mathematics need  not   be   limited   to
scientific ones.
\smallskip
Note:   In  this  first  issue,  there is one of each  type  of
article--this need not be the case in the future.
\smallskip
4.  REGULAR FEATURES
\smallskip
\item{i.} {\bf Theorem Museum} ---
Desargues's Two Triangle Theorem of projective geometry.
\item{ii.} {\bf Construction Zone} ---
a  centrally symmetric hexagon is derived from an  arbitrary
convex hexagon.
\item{iii.} {\bf Reference Corner} ---
Point set theory and topology.
\item{iv.} {\bf Games and other educational features} ---
Crossword puzzle focused on spices.
\item{v.} {\bf Coming attractions} ---
Indication of topics for the REGULAR FEATURES" section
in forthcoming issues.
\smallskip
\item{vi.}{\bf Solution to puzzle}
\smallskip
\smallskip
This section shows the exact set of commands that  work  to
Because different universities will have different installations
of {\TeX}, this is only a rough guideline which {\sl might\/} be
\vfill\eject
\noindent 1.  REPRINT
\smallskip
\centerline{\bf THE POSTULATES OF THE SCIENCE OF SPACE}
\vskip.2cm
\centerline{William Kingdon Clifford}
\smallskip
From a set of lectures given before the Royal Institution, 1873 --
The Philosophy of the Pure Sciences."  Reprinted excerpt
longer than this one appears in
{\sl The  World of Mathematics\/}, edited by James R. Newman, New York:
Simon and Schuster, 1956.
\smallskip
\noindent   In my first lecture I said that,  out of the pictures
which are all that we can really see, we imagine a world of solid
things;  and  that  this world is constructed so as to  fulfil  a
certain  code  of rules,  some called  axioms,  and  some  called
definitions, and some called postulates, and some assumed  in the
course of demonstration, but all laid down in one form or another
in Euclid's Elements of Geometry.   It is this code of rules that
we have to consider to--day.   I do not, however, propose to take
this book that I have mentioned, and to examine one after another
the  rules as Euclid has laid them down or unconsciously  assumed
them; notwithstanding that many things might be said in favour of
such  a  course.   This  book  has been  for  nearly  twenty--two
centuries the encouragement and guide of that scientific  thought
which  is  one thing with the progress of man from a worse  to  a
better  state.   The  encouragement;  for it contained a body  of
knowledge that was really known and could be relied on,  and that
moreover was growing in extent and application.   For even at the
time this book was written---shortly after the foundation of  the
Alexandrian  Museum--Mathematic  was no longer the  merely  ideal
science of the Platonic school,  but had started on her career of
conquest over the whole world of Phenomena.   The guide;  for the
aim of every scientific student of every subject was to bring his
knowledge  of  that subject into a form as perfect as that  which
geometry  had attained.   Far up on the great mountain of  Truth,
which all the sciences hope to scale, the foremost of that sacred
sisterhood was seen,  beckoning to the rest to follow  her.   And
hence  she was called,  in the dialect of the Pythagoreans,  the
purifier  of the reasonable soul.'  Being thus in itself at  once
the  inspiration and the aspiration of scientific  thought,  this
Book of Euclid's has had a history as chequered as that of  human
progress itself.  [Deleted text.]  The geometer of to--day  knows
nothing  about  the  nature  of actually  existing  space  at  an
infinite distance;  he knows nothing about the properties of this
present space in a past or a future eternity.   He knows, indeed,
that the laws assumed by Euclid are true with an accuracy that no
direct experiment can approach,  not only in this place where  we
are,  but  in places at a distance from us that no astronomer has
conceived; but he knows this as of Here and Now; beyond his range
is a There and Then of which he knows nothing at present, but may
ultimately  come to know more.   So,  you see,  there is  a  real
parallel between the work of Copernicus and his successors on the
one hand,  and the work of Lobatchewsky and his successors on the
other.   In both of these the knowledge of Immensity and Eternity
is replaced by knowledge of Here and Now.  And in virtue of these
two revolutions the idea of the Universe, the Macrocosm, the All,
as  subject of human knowledge,  and therefore of human interest,
has fallen to pieces.

It will now,  I think, be clear to you why it will not do to
take for our present consideration the postulates of geometry  as
Euclid  has laid them down.   While they were all certainly true,
there  might  be  substituted  for  them  some  other  group   of
equivalent propositions;  and the choice of the particular set of
statements  that should be used as the groundwork of the  science
was  to  a  certain  extent  arbitrary,   being  only  guided  by
convenience  of exposition.   But from the moment that the actual
truth  of  these  assumptions  becomes  doubtful,  they  fall  of
themselves into a necessary order and classification; for we then
begin  to  see  which of them may be true  independently  of  the
others.   And  for  the purpose of criticizing the  evidence  for
them,  it is essential that this natural order should  be  taken;
for  I  think you will see presently that any other  order  would
bring hopeless confusion into the discussion.

Space  is divided into parts in many ways.   If we  consider
any material thing,  space is at once divided into the part where
that  thing is and the part where it is not.   The water in  this
glass,  for example,  makes a distinction between the space where
it is and the space where it is not.   Now,  in order to get from
one  of these to the other you must cross the {\it surface\/}  of
the  water;  this surface is the boundary of the space where  the
water  is  which  separates it from the space where  it  is  not.
Every {\it thing\/},  considered as occupying a portion of space,
has  a  surface which separates that space where it is  from  the
space where it is not.  But, again, a surface may be divided into
parts  in  various ways.   Part of the surface of this  water  is
against  the air,  and part is against the glass.   If you travel
over the surface from one of these parts to the other,  you  have
to cross the {\it line\/} which divides them; it is this circular
edge where water,  air,  and glass meet.  Every part of a surface
is separated from the other parts by a line which bounds it.  But
now  suppose,  further,  that this glass had been so  constructed
that  the  part towards you was blue and the part towards me  was
white,  as it is now.   Then this line, dividing two parts of the
surface  of the water,  would itself be divided into  two  parts;
there would be a part where it was against the blue glass,  and a
part  where  it was against the white glass.   If you  travel  in
thought along that line, so as to get from one of these two parts
to  the other,  you have to cross a {\it point\/} which separates
them,  and is the boundary between them.  Every part of a line is
separated from the other parts by points which bound it.   So  we
may say altogether ---
\vskip.1cm
The boundary of a solid ({\it i.e.\/},  of a part of space) is a
surface.
\vskip.1cm
The boundary of a part of a surface is a line.
\vskip.1cm
The boundaries of a part of a line are points.

And  we are only settling the meanings in which words are to
be  used.   But here we may make an observation which is true  of
all  space that we are acquainted with:   it is that the  process
ends  here.   There are no parts of a point which  are  separated
from  one another by the next link in the series.   This is  also
indicated by the reverse process.

For  I shall now suppose this point --- the last thing  that we
got  to --- to move round the tumbler so as to trace out  the
line,  or edge,  where air, water, and glass meet.  In this way I
get  a series of points,  one after another;  a series of such  a
nature that,  starting from any one of them, only two changes are
possible  that  will  keep it within  the  series:   it  must  go
forwards or it must go backwards,  and each of these if perfectly
definite.   The  line  may  then be regarded as an  aggregate  of
points.   Now let us imagine,  further, a change to take place in
this  line,  which  is nearly a circle.   Let us  suppose  it  to
contract  towards  the  centre of the circle,  until  it  becomes
indefinitely small,  and disappears.   In so doing it will  trace
out the upper surface of the water, the part of the surface where
it  is  in   contact with the air.   In this way we shall  get  a
series of circles one after another --- a series of such a nature
that,  starting  from  any  one of them,  only  two  changes  are
possible that will keep it within the series:   it must expand or
it must contract.   This series,  therefore,  of circles, is just
similar to the series of points that make one circle; and just as
the line is regarded as an aggregate of points,  so we may regard
this surface as an aggregate of lines.   But this surface is also
in another sense an aggregate of point,  in being an aggregate of
aggregates of points.  But, starting from a point in the surface,
more  than two changes are possible that will keep it within  the
surface, for it may move in any direction.  The surface, then, is
an  aggregate of points of a different kind from  the  line.   We
speak  of  the  line  as a  point--aggregate  of  one  dimension,
because,  starting  from one point,  there are only two  possible
directions  of change;  so that the line can be traced out in one
motion.   In the same way,  a surface is a line--aggregate of one
dimension,  because  it  can be traced out by one motion  of  the
line; but it is a point--aggregate of two dimensions, because, in
order to build it up of points, we have first to aggregate points
into  a line,  and then lines into a surface.   It  requires  two
motions of a point to trace it out.

