\hsize = 6.5 true in %THE FIRST 14 LINES ARE TYPESETTING CODE.
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\def\righthead{\sl\hfil SOLSTICE }
\def\lefthead{\sl Summer, 1990 \hfil}
\def\ref{\noindent\hang}
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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 Business} \hfil}
\line{{\bf Robert F. Austin},
Director, Automated Mapping and Facilities Management, CDI. \hfil}
\smallskip
\hrule
\smallskip
The purpose of {\sl Solstice\/} is to promote interaction
between geography and mathematics. Articles in which elements
of one discipline are used to shed light on the other are
particularly sought. Also welcome, are original contributions
that are purely geographical or purely mathematical. These may
be prefaced (by editor or author) with commentary suggesting
directions that might lead toward the desired interaction.
Individuals wishing to submit articles, either short or full--
length, as well as contributions for regular features, should
send them, in triplicate, directly to the Editor--in--Chief.
Contributed articles will be refereed by geographers and/or
mathematicians. Invited articles will be screened by suitable
members of the editorial board. IMaGe is open to having authors
suggest, and furnish material for, new regular features.
\vskip2in
\noindent {\bf Send all correspondence to:}
\vskip.1cm
\centerline{\bf Institute of Mathematical Geography}
\centerline{\bf 2790 Briarcliff}
\centerline{\bf Ann Arbor, MI 48105-1429}
\vskip.1cm
\centerline{\bf (313) 761-1231}
\centerline{\bf IMaGe@UMICHUM}
\vfill\eject
This document is produced using the typesetting program,
{\TeX}, of Donald Knuth and the American Mathematical Society.
Notation in the electronic file is in accordance with that of
Knuth's {\sl The {\TeX}book}. The program is downloaded for
hard copy for on The University of Michigan's Xerox 9700 laser--
printing Xerox machine, using IMaGe's commercial account with
that University.
Unless otherwise noted, all regular features are written by the
Editor--in--Chief.
\smallskip
{\nn Upon final acceptance, authors will work with IMaGe
to get manuscripts into a format well--suited to the
requirements of {\sl Solstice\/}. Typically, this would mean
that authors would submit a clean ASCII file of the
manuscript, as well as hard copy, figures, and so forth (in
camera--ready form). Depending on the nature of the document
and on the changing technology used to produce {\sl
Solstice\/}, there may be other requirements as well.
Currently, the text is typeset using {\TeX}; in that way,
mathematical formul{\ae} can be transmitted as ASCII files and
downloaded faithfully and printed out. The reader
inexperienced in the use of {\TeX} should note that this is
not a ``what--you--see--is--what--you--get" display; however,
we hope that such readers find {\TeX} easier to learn after
exposure to {\sl Solstice\/}'s e-files written using {\TeX}!}
{\nn Copyright will be taken out in the name of the
Institute of Mathematical Geography, and authors are required to
transfer copyright to IMaGe as a condition of publication.
There are no page charges; authors will be given permission to make
reprints from the electronic file, or to have IMaGe make a
single master reprint for a nominal fee dependent on manuscript
length. Hard copy of {\sl Solstice\/} will be sold (contact
IMaGe for price--{\sl Solstice\/} will be priced to cover
expenses of journal production); it is the desire of IMaGe to
offer electronic copies to interested parties for free--as a
kind of academic newsstand at which one might browse, prior to
making purchasing decisions. Whether or not it will be feasible
to continue distributing complimentary electronic files remains
to be seen.}
\vskip.5cm
Copyright, August, 1990, Institute of Mathematical Geography.
All rights reserved.
\vskip1cm
ISBN: 1-877751-51-0
\vfill\eject
\centerline{\bf SUMMARY OF CONTENT}
\smallskip
Note: in this first issue, there is one of each type of article--this
need not be the case in the future.
\vskip.2cm
1. REPRINT.
\smallskip
William Kingdon Clifford, {\sl Postulates of the Science of Space\/}.
This reprint of a portion of Clifford's lectures to the
Royal Institution in the 1870's suggests many geographic topics
of concern in the last half of the twentieth century. Look for
connections to boundary issues, to scale problems, to self--
similarity and fractals, and to non--Euclidean geometries (from
those based on denial of Euclid's parallel postulate to those
based on a sort of mechanical ``polishing"). What else did, or
might, this classic essay foreshadow?
\smallskip
2. FULL--LENGTH ARTICLE.
\smallskip
Sandra L. Arlinghaus, {\sl Beyond the Fractal\/}.
Figures are transmitted in this e-file only for the half of the
article described in the first paragraph below.
An original article. The fractal notion of self--
similarity is useful for characterizing change in scale; the
reason fractals are effective in the geometry of central place
theory is because that geometry is hierarchical in nature.
Thus, a natural place to look for other connections of this sort
is to other geographical concepts that are also hierarchical.
Within this fractal context, this article examines the case of
spatial diffusion.