Lastly,  let  us suppose this upper surface of the water to
move downwards,  remaining always horizontal till it becomes  the
under  surface.   In so doing it will trace out the part of space
occupied  by the water.   We shall thus get a series of  surfaces
one  after another,  precisely analogous to the series of  points
which make a line,  and the series of lines which make a surface.
The  piece  of solid space is an aggregate of  surfaces,  and  an
aggregate  of  the same kind as the line is of points;  it  is  a
surface--aggregate of one dimension.   But at the same time it is
a  line--aggregate of two dimensions,  and a point--aggregate  of
three  dimensions.   For if you consider a particular line  which
has gone to make this solid,  a circle partly contracted and part
of the way down,  there are more than two opposite changes  which
it  can  undergo.   For it can ascend or descend,  or  expand  or
contract,  or do both together in any proportion.  It has just as
great  a  variety of changes as a point in a  surface.   And  the
piece  of space is called a point--aggregate of three dimensions,
because  it takes three distinct motions to get it from a  point.
We  must first aggregate points into a line,  then lines  into  a
surface, then surfaces into a solid.
At this step it is clear,  again, that the process must stop
in all the space we know of.  For it is not possible to move that
piece  of  space  in such a way as to change every point  in  it.
When  we moved our line or our surface,  the new line or  surface
contained no point whatever that was in the old one;  we  started
with one aggregate of points, and by moving it we got an entirely
new aggregate, all the points of which were new.  But this cannot
be  done with the solid;  so that the process is at an  end.   We
arrive,  then,  at  the  result  that  {\it  space  is  of  three
dimensions\/}.

Is  this,  then,  one of the postulates of the  science  of
space?   No;  it  is not.   The science of space,  as we have it,
deals  with relations of distance existing in a certain space  of
three  dimensions,  but it does not at all require us  to  assume
that  no relations of distance are possible in aggregates of more
than  three  dimensions.   The  fact that there  are  only  three
dimensions does regulate the number of books that we  write,  and
the  parts of the subject that we study:   but it is not itself a
postulate  of  the science.   We investigate a certain  space  of
three  dimensions,   on  the  hypothesis  that  it  has   certain
elementary  properties;  and  it  is  the  assumptions  of  these
elementary properties that are the real postulates of the science
of space.  To these I now proceed.

The first of them is concerned with {\it points\/}, and with
the  relation  of  space  to them.   We spoke of  a  line  as  an
aggregate  of  points.   Now there are two kinds  of  aggregates,
which  are called respectively continuous and discrete.   If  you
consider  this line,  the boundary of part of the surface of  the
water,  you  will  find yourself believing that between  any  two
points of it you can put more points of division, and between any
two of these more again,  and so on; and you do not believe there
can be any end to the process.  We may express that by saying you
believe  that  between  any two points of the line  there  is  an
infinite number of other points.  But now here is an aggregate of
marbles,  which, regarded as an aggregate, has many characters of
resemblance  with  the aggregate of points.   It is a  series  of
marbles,  one  after  another;  and if we take into  account  the
relations  of  nextness or contiguity which  they  possess,  then
there are only two changes possible from one of them as we travel
along  the series:   we must go to the next in front,  or to  the
next behind.  But yet it is not true that between any two of them
here  is an infinite number of other marbles;  between these two,
for example, there are only three.  There, then, is a distinction
at  once  between  the two kinds of  aggregates.   But  there  is
another,  which  was pointed out by Aristotle in his Physics  and
made the basis of a definition of continuity.   I have here a row
of  two different kinds of marbles,  some white and  some  black.
This aggregate is divided into two parts, as we formerly supposed
the line to be.  In the case of the line the boundary between the
two parts is a point which is the element of which the line is an
aggregate.   In this case before us, a marble is the element; but
here  we cannot say that the boundary between the two parts is  a
marble.   The boundary of the white parts is a white marble,  and
the  boundary  of the black parts is a black  marble;  these  two
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
by  turning it round the point:   we find thus a single series of
directions in which we may start from the point.   According   to
our  first  postulate,  it is a continuous series of  directions.
Now  when  I  speak  of a direction from  the  point,  I  mean  a
direction of starting;  I say nothing about the subsequent  path.
Two  different paths may have the same direction at starting;  in
this case they will touch at the point;  and there is an  obvious
difference between two paths which touch and two paths which meet
and form an angle.   Here,  then,  is an aggregate of directions,
and they can be changed into one another.   Moreover, the changes
by  which  they  pass  into  one  another  have  magnitude,  they
constitute   distance--relations;   and  the  amount  of   change
necessary  to  turn one of them into another is called the  angle
between  them.   It  is involved in this postulate  that  we  are
considering,   that   angles  can  be  compared  in  respect   of
magnitude.   But  this  is  not all.   If we  go  on  changing  a
direction of start,  it will,  after a certain amount of turning, come
round  into itself again,  and be the same  direction.   On
every surface which has the property of elementary flatness,  the
amount  of turning necessary to take a direction all  round  into
its first position is the same for all points of the surface.   I
will now show you a surface which at one point of it has not this
property.   I  take this circle of paper from which a sector  has
been cut out,  and bend it round so as to join the edges; in this
way I form a surface which is called a {\it cone\/}.   Now on all
points  of this surface but one,  the law of elementary  flatness
holds good.   At the vertex of the cone, however, notwithstanding
that  there is an aggregate of directions in which you may start,
such  that  by continuously changing one of them you may  get  it
round into its original position,  yet the whole amount of change
necessary  to effect this is not the same at the vertex as it  is
at any other point of the surface.   And this you can see at once
when  I unroll it;  for only part of the directions in the  plane
have been included in the cone.  At this point of the cone, then,
it does not possess the property of elementary flatness;  and  no
amount  of  magnifying  would ever make a cone seem flat  at  its
vertex.

To apply this to solid space, we must notice that here also
there is a choice of directions in which you may go out from  any
point;  but it is a much greater choice than a surface gives you.
Whereas  in a surface the aggregate of directions is only of  one
dimension, in solid space it is of two dimensions.  But here also
there  are distance--relations,  and the aggregate of  directions
may be divided into parts which have quantity.   For example, the
directions  which start from the vertex of this cone are  divided
into those which go  inside the cone,  and those which go outside
the cone.   The part of the aggregate which is inside the cone is
called  a solid angle.   Now in those spaces of three  dimensions
which have the property of elementary flatness,  the whole amount
of solid angle round one point is equal to the whole amount round
another point.  Although the space need not be exactly similar to
itself  in all parts,  yet the aggregate of directions round  one
point  is  exactly similar to the aggregate of  directions  round
another  point,  if  the  space has the  property  of  elementary
flatness.

How   does  Euclid  assume  this  postulate  of  Elementary
Flatness?   In his fourth postulate he has expressed it so simply
and clearly that you will wonder how anybody could make all  this
fuss.  He says, All right angles are equal.'

Why  could  I not have adopted this at once,  and  saved  a
great  deal  of trouble?   Because it assumes the knowledge of  a
surface  possessing the  property of elementary flatness  in  all
its  points.   Unless such a surface is first made out to  exist,
and  the definition of a right angle is restricted to lines drawn
upon it---for there is no necessity for the word {\it straight\/}
in that definition---the postulate in Euclid's form is  obviously
not true.   I can make two lines cross at the vertex of a cone so
that the four adjacent angles shall be equal,  and yet not one of
them equal to a right angle.
I  pass on to the third postulate of the science of space---
the  postulate of Superposition.   According to this postulate  a
body  can  be moved about in space without altering its  size  or
shape.   This  seems  obvious enough,  but it is worth  while  to
examine  a little more closely into the meaning of it.   We  must
define  what we mean by size and by shape.   When we say  that  a
body  can be moved about without altering its size,  we mean that
it  can  be so moved as to keep unaltered the length of  all  the
lines in it.  This postulate therefore involves that lines can be
compared  in  respect of magnitude,  or that they have  a  length
independent of position; precisely as the former one involved the
comparison  of angular magnitudes.   And when we say that a  body
can  be moved about without altering its shape,  we mean that  it
can be so moved as to keep unaltered all the angles in it.  It is
not necessary  to make mention of the motion of a body,  although
that  is  the easiest way of expressing and  of  conceiving  this
postulate;  but we may,  if we like, express it entirely in terms
which  belong  to  space,  and  that we should do  in  this  way.
Suppose  a  figure to have been constructed in  some  portion  of
space;  say  that  a triangle has been drawn whose sides are  the
shortest  distances between its angular points.   Then if in  any
other  portion  of  space two points  are  taken  whose  shortest
distance  is equal to a side of the triangle,  and at one of them
an  angle  is made equal to one of the  angles adjacent  to  that
side,  and  a  line  of  shortest distance  drawn  equal  to  the
corresponding  side of the original triangle,  the distance  from
the  extremity  of this to the other of the two  points  will  be
equal  to  the third side of the original triangle,  and the  two
will be equal in all respects; or generally, if a figure has been
constructed anywhere,  another figure, with all its lines and all
its  angles  equal to the corresponding lines and angles  of  the
first,  can  be constructed anywhere else.   Now this is  exactly
what   is  meant by the principle of  superposition  employed  by
Euclid to prove the proposition that I have just mentioned.   And
we may state it again in this short form---All parts of space are
exactly alike.

But   this  postulate  carries  with  it  a  most  important
consequence.   In enables  us to make a pair of most  fundamental
definitions---those  of the plane and of the straight  line.   In
order to explain how these come out of it when it is granted, and
how  they cannot be made when it is not granted,  I must here say
something  more about the nature of the postulate  itself,  which
might otherwise have been left until we come to criticize it.

We  have stated the postulate as referring to solid  space.
But  a  similar  property  may  exist  in  surfaces.   Here,  for
instance,  is  part of the surface of a sphere.   If I  draw  any
figure  I like upon this,  I can suppose it to be moved about  in
any way upon the sphere, without alteration of its size or shape.
If  a  figure  has  been drawn on any part of the  surface  of  a
sphere,  a figure equal to it in all respects may be drawn on any
other part of the surface.   Now I say that this property belongs
to the surface itself, is a part of its own internal economy, and does
not depend in any way upon its relation to space  of  three
dimensions.   For  I can pull it about and bend it in all  manner
of ways,  so as altogether to alter its relation to solid  space;
and yet, if I do not stretch  it or tear it, I make no difference
whatever  in the length of any lines upon it,  or in the size  of
any angles upon it.   I do not in any way alter the figures drawn
upon it,  or the possibility of drawing figures upon it,  {\it so
far  as their relations with the surface itself are concerned\/}.
This  property  of the surface,  then,  could be  ascertained  by
people who lived entirely in it,  and were absolutely ignorant of
a third dimension.   As a point--aggregate of two dimensions,  it
has  in itself properties determining the distance--relations  of
the  points  upon it,  which  are absolutely independent  of  the
existence of any points which are not upon it.

Now  here  is a surface which has not  that  property.   You
observe that it is not of the same shape all over,  and that some
parts  of  it are more curved than other parts.   If you  drew  a
figure upon this surface,  and then tried to move it  about,  you
would  find that it was impossible to do so without altering  the
size  and  shape of the figure.   Some parts of it would have  to
expand,  some to contract, the lengths of the lines could not all
be  kept the same,  the angles would not hit off  together.   And
this property of the surface---that its parts are different  from
one another---is a property of the surface itself,  a part of its
internal economy,  absolutely independent of any relations it may
have with space outside of it.  For, as with the other one, I can
pull  it  about in all sorts of ways,  and,  so long as I do  not
stretch  it  or tear it,  I make no alteration in the  length  of
lines drawn upon it or in the size of the angles.

Here,  then,  is an intrinsic difference between these  two
surfaces,  as  surfaces.   They are both point--aggregates of two
dimensions;  but  the  points in them have certain  relations  of
distance (distance measured always {\it on\/} the  surface),  and
these  relations of distance are not the same in one case as they
are in the other.

The  supposed  people living in the surface and  having  no
idea  of a third dimension might,  without suspecting that  third
dimension  at  all,  make  a very accurate determination  of  the
nature of their {\it locus in quo\/}.  If the people who lived on
the  surface  of  the  sphere were to measure  the  angles  of  a
triangle,  they  would find them to exceed two right angles by  a
quantity proportional to the area of the triangle.   This  excess
of  the angles above two right angles,  being divided by the area
of the triangle, would be found to give exactly the same quotient
at all parts of the sphere. That quotient is called the curvature
of the surface;  and we say that a sphere is a surface of uniform
curvature.   But  if the people living on this irregular  surface
were  to do the same thing,  they would not find quite  the  same
result.   The  sum of the angles would,  indeed,  differ from two
right angles,  but sometimes in excess,  and sometimes in defect,
according  to  the  part of the surface  where  they  were.   And
though  for small triangles in any on neighbourhood the excess or defect
would be nearly proportional to the area of the  triangle,
yet  the  quotient obtained by dividing this excess or defect  by
the area of the triangle would vary from one part of the  surface
to another.  In other words, the curvature of this surface varies
from  point  to  point;   it  is  sometimes  positive,  sometimes
negative, sometimes nothing at all.

But now comes the important difference.   When I speak of a
triangle, what do I suppose the sides of that triangle to be?

If  I take two points near enough together upon  a  surface,
and  stretch a string between them,  that string will take  up  a
certain  definite position upon the surface,  marking the line of
shortest distance from one point to  the other.   Such a line  is
called a geodesic line.  It is a line determined by the intrinsic
properties of the surface, and not by its relations with external
space.   The  line would still be the shortest line,  however the
surface  were  pulled about without  stretching  or  tearing.   A
geodesic  line  may be {\it produced\/},  when a piece of  it  is
given; for we may take one of the points, and, keeping the string
stretched,  make  it go round in a sort of circle until the other
end has turned through two right angles.   The new position  will
then be a prolongation of the same geodesic line.

In speaking of a triangle,  then,  I meant a triangle whose
sides  are  geodesic  lines.   But  in the case  of  a  spherical
surface---or,   more  generally,   of   a  surface  of   constant
curvature---these  geodesic lines have another and most important
property.   They are {\it straight\/},  so far as the surface  is
concerned.   On  this surface a figure may be moved about without
altering its  size or shape.   It is possible, therefore, to draw
a  line  which shall be of the same shape all along and  on  both
sides.  That is to say, if you take a piece of the surface on one
side of such a line,  you may slide it all along the line and  it
will  fit;  and  you may turn it round and apply it to the  other
side,  and it will fit there also.  This is Leibniz's  definition
of a straight line, and, you see, it has no meaning except in the
case of a  surface of constant curvature,  a surface all parts of
which are alike.

Now let us consider the corresponding things in solid space.
In this also we may have geodesic lines;  namely, lines formed by
stretching  a string between two points.   But we may  also  have
geodesic surfaces; and they are produced in this manner.  Suppose
we  have  a point on a surface,  and this surface  possesses  the
property of elementary flatness.   Then among all the  directions
of  starting from the point,  there are some which start {\it  in
the surface\/},  and do not make an angle with it.  Let all these
be  prolonged  into geodesics;  then we may imagine one of  these
geodesics  to  travel round and coincide with all the  others  in
turn.   In so doing it will trace out a surface which is called a
geodesic  surface.   Now in the particular case where a space  of
three  dimensions has the property of superpositoin,  or  is  all
over alike,  these geodesic surfaces are {\it planes\/}.  That is
to  say,  since the space is all over alike,  these surfaces  are also
of  the  same shape all over and on both  sides;  which  is
Leibniz's  definition of a plane.   If you take a piece of  space
on one side of such a plane, partly bounded by the plane, you may
slide it all over the plane, and it will fit; and you may turn it
round and apply it to the other side, and it will fit there also.
Now  it is clear that this definition will have no meaning unless
the  third  postulate be granted.   So we may say that  when  the
postulate  of Superposition is true,  then there are  planes  and
straight  lines;  and they are defined as being of the same shape
throughout and on both sides.

It  is  found that the whole geometry of a space  of  three
dimensions is known when we know the curvature of three  geodesic
surfaces  at every point.  The third postulate requires that  the
curvature  of all geodesic surfaces should be everywhere equal to
the same quantity.

I  pass to the fourth postulate,  which I call the postulate
of Similarity.   According to this postulate,  any figure may  be
magnified or diminished in any degree without altering its shape.
If  any figure has been constructed in one part of space,  it may
be  reconstructed  to any scale whatever in any   other  part  of
space,  so that no one of the angles shall be altered through all
the lengths of lines will of course be altered.  This seems to be
a sufficiently obvious induction from experience; for we have all
frequently seen different sixes of the same shape; and it has the
advantage of embodying the fifth and sixth of Euclid's postulates
in a single principle, which bears a great resemblance in form to
that of Superposition, and may be used in the same manner.  It is
easy to show that it involves the two postulates of Euclid:  Two
straight  lines cannot enclose a space,' and Lines in one  plane
which never meet make equal angles with every other  line.'

This  fourth postulate is equivalent to the assumption  that
the  constant curvature of the geodesic surfaces is zero;  or the
third and fourth may be put together,  and we shall then say that
the  three  curvatures  of space are all of them  zero  at  every
point.

The  supposition  made by Lobatchewsky was,  that the  three
first  postulates  were true,  but not the fourth.   Of  the  two
Euclidean  postulates included in this,  he  admitted  one,  {\it
viz\/}.,  that two straight lines cannot enclose a space, or that
two  lines which once diverge go on diverging for ever.   But  he
left  out the postulate about parallels,  which may be stated  in
this  form.   If through a point outside of a straight line there
be drawn another, indefinitely produced both ways; and if we turn
this  second one round so as  to make the point  of  intersection
travel  along the first line,  then at the very instant that this
point  of intersection disappears at one end it will reappear  at
the other,  and there is only one position in which the lines  do
not intersect.  Lobatchewsky supposed, instead, that there was  a
finite  angle through which the second line must be turned  after
the  point of intersection had disappeared at one end,  before it
reappeared  at the other.   For all positions of the second  line
within  this angle there is then no intersection.    In  the  two
limiting positions,  when the lines have just done meeting at one
end,  and when they are just going to meet at the other, they are
called  parallel;  so that two lines can be drawn through a fixed
point parallel to a given straight line.  The angle between these
two depends in a  certain way upon the distance of the point from
the line.   The sum of the  angles of a triangle is less than two
right  angles  by  a quantity proportional to  the  area  of  the
triangle.   The whole of this geometry is worked out in the style
of  Euclid,  and the most interesting conclusions are arrived at;
particularly  in the theory of solid space,  in which  a  surface
turns up which is not plane relatively to that space,  but which,
for  purposes of drawing figures upon it,  is identical with  the
Euclidean plane.

It was Riemann, however, who first accomplished the task of
analysing  all the assumptions of geometry,  and showing which of
them were independent.  This very disentangling and separation of
them  is  sufficient to deprive them for the  geometer  of  their
exactness and necessity;  for the process by which it is effected
consists   in  showing  the  possibility  of  conceiving    these
suppositions one by one to be untrue;  whereby it is clearly made
out  how  much is supposed.   But it may be worth while to  state
formally the case for and against them.

When  it is maintained that we know these postulates  to  be
universally  true,  in  virtue  of certain  deliverances  of  our
consciousness,  it  is implied that these deliverances could  not
exist,  except upon the supposition that the postulates are true.
If it can be shown,  then, from experience that our consciousness
would  tell us exactly the same things if the postulates are  not
true,  the ground of their validity will be taken away.  But this
is a very easy thing to show.

That  same faculty which tells you that space is continuous
tells  you  that this water is continuous,  and that  the  motion
perceived  in  a wheel of life is continuous.   Now we happen  to
know  that  if we could magnify this water as much again  as  the
best microscopes can magnify it,  we should perceive its granular
structure.   And what happens in a wheel of life is discovered by
stopping the machine.   Even apart,  then,  from our knowledge of
the way nerves act in carrying messages,  it appears that we have
no means of knowing anything more about an aggregate than that it
is too fine--grained for us to perceive its discontinuity,  if it
has any.

Nor can we,  in general,  receive a conception as  positive
knowledge which is itself founded merely upon inaction.   For the
conception of a continuous thing is of that which looks just  the
same however much you magnify it.  We may conceive the magnifying
to  go  on to a certain extent without change,  and then,  as  it
were,  leave  it going on,  without taking the  trouble to  doubt
about the changes that may ensue.

In regard to the second postulate, we have merely to point to
the  example of polished surfaces.  The smoothest surface that
can be made is the one  most completely covered with the minutest
ruts and furrows.  Yet geometrical constructions can be made with
extreme accuracy upon such a surface,  on the supposition that it
is an exact plane.   If,  therefore, the sharp points, edges, and
furrows of space are only small enough,  there will be nothing to
hinder  our conviction of its elementary flatness.   It has  even
been  remarked  by  Riemann that we must  not  shrink  from  this
supposition   if  it  is  found  useful  in  explaining  physical
phenomena.

The  first  two postulates may therefore be doubted  on  the
side  of  the   very small.   We may put  the  third  and  fourth
together,  and doubt them on the side of the very great.   For if
the  property  of elementary flatness exist on the  average,  the
deviations from it being,  as we have supposed,  too small to  be
perceived,  then,  whatever  were  the true nature of  space,  we
should have exactly the conceptions of it which we now  have,  if
only  the regions we can get at were small in comparison with the
areas  of curvature.   If we suppose the curvature to vary in  an
irregular manner,  the effect of it might be very considerable in
a triangle formed by the nearest fixed stars;  but if we  suppose
it  approximately  uniform to the limit of telescopic  reach,  it
will  be  restricted  to very much  narrower  limits.   I  cannot
perhaps do better than conclude by describing to you as well as I
can  what  is the nature of things on the  supposition  that  the
curvature of all space is nearly uniform and positive.

In this case the Universe,  as known, becomes again a valid
conception;  for  the extent of space is a finite number of cubic
miles.   And this comes about in a curious way.   If you were  to
start in any direction whatever,  and move in that direction in a
perfect  straight  line according to the definition  of  Leibniz;
after  travelling  a  most  prodigious  distance,  to  which  the
parallactic  unit---200,000  times the diameter  of  the  earth's
orbit---would  be  only a few steps,  you would arrive  at---this
place.   Only,  if you had started upwards, you would appear from
below.   Now,  one of two things would be true.  Either, when you
had  got half--way on your journey,  you came to a place that  is
opposite to this,  and which you must have gone through, whatever
direction you started in;  or else all paths you could have taken
diverge  entirely  from each other till they meet again  at  this
place.   In the former case,  every two straight lines in a plane
meet in two points,  in the latter they meet only in  one.   Upon
this  supposition of a positive curvature,  the whole of geometry
is far more complete and interesting;  the principle of  duality,
instead  of half breaking down over metric relations,  applies to
all  propositions  without exception.   In fact,  I  do  no  mind
confessing  that  I personally have often found relief  from  the
dreary infinities of homaloidal space in the consoling hope that,
after all, this other may be the true state of things.
\vfill\eject
\noindent 2.  FULL--LENGTH ARTICLE
\vskip.5cm
\centerline{\bf BEYOND THE FRACTAL} \vskip.1cm
\centerline{\sl Sandra Lach Arlinghaus}
\vskip.5cm
\centerline{I never saw a moor,}
\centerline{I never saw the sea;}
\centerline{Yet know I how the heather looks,}
\centerline{And what a wave must be."}
\vskip.1cm
{\sl Emily Dickinson, Chartless."}
\vskip.5cm
\centerline{\bf Abstract.}

{\nn  The   fractal   notion   of   self--similarity    is
useful   for characterizing change in scale; the reason fractals
are   effective  in  the  geometry of central  place  theory  is
because   that  geometry is hierarchical  in  nature.   Thus,  a
natural   place  to  look  for other connections of this sort is
to  other   geographical   concepts  that  are  hierarchical  in
nature.    Within  this  fractal  context, this chapter examines
the  case of spatial diffusion.

When the idea of diffusion is extended to see  adopters"
of  an innovation as attractors" of new  adopters,   a   Julia
set   is introduced as a possible axis against which to  measure
one   class  of  geographic phenomena.    Beyond   the   fractal
context,  fractal concepts, such as compression" and space--
filling" are considered in a broader graph--theoretic context.}

\centerline{\bf Introduction.}

Because  a  fractal  may  be  considered  as  a   randomly
generated statistical image  (Mandelbrot,   1983),   one   place
to   look   for  geometric fractals tailored to  fit  geographic
concepts   is   within  the  set   of   ideas   behind   spatial
The   spatial   diffusion  of  an innovation is one  such  case;
H\"agerstrand   characterized it using probabilistic  simulation
techniques   (H\"agerstrand,   1967).    This  chapter    builds
directly   on  H\"agerstrand's  work  in  order to  demonstrate,
in some detail, how fractals might arise  in  spatial diffusion.
From  there,  and  with   a   view   of   an   adopter   of   an
innovation  as  an  attractor"   of   other   adopters,    the
connected Julia set $z = z^2-1$ is examined,  only broadly,  for
its  potential to serve as an axis from which to measure spatial
attraction."

More generally,  it is not necessary to consider  fractal--
like  concepts  such  as  attraction,"  space--filling,"  or
compression"  relative  to any metric,  as  in  the  diffusion
example,   or   relative  to any axis,  as in  the   Julia   set
case.    These   broad   fractal notions are examined,  in  some
detail,  in a graph--theoretic  realm,  free  from   metric/axis
encumbrance,  as  one  step  beyond  the fractal.  An effort has
been made to explain key geographical  and mathematical concepts
so that much of the material, and  the  flow of ideas, is self--
contained and accessible to readers from various disciplines.

\centerline{\bf A fractal connection to spatial diffusion}

The  diffusion   of  the  knowledge   of   an   innovation
across  geographic  space  may be simulated  numerically   using
Monte    Carlo   techniques   based   in   probability    theory
(H\"agerstrand,  1967).   A simple example illustrates the basic
mechanics of H\"agerstrand's procedure.

Consider  a geographic region and cover  it  with  a  grid
of  uniform  cell  size suited to the scale  of  the   available
empirical information about the innovation.   Enter  the  number
of  initial adopters of the innovation in the grid:  an entry of
$1$"  means  one  person   (household,   or   other   set   of
people)  knows   of   the innovation.   Over time,  this  person
will tell others.  Assume that the spread of the news, from this
person to  others,   decays  with distance.   To  simulate  this
spread,   probabilities   of   the likelihood of contact will be
assigned  to  each  cell  surrounding each initial  adopter.   A
table   of   random  numbers  is  used  in conjunction with  the
probabilities, as follows.

Given  a gridded geographic region and a  distribution  of
three  initial adopters of an innovation  (Figure  1).    Assume
that  an initial telling occurs no  more  than  two  cells  away
from   the initial adopters' cells.   This assumption creates  a
five--by--five  grid in which interchange can occur  between  an
initial  adopter   in  the  central  cell  and  others.   Assign
probabilities  of  contact  to each of these twenty--five  cells
as  a  percentage likelihood that a randomly chosen  four  digit
number falls within a given interval of numbers assigned to each
cell  (Figure  2).    Because   the   intervals  in   Figure   2
partition   the  set  of  four  digit  numbers,   the percentage
probabilities assigned to each cell add to 100\%.   Pick up  the
five--by--five  grid  and center it on the original  adopter  in
cell H3 (Figure 1).   Choose the first number, 6248, in the list
of  random  numbers  (Figure 2).   It falls in the  interval  of
numbers in the central cell.  Enter a $+1$" in the  associated
cell,  H3,  to represent this new adopter.   Move the five--by--
five grid across the distribution of original adopters, stopping
it and repeating  this procedure with the next random number  in
the  list  each  time a  new original  adopter  is  encountered.
Center the five--by--five grid on H4;  the next random number is
0925  which  falls  in  the interval  in  the  cell  immediately
northwest  of  center (Figure 2).   Enter a $+1$" in  cell  G3
(Figure 1),  the  cell  immediately  northwest  of  H4. Finally,
center the moving grid on H5.   The  next  random  number, 4997,
falls in the center cell; therefore, enter a $+1$" in cell H5.
Once  this procedure has been applied to all original  adopters,
the   population    of   adopters  doubles   and    a    first
newer   adopters represented as $+1$'s", emerges  (Figure  1).
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.
TYPESETTING, USING TeX, FOR THIS IMAGE APPEARS BELOW

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

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

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\smallskip A random set of numbers (source: {\sl CRC Handbook of Standard Mathematical Tables\/}): \smallskip \line{6248, 0925, 4997, 9024, 7754 \hfil} \smallskip \line{7617, 2854, 2077, 9262, 2841 \hfil} \smallskip \line{9904, 9647, \hfil} \smallskip \noindent and so forth. \vskip.5cm Random number assignment to matrix cells, with cell number given as an ordered pair whose first entry refers to the reference number on the left of the matrix in this figure and whose second entry refers to the reference number at the top of that matrix. \vskip.2cm \line{(1,1): 0000-0095; (1,2): 0096-0235; (1,3): 0236-0403 \hfil} \line{(1,4): 0404-0543; (1,5): 0544-0639 \hfil} \smallskip \line{(2,1): 0640-0779; (2,2): 0780-1080; (2,3): 1081-1627 \hfil} \line{(2,4): 1628-1928; (2,5): 1929-2068 \hfil} \smallskip \line{(3,1): 2069-2236; (3,2): 2237-2783; (3,3): 2784-7214 \hfil} \line{(3,4): 7215-7761; (3,5): 7762-7929 \hfil} \smallskip \line{(4,1): 7930-8069; (4,2): 8070-8370; (4,3): 8371-8917 \hfil} \line{(4,4): 8918-9218; (4,5): 9219-9358 \hfil} \smallskip \line{(5,1): 9359-9454; (5,2): 9455-9594; (5,3): 9595-9762 \hfil} \line{(5,4): 9763-9902; (5,5): 9903-9999 \hfil} \endinsert There are numerous side issues, which are important, that may complicate this basic procedure (H\"agerstrand, 1967; Haggett {\it et al.\/}, 1977). How are the percentages for the five--by--five grid chosen? Indeed, how is the dimension of five" chosen for a side of this grid? Should the choices of percentages and of dimension be based on empirical data, on other abstract considerations, or on a mix of the two? What sorts of criteria should there be in judging suitability of empirical data? What if a random entry falls outside the given grid; what sorts of boundary/barrier considerations, both in terms of the position of new adopters relative to the regional boundary and of the symmetry of the probabilities within the five--by--five grid, should be taken into account? Independent of how many generations are calculated using this procedure, the pattern of filling in" of new adopters is heavily influenced by the shape of the set of original adopters. Indeed, over time, knowledge of the innovation diffuses slowly initially, picks up in speed of transmission, tapers off, and eventually the population becomes saturated with the knowledge. Typically this is characterized as a continuous phenomenon using a differential equation of inhibited growth that has as an initial supposition that the population may not exceed $M$, an upper bound, and that $P(t)$, the population $P$ at time $t$, grows at a rate proportional to the size of itself and proportional to the fraction left to grow (Haggett {\it et al.\/}, 1977; Boyce and DiPrima, 1977). An equation such as $${dP(t) \over dt} = k\, P(t)(1- (P(t)/M))$$ serves as a mathematical model for this sort of growth in which $k >0$ is a growth constant and the fraction $(1-(P(t)/M)$ acts as a damper on the rate of growth (Boyce and DiPrima, 1977). The graph of the equation is an $S$--shaped (sigmoid) logistic curve with horizontal asymptote at $P(t)=M$ and inflection point at $P(t)=M/2$. When $dP/dt > 0$ the population shows growth; when $d^2 P/dt^2 > 0$ (below $P(t)=M/2$) the rate of growth is increasing; when $d^2P/dt^2<0$ (above $P(t)=M/2$) the rate of growth is decreasing. The differential equation model thus yields information concerning the rate of change of the total population and in the rate of change in growth of the total population. It does not show how to determine $M$; the choice of $M$ is given {\it a priori\/}. Iteration of the H\"agerstrand procedure gives a position for $M$ once the procedure has been run for all the generations desired. For, it is a relatively easy matter to accumulate the distributions of adopters and stack them next to each other, creating an empirical sigmoid logistic curve based on the simulation (Haggett {\it et al.\/}, 1977). Finding the position for the asymptote (or for {\it an\/} upper bound close to the asymptotic position) is then straightforward. Neither the H\"agerstrand procedure nor the inhibited growth model provides an estimate of saturation level (horizontal asymptote position) (Haggett, {\it et al.\/}, 1977) that can be calculated early in the measurement of the growth. The fractal approach suggested below offers a means for making such a calculation when self--similar hierarchical data are involved; allometry is a special case of this procedure (Mandelbrot, 1983). The reasons for wanting to make such a calculation might be to determine where to position adopter seeds" in order to produce various levels of innovation saturation. As is well--known, not all innovations diffuse in a uniform manner; Paris fashions readily available in major U. S. cities up and down each coast might seldom be seen in rural midwestern towns. To determine how the ideas of fractal space--filling" and this sort of diffusion--related space-- filling" might be aligned, consider the following example. Given a distribution of three original adopters occupying cells H3, H4, and H5 in a linear pattern (Figure 3.A). The probabilities for positions for new adopters are encoded within each cell surrounding each of these (as determined from the five--by--five grid of Figure 2). Thus, for example, when the grid of Figure 2 is superimposed and centered on the original adopter in cell H3, a probability of 3.01\% is assigned to the likelihood for contact from H3 to G4; when it is superimposed and centered on the original adopter in H4, there is a 5.47\% likelihood for contact from H4 to G4; and, when it is superimposed and centered on the original adopter in H5, there is a 3.01\% likelihood for contact from H5 to G4. Therefore, the percentage likelihood of a new first--generation adopter in cell G4, given this initial configuration of adopters, is the sum of the percentages divided by the number of initial adopters, or 11.49/3. For ease in inserting fractions into the grid, only the numerator, 11.49, is shown as the entry (Figure 3.A). It would be useful, for purposes of comparison of this distribution to those with sets of initial adopters of sizes other than 3, to divide by the number of initial adopters in order to derive a percentage that is independent of the size of the initial distribution ({\it i.e.\/}, to normalize the numerical entries). \topinsert \noindent{\bf Figure 3.A}. \smallskip The simulation is run on three original adopters with positions given below. Numerical entries show the likelihood, out of 300, that a new adopter will fall into a given cell. Zones of interaction between overlapping five--by--five grids are outlined by a heavy line (begin at the upper left--hand corner (ulhc) of cell F2; move horizontally to the upper right--hand corner (urhc) of cell F6; vertically to lower right--hand corner (lrhc) of cell J6; horizontally to lower left--hand corner (llhc) of cell J2; vertically to ulhc of F2 --- should be a rectangular enclosure that you have added to this figure). {\bf Original adopters are in cells H3, H4, H5.} \smallskip North at the top.LINES DESCRIBED ABOVE WERE ADDED TO THE SCANNED IMAGE IN COREL PHOTO-PAINT.
Figure 3A.

TYPESETTING USING TeX OF THE FIGURE.
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\endinsert

It is easy to see that the values in the cells of Figure 3.A must add to a total of 300 if one views them as derived from each of three five--by--five grids centered on each original adopter. A zone of interaction" of entries from two or more five--by--five grids is outlined by a heavy line; 25 cells are enclosed in it in Figure 3.A. The pattern of numbers exhibits bilateral symmetry, insofar as is possible (allowing for the appendix" of .01 required to make the numerical partition associated with Figure 2 complete) with respect to both North--South and East--West axes (with the origin in cell H4). Sum and column totals are calculated; as the shape of the distribution of initial adopters is altered (below), these totals will tag sets of cells to demonstrate how changes in the zone of interaction are occurring. Next consider a distribution of three initial adopters derived from the linear one by moving the middle adopter one unit to the North (Figure 3.B). When interaction values are calculated as they were for the initial distribution in Figure 3.A, a comparable, but different numerical pattern emerges (Figure 3.B). Here, the column totals are the same as those in Figure 3.A, but the row totals are different. The zone of interaction contains 23 cells; the highest individual cell value of 50.33 is less than that of the highest cell value, 55.25, in Figure 3.A. Because both sets of values are partitions of the number 300, and because there are more cells with potential for contact in Figure 3.B than in Figure 3.A, the concentration of entries in Figure 3.B is not as compressed as in Figure 3.A. This is reflected in the row totals; a visual device useful for tracking this compression is to think of the five--by--five grid centered on 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.
(HERE, AND IN SUBSEQUENT RELATED FIGURES,
DOTTED LINE TO INDICATE T-SHAPED POLYGON ADDED IN SCANNED IMAGE USING COREL PHOTO-PAINT, 5.0.)
Figure 3B.

TYPESETTING THAT PRODUCED FIGURE ABOVE
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\endinsert

Naturally, as the middle initial adopter is pulled successively one unit to the north in the configuration of original adopters, the middle five--by--five grid is also pulled one unit to the north (Figures 3.C, 3.D, 3.E, and 3.F). The numerical consequence is to reduce the size of the zone of interaction among the initial adopters and to spread the range of cells over which the value of 300 is partitioned. This implies less concentration near the original adopters and less filling in" around them as one proceeds from Figure 3.A to Figure 3.F. Thus, in Figure 3.C the zone of interaction shrinks to 21 cells with a largest individual cell entry of 47.39. At the stage shown in Figure 3.D, the largest cell entry is 45.99; because the cells associated with this value are not overlapped by the five--by-- five grid centered on the middle adopter, this largest value will not change as the middle adopter is pulled more to the north. Table 1 shows the sizes of the zones of interaction of the largest individual cell entry for each of Figures 3.A to 3.F. No new information arises from moving the middle cell to the north beyond the position in Figure 3.F; the five-- by--five grid is revealed and no longer intersects the two overlapping grids associated with the other two initial 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.

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\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.

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

TYPESETTING, USING TeX, THAT PRODUCED THIS FIGURE.
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\endinsert

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

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

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

TYPESETTING, USING TeX, THAT PRODUCED TABLE 1.
\settabs\+\indent \quad&Figure 3.B:  middle cell 2 units north
\smallskip
\+&Figure number:     &Number of cells &Largest value   \cr
\+&Position of three  &in interaction  &(out of 300) in \cr
\smallskip
\+&Figure 3.A: linear arrangement         &25  &55.25\cr
\+&Figure 3.B: middle cell 1 unit north   &23  &50.33\cr
\+&Figure 3.C: middle cell 2 units north  &21  &47.39\cr
\+&Figure 3.D: middle cell 3 units north  &19  &45.99\cr
\+&Figure 3.E: middle cell 4 units north  &17  &45.99\cr
\+&Figure 3.F: middle cell 5 units north  &15  &45.99\cr
\smallskip
\hrule
\endinsert

The example depicted in Figure 3 shows that even as early  as
the first generation, the pattern of the positions of the  initial
adopters affects significantly  the  configuration  of  the  later
adopters.  Figure 3.A with the heaviest possible filling of  space
using three initial adopters represents  a  most  saturated  case,
which,  taken  together  with  an  underlying  symmetry  that   is
bilateral relative to mutually perpendicular axes,  suggests  that
an associated space--filling curve should have dimension 2, should
have a  rectilinear  appearance,  and  should  be  formed  from  a
generator whose shape is related to the pattern  of  placement  of
the original adopters.  One space--filling curve that  meets these
requirements  is  the  rectilinear  curve  of  Figure  4.A.    The
generator is composed of three nodes hooked together by two  edges
in a straight path.  This is scaled--down, by a factor of 1/2, and
hooked to the endpoints of the original generator.   Iteration  of
this procedure leads  to  a  rectilinear  tree  with  the  desired
properties.  The approach of looking for a geometric form to fit a
given set of conditions is like the calculus approach  of  looking
for a differential equation to fit a given set of conditions.  The
difference here is that the  shape  of  the  generator  and  other
information from early stages may be used to estimate the relative
saturation or space--filling level.

The  spatial position of the original adopters in   Figure
3.B  suggests  a fractal generator in  the  shape  of   a   V"
with  an interbranch angle, ${\theta}$, of 90 degrees, while the
V  in  Figure  3.C suggests a  generator  with  $\theta \approx 53^{\circ}$,  that  of  Figure  3.D  one  with  $\theta \approx 37^{\circ}$,  that  of  Figure  3.E  one  with  $\theta \approx 28^{\circ}$,  and  that of  Figure  3.F one with $\theta \approx 23^{\circ}$.   Figures 4.B,  4.C,  4.D,  4.E,  and  4.F  suggest
trees that can be generated using these values for $\theta$.

\topinsert \vskip 5in
\noindent {\bf Figure 4}.
\smallskip
THIS FIGURE DOES NOT TRANSMIT---AVAILABLE ONLY AS HARD COPY
(THUS, SCANNED IMAGE ONLY IS AVAILABLE HERE...NO CORRESPONDING TeX CODE.)

Figure 4.

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

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

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

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

Figure 5.

\endinsert

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

TYPESETTING, USING TeX, THAT PRODUCED TABLE 2. \smallskip \settabs\+\noindent &Figure 3.C: middle cell 2 units north\quad &Figure 4.C: $\theta \approx 53.13^{\circ}$\quad &$=(180-(\theta /2))/2$ \quad &$(\hbox{ln}\,(1/\hbox{cos}\,\phi))$ &\cr \smallskip \+&Figure number: &Size of interbranch &Size &$D$--value:\cr \+&Position of three &angle, $\theta$, in &of $\phi$ &$D=(\hbox{ln}\, 2)/$\cr \+&original adopters. &associated tree. &$=(180-(\theta /2))$ &$(\hbox{ln}\, (1/\hbox{cos}\,\phi))$\cr \smallskip \+&Figure 3.A: linear arrangement &Figure 4.A: $\theta = 180^{\circ}$ &$45^{\circ}$ &2 \cr \+&Figure 3.B: middle cell 1 unit north &Figure 4.B: $\theta = 90^{\circ}$ &$67.5^{\circ}$&0.721617\cr \+&Figure 3.C: middle cell 2 units north&Figure 4.C: $\theta \approx 53.13^{\circ}$ &76.78 &0.471288\cr \+&Figure 3.D: middle cell 3 units north&Figure 4.D: $\theta \approx 36.87^{\circ}$ &80.78 &0.378471\cr \+&Figure 3.E: middle cell 4 units north&Figure 4.E: $\theta \approx 28.07^{\circ}$ &82.98 &0.32971 \cr \+&Figure 3.F: middle cell 5 units north&Figure 4.F: $\theta \approx 22.62^{\circ}$ &84.35 &0.299116\cr \smallskip \hrule \endinsert
This decreasing sequence of $D$--values corresponds only loosely to Mandelbrot's measurements of fractal dimensions of trees (Mandelbrot, 1983); here, however, when $D=1$ the corresponding tree is one with an interbranch angle of $120^{\circ}$. This has some appeal if one notes that then the tree associated with $D=1$ might therefore represent a Steiner network (tree of shortest total length under certain circumstances) or part of a central place net. The numerical unit $D$--value would thus correspond to optimal forms for transport networks or for urban arrangements in abstract geographic space (in which H\"agerstrand's diffusion procedure also exists). One use for these $D$--values, which measure the relative space--filling by trees, might be as units fundamental to developing an algebraic structure for planning the eventual saturation level to arise in communities into which an innovation is introduced to selected adopters. By choosing judiciously the pattern of initial adopters, the relative space--filling of associated trees might be guided by local municipal authorities so as not to conflict with, or to interfere with, other issues of local concern. The $D$--values associated with triads of original adopters (as in Table 2) might serve as irreducible elements of this algebra, into which larger sets could be decomposed (much as positive integers ($> 1$) can be decomposed into a product of powers of prime numbers). The manner in which the decomposition is to take place would likely be an issue of considerable algebraic difficulty, no doubt requiring the use of geographic constraints to limit it. (For, unlike the parallel with integer decomposition, this one would seem not to be unique.) An initial direction for such a diffusion--algebra might therefore be to exploit the parallel with the Fundamental Theorem of Arithmetic. Another use might involve a self--study by the National Center for Geographic Information and Analysis (NCGIA) in order to monitor the diffusion of Geographic Information System (GIS) technology through the various programs designed to promote this technology in the academic arena. University test--sites for the materials of the NCGIA, for example, might be selected as seeds" with deliberate plans for using a diffusion structure based on these seeds to bring later adopters up to date. Another use might involve the determination of sites for locally unwanted land uses such as waste sites, prisons, and so forth. Regions expected to experience high concentrations of population coming from the totality of innovations already introduced, or to be introduced, might be overburdened by such a landuse. When relative fractal saturation estimates are run on a computer in conjunction with a GIS, local municipal authorities might examine issues such as this for themselves. \centerline{\bf Attraction: the Julia set $z = z^2 - 1$} A different way to view the space--filling characteristics of the diffusion example is to consider each initial adopter as an attractor" of other adopters, once again suggesting a fractal 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.
equation

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

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

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

Figure 6.

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

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

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

Because  the  Fibonacci  sequence  can  be  expressed  using   the
logarithmic spiral, this particular Julia set with these values as
units" might therefore serve as an axis  from  which  to measure
spiral phenomena at various scales ranging from the global to  the
local:   from,   for   example,   the   climatological   to    the
meteorological.

The mechanics of using this curve as an axis might involve an
approach different from  that  customarily  employed.   The  curve
might, for  example,  be  mounted  as  an  equator  on  the  globe
partitioning the earth into two pieces in much the way that a seam
serves as an equatorial line to partition the hide on a  baseball.
In this circumstance, there would be freedom  to  choose  how  the
equator partitions the earth's landmass.  It might be  located  in
such a way that exactly half of the earth's water and half of  the
earth's land lie on either side of the Julia set  (using  theorems
from algebraic topology (Lefschetz, 1949; Dugundji, 1966; Spanier,
1966)).

\centerline{\bf Beyond  the  fractal:  a graph theoretic connection.}

The notions of attraction" and repulsion"  have also been
expressed in the physical world, using graph theory (Harary, 1969;
Uhlenbeck, 1960).  Fractals rely on distance, angle, or some other
quantifier; graphs do not, and in that respect, are  more  general
than are fractals.  Fractal--like concepts, such as space--filling
and the associated image  compression  (Barnsley,  1988),  may  be
characterized using graphs, as below (Arlinghaus, 1977; 1985).

This strategy will be expressed in terms of cubic trees  (all
nodes are of degree three, unless they are at the tip of a branch)
of shortest total length (Steiner trees) of maximal branching.  It
could be expressed in terms of graphs of various linkage patterns;
what is important is to begin with  some  systematic  process  for
forming graphs.

Given  a geographic region  whose  periphery  is   outlined
by landmark positions at $P_1$,  $P_2$,  $P_3$, $P_4$, and $P_5$
(Figure  7.A).    View the landmarks as the nodes of a graph and
the  peripheral outline as the edges linking these nodes (Figure
7.A).   A  global" network within the entire pentagonal region
might  lie along   lines of a Steiner (shortest total  distance)
tree  (Figure  7.A)  (Arlinghaus,  1977;  1985) attached to  the
pentagonal hull joining neighboring branch tips  (Balaban,  {\it
et al.\/}, 1970).

\topinsert \vskip 6in
\noindent{\bf Figure 7}.
\smallskip
THIS FIGURE DOES NOT TRANSMIT---AVAILABLE ONLY AS HARD COPY.
(THUS SHOWN HERE AS A SCANNED IMAGE ONLY, WITHOUT THE ACCOMPANYING TeX.)
Figure 7.

\smallskip Network location within geographic regions. Points of the pentagonal hull have P" as a notational base; Steiner points have S" as a notational base. A. A Steiner (shortest total distance) tree linking five locations. B. Partition into three distinct, contiguous geographic regions. C. Steiner networks in each geographic region; boundaries separating regions are removed. D. Steiner networks in two quadrangular circuits; circuit boundaries removed. E. Process repeated on remaining quadrangular cell; the result is a tree with local Steiner characteristics that provides global linkage following the basic pattern of the global Steiner tree (Figure 7.A)). \endinsert Figure 7.B will be used as an initial figure from which to produce a network that penetrates triangular geographic subregions (introducing edges $P_2P_5$ and $P_2P_4$) more deeply than does the global network of Figure 7.A, yet retains the Steiner characteristic locally within each geographic subregion. An iterative process using Steiner trees (as a Steiner transformation") will be applied to Figure 7.B (Arlinghaus, 1977; 1983), as follows. Introduce Steiner networks into each of the three triangular regions and remove the edges $P_2P_5$ and $P_2P_4$ so that a new network, containing two quadrangular cells, is hooked into the pentagon $P_1P_2P_3P_4P_5$ (Figure 7.C). Repeat this procedure in the network of Figure 7.C, introducing Steiner networks into all circuits that do not have an edge in common with the pentagon $P_1P_2P_3P_4P_5$. Thus, the two four--sided circuits, $P_5S_1P_2S_2$; $P_2S_2P_4S_3$, in Figure 7.C are replaced with the lines of the network, $P_5S_1'$, $S_1S_1'$, $S_1'S_2'$, $P_2S_2'$, $S_2'S_2$; $S_2S_3'$, $P_2S_3'$, $S_3'S_4'$, $S_4'P_4$, $S_4'S_3$, shown in Figure 7.D. Repeat this process in Figure 7.D, using a Steiner tree, $S_2S_1''$, $S_2'S_1''$, $S_1''S_2''$, $S_2''P_2$, $S_2''S_3'$, to replace the single four--sided cell, $P_2S_2'S_2S_3'$, not sharing an edge with $P_1P_2P_3P_4P_5$. The result, shown in Figure 7.E, is a tree which cannot be further reduced using the Steiner transformation. It satisfies the initial conditions of generating a tree more local than the Steiner network of maximal branching on $P_1P_2P_3P_4P_5$ (but with local Steiner characteristics), while retaining the global structure of a graph--theoretic tree hooked into $P_1P_2P_3P_4P_5$ in a pattern similar to that of the global Steiner tree (with only local variation as along the edge $S_2S_1''$). This process attempts to integrate local with global concerns. In this case, the process terminates after a finite number of steps; were it to continue, greater space--filling of the geographic region by lines of the network would occur (Arlinghaus, 1977; 1985). A natural question to ask is whether or not this process necessarily terminates; do successive applications generate a finite reduction sequence of the cellular" structure into a tree" structure within $P_1P_2P_3P_4P_5$? Or, is it possible that this transformation, applied iteratively, might fill enough space to choke the entire region with an infinite regeneration of cells and of lines bounding those cells (Arlinghaus, 1977; 1985)? In this vein, take Figure 7.B and add one edge to it, creating four triangular geographic regions (Figure 8.A). Apply the same process to it as above, producing the networks shown in Figures 8.B and 8.C. Clearly, further iteration would simply produce a greater number of polygonal cells, tightly compressed around the node $P_2$. Discovering a means to calculate the dimension of this compression is an open issue. It is not difficult, however, to understand under what conditions this sequence might, or might not, terminate (Comments (based on material in Arlinghaus, 1977; 1985) below). \vskip.1cm \noindent Definition (Harary, 1969; Tutte, 1966), \vskip.1cm A wheel $W_n$ of order $n$, $n>3$, is a graph obtained from an $n$--gon by inserting one new vertex, the hub, and by joining the hub to at least two of the vertices of the $n$--gon by a finite sequence of edges ($P_2$ is the hub of a wheel formed in Figure 8.A).

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

Figure 8.

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

\noindent Comment 1 \vskip.1cm Hubs of wheels are invariant, as hubs of wheels, under a sequence of successive applications of the Steiner transformation described above. \vskip.1cm \noindent Comment 2 \vskip.1cm Suppose that there exists a finite set of contiguous triangles, $T$. If $T$ contains a wheel, then a sequence of successive applications of the Steiner transformation to $T$ fails to produce an irreducible tree. The sequence fails to terminate (as long as the Steiner trees produced at each stage are not degenerate). \vskip.1cm \noindent Comment 3 \vskip.1cm Suppose that there exists a finite set of contiguous triangles $T = \{L_1 \ldots L_m \}$ with vertex set $V = \{P_1 \ldots P_n \}, \quad n>m$ (as in Figure 7.B, $m = 3$, $n = 5$). Suppose that $T$ does not contain a wheel. The number of steps $M$, in the sequence of successive applications of the Steiner transformation to $T$ required to reduce $T$ to a tree is $$M = (\hbox{max} (\hbox{degree} (P_i ))) - 1.$$ Since $T$ does not contain a wheel, it follows from Comment 2 that the reduction sequence is finite. The actual size of $M$ might be found using mathematical induction on the number of cells in $T$ and on the graph--theoretic degree of $P_i$. The examples shown in Figures 7 and 8, together with the Comments above, suggest that a sequence of successive applications of the Steiner transformation to such geo-- graphs" resolves scale problems in the same manner as fractals. A natural next step beyond the fractal might be to note that a graph is a simplicial complex of dimension 0 or 1 (Harary, 1969). Thus, similar strategy might be applied there: the triangles of Figure 7.B might represent simplexes of arbitrary dimension in a simplicial complex of higher dimension. Theorems from algebraic topology might then be turned back on the mapping of geographic information using a computer. This notion is already in evidence: because point," line," and area" translate into the topological notions of 0--cell," 1--cell," and 2--cell" in a Geographic Information System, cells in the underlying computerized sim--pixel" complex can then be colored as inside" or outside" a given data set. This follows from the Jordan Curve Theorem (of algebraic topology). Independent of choice of theoretical tool---from fractal to graph to simplicial complex---the resolution of scale is achieved by uniting local and global mathematical structures: within fractal geometry as well as beyond it. \vskip.1cm {\narrower\noindent In nature, parts clearly do fit together into real structures, and the parts are affected by their environment. The problem is largely one of understanding. The mystery that remains lies largely in the nature of structural hierarchy, for the human mind can examine nature on many different scales sequentially, but not simultaneously." \smallskip} {\sl C. S. Smith, in Arthur L. Loeb, 1976\/}. \vfill\eject \centerline{\bf References.} \ref Arlinghaus, S. L. 1985. {\sl Essays on Mathematical Geography\/}. Institute of Mathematical Geography, Monograph \#3. Ann Arbor: Michigan Document Services. \ref Arlinghaus, S. L. 1977. On Geographic Network Location Theory." Unpublished Ph.D. dissertation, Department of Geography, The University of Michigan. \ref Balaban, A. T.; Davies, R. O.; Harary, F.; Hill, A.; and Westwick, R. 1970. Cubic identity graphs and planar graphs derived from trees. {\sl Journal\/}, Australian Mathematical Society 11:207-215. \ref Barnsley, M. 1988. {\sl Fractals Everywhere\/}. New York: Academic Press. \ref Boyce, W. E. and DiPrima, R. C. 1977. {\sl Elementary Differential Equations\/}. New York: Wiley. \ref Dugundji, J. 1966. {\sl Topology \/}. Boston: Allyn and Bacon. \ref H\"agerstrand, T. 1967. {\sl Innovation Diffusion as a Spatial Process\/}. Postscript and translation by Allan Pred. Chicago: University of Chicago Press. \ref Haggett, P.; Cliff, A. D.; and Frey, A. 1977. {\sl Locational Analysis in Human Geography\/}. New York: Wiley. \ref Harary, F. 1969. {\sl Graph Theory\/}. Reading, MA: 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.

\endinsert

From a geographic viewpoint, this says that if a rigid tetrahedron were built of metal rods with apex at point O, that any two triangles that fit perfectly inside this structure would have this property. One triangle projects" from a point (as for example in gnomonic or stereographic map projection) to the other. This might suggest a way to deform cells of a triangulation of a region of the earth into one another in such a way that this Desargues's line serves as some sort of an invariant of the deformation. This observation might then make one wonder what sorts of geometries exist that do not obey Desargues's Theorem. There is a whole class of Combinatorial geometries" or finite projective planes that do not. \smallskip References \smallskip \ref Coxeter, H. S. M. {\sl Introduction to Geometry\/}, New York: Wiley, 1961. \ref Coxeter, H. S. M. {\sl Projective Geometry\/}, Toronto: University of Toronto Press, 1974. \vfill\eject \noindent{\bf Construction Zone} --- One possible direction for application of Desargues's Theorem is to deform one tesselation of a region into another, leaving something invariant. Another related issue with tesselations is to try to regularize a tesselation composed of irregularly shaped cells. The following construction shows how to derive a centrally symmetric hexagon from an arbitrary convex hexagon. Given an arbitrary convex hexagon, $V_1V_2V_3V_4V_5V_6$. Join alternate vertices to inscribe a six--pointed star within this hexagon---that is, draw lines $V_1V_3$, $V_2V_4$, $V_3V_5$, $V_4V_6$, $V_5V_1$, $V_6V_2$ (it is suggested that you do so on a separate sheet of paper, at this point).

\topinsert \vskip5.5in Figure to accompany construction of centrally symmetric hexagon.
Figure 10.

\endinsert

This produces six distinct triangles (of interest here--of course there are more): $$\triangle V_1V_2V_3; \quad \triangle V_2V_3V_4; \quad \triangle V_3V_4V_5; \quad \triangle V_4V_5V_6; \quad \triangle V_5V_6V_1; \quad \triangle V_6V_1V_2.$$ To find the center of gravity of any triangle, find the point at which the medians are concurrent (the median is the line joining a vertex to the midpoint of the opposite side). This point is the center of gravity. Find the centers of gravity $$G_1, \quad G_2, \quad G_3, \quad G_4, \quad G_5, \quad G_6$$ of each of the triangles distinguished above (in the order suggested). The hexagon determined by these centers of gravity will be centrally symmetric. That is, opposite sides will be equal in length and parallel to each other: $$G_1G_2 \parallel G_4G_5; \quad |G_1G_2|=|G_4G_5|;$$ $$G_2G_3 \parallel G_5G_6; \quad |G_2G_3|=|G_5G_6|;$$ $$G_3G_4 \parallel G_6G_1; \quad |G_3G_4|=|G_6G_1|.$$ Another way of visualizing the symmetry is to observe that the three lines joining $G_1G_4$, $G_2G_5$, $G_3G_6$ are concurrent at a single point (call it $O$). In this way, one might also determine a center" for this symmetric hexagon which might then serve as a point to which a reference value might be attached for the arbitrary hexagon from which it was derived. This centrally symmetric hexagon is called the Dirichlet region of the arbitrary convex hexagon. This construction can be proved using Euclidean geometry (if requests come in, I'll put it in a later issue). \smallskip This feature is based on discussions in \smallskip \ref Kasner, Edward, and Newman, James R. New names for old," in {\sl The World of Mathematics\/}, edited by James R. Newman, Volume III, 1996-2010. New York: Simon and Schuster, 1956. \ref Coxeter, H. S. M. {\sl Introduction to Geometry\/}, New York: Wiley, 1961. \vfill\eject \noindent{\bf Reference Corner} --- \vskip.5cm Point set theory and topology. A recent pleasant evening spent with Hal Moellering had him questioning me and Bill Arlinghaus as to what might be reasonable, or useful, references from which graduate students in geography could get some sort of grasp of the elements of point set topology. A few references are listed below; send in your favorites and they will be printed next time. Hope that mathematicians as well as geographers will do so. Future topics to include graph theory and number theory as well as others suggested by reader input. Thanks Hal for the idea (generated by your questions) of doing this feature! \noindent Some long--time favorites and classics: \ref Dugundji, James. {\sl Topology\/}. Boston: Allyn and Bacon, 1960. \ref Hall, Dick Wick and Guilford L. Spencer II, {\sl Elementary Topology\/}, New York: Wiley, 1955. \ref Halmos, Paul R., {\sl Na\"{\i}ve Set Theory\/}, Princeton: D. Van Nostrand, 1960. \ref Hausdorff, Felix, {sl Mengenlehre\/}, Berlin: Walter de Gruyer, 1935. \ref Hocking, John G. and Gail S. Young, {\sl Topology}, 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.

TYPESETTING, USING TeX, THAT PRODUCED THIS CROSSWORD PUZZLE.
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{{\phantom{0} \phantom{0} \atop *}}& {{19 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}&\cr {{20 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{21 \atop \phantom{0}*}}& {{22 \atop \phantom{0}*}}& {{23 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{24 \atop \phantom{0}*}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{25 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{26 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\phantom{0} \phantom{0} \atop *}}&\cr {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\bullet \atop \bullet}{\bullet 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{{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}&\cr {{39 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{40 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{41 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{42 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{43 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}&\cr {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{44 \atop \phantom{0}*}}& {{45 \atop \phantom{0}*}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{46 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{47 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{48 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{49 \atop \phantom{0}*}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{50 \atop \phantom{0}*}}& {{51 \atop \phantom{0}*}}&\cr {{52 \atop \phantom{0}*}}& {{53 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\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}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{55 \atop \phantom{0}*}}& {{56 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}&\cr {{57 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{58 \atop \phantom{0}*}}& {{59 \atop \phantom{0}*}}& {{60 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{61 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}&\cr {{62 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{63 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}& {{64 \atop \phantom{0}*}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\phantom{0} \phantom{0} \atop *}}& {{\bullet \atop \bullet}{\bullet \atop \bullet}}&\cr }$$
\endinsert

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

TYPESETTING, USING TeX, THAT PRODUCED THE CROSSWORD PUZZLE SOLUTION.
\vskip.5cm
\topinsert
\hsize = 6.5 true in
\input fontmac
\setpointsize{9}{9}{8}
\parskip=3pt
\baselineskip=11 pt
\mathsurround=1pt
$$\matrix{ {{\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 }$$
\endinsert
\vfill\eject

\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