When the idea of diffusion is extended to see ``adopters"
of an innovation as ``attractors" of new adopters, a Julia set
is introduced as a possible axis against which to measure one
class of geographic phenomena. Beyond the fractal context,
fractal concepts, such as ``compression" and ``space--filling"
are considered in a broader graph--theoretic context. \smallskip
3. SHORT ARTICLE.
\smallskip
William C. Arlinghaus, {\sl Groups, graphs, and God\/}
An original article based on a talk given before a MIdwest
GrapH TheorY (MIGHTY) meeting. The author, an algebraic graph
theorist, ties his research interests to a broader philosophical
realm, suggesting the breadth of range to which algebraic
structure might be applied.
The fact that almost all graphs are rigid (have trivial
automorphism groups) is exploited to argue probabilistically
for the existence of God. This is presented in the context
that applications of mathematics need not be limited to
scientific ones.
\smallskip
Note: In this first issue, there is one of each type of
article--this need not be the case in the future.
\smallskip
4. REGULAR FEATURES
\smallskip
\item{i.} {\bf Theorem Museum} ---
Desargues's Two Triangle Theorem of projective geometry.
\item{ii.} {\bf Construction Zone} ---
a centrally symmetric hexagon is derived from an arbitrary
convex hexagon.
\item{iii.} {\bf Reference Corner} ---
Point set theory and topology.
\item{iv.} {\bf Games and other educational features} ---
Crossword puzzle focused on spices.
\item{v.} {\bf Coming attractions} ---
Indication of topics for the ``REGULAR FEATURES" section
in forthcoming issues.
\smallskip
\item{vi.}{\bf Solution to puzzle}
\smallskip
5. SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE
\smallskip
This section shows the exact set of commands that work to
download this file on The University of Michigan's Xerox 9700.
Because different universities will have different installations
of {\TeX}, this is only a rough guideline which {\sl might\/} be
of use to the reader or to the reader's computing center.
\vfill\eject
\noindent 1. REPRINT
\smallskip
\centerline{\bf THE POSTULATES OF THE SCIENCE OF SPACE}
\vskip.2cm
\centerline{William Kingdon Clifford}
\smallskip
From a set of lectures given before the Royal Institution, 1873 --
``The Philosophy of the Pure Sciences." Reprinted excerpt
longer than this one appears in
{\sl The World of Mathematics\/}, edited by James R. Newman, New York:
Simon and Schuster, 1956.
\smallskip
\noindent In my first lecture I said that, out of the pictures
which are all that we can really see, we imagine a world of solid
things; and that this world is constructed so as to fulfil a
certain code of rules, some called axioms, and some called
definitions, and some called postulates, and some assumed in the
course of demonstration, but all laid down in one form or another
in Euclid's Elements of Geometry. It is this code of rules that
we have to consider to--day. I do not, however, propose to take
this book that I have mentioned, and to examine one after another
the rules as Euclid has laid them down or unconsciously assumed
them; notwithstanding that many things might be said in favour of
such a course. This book has been for nearly twenty--two
centuries the encouragement and guide of that scientific thought
which is one thing with the progress of man from a worse to a
better state. The encouragement; for it contained a body of
knowledge that was really known and could be relied on, and that
moreover was growing in extent and application. For even at the
time this book was written---shortly after the foundation of the
Alexandrian Museum--Mathematic was no longer the merely ideal
science of the Platonic school, but had started on her career of
conquest over the whole world of Phenomena. The guide; for the
aim of every scientific student of every subject was to bring his
knowledge of that subject into a form as perfect as that which
geometry had attained. Far up on the great mountain of Truth,
which all the sciences hope to scale, the foremost of that sacred
sisterhood was seen, beckoning to the rest to follow her. And
hence she was called, in the dialect of the Pythagoreans, `the
purifier of the reasonable soul.' Being thus in itself at once
the inspiration and the aspiration of scientific thought, this
Book of Euclid's has had a history as chequered as that of human
progress itself. [Deleted text.] The geometer of to--day knows
nothing about the nature of actually existing space at an
infinite distance; he knows nothing about the properties of this
present space in a past or a future eternity. He knows, indeed,
that the laws assumed by Euclid are true with an accuracy that no
direct experiment can approach, not only in this place where we
are, but in places at a distance from us that no astronomer has
conceived; but he knows this as of Here and Now; beyond his range
is a There and Then of which he knows nothing at present, but may
ultimately come to know more. So, you see, there is a real
parallel between the work of Copernicus and his successors on the
one hand, and the work of Lobatchewsky and his successors on the
other. In both of these the knowledge of Immensity and Eternity
is replaced by knowledge of Here and Now. And in virtue of these
two revolutions the idea of the Universe, the Macrocosm, the All,
as subject of human knowledge, and therefore of human interest,
has fallen to pieces.
It will now, I think, be clear to you why it will not do to
take for our present consideration the postulates of geometry as
Euclid has laid them down. While they were all certainly true,
there might be substituted for them some other group of
equivalent propositions; and the choice of the particular set of
statements that should be used as the groundwork of the science
was to a certain extent arbitrary, being only guided by
convenience of exposition. But from the moment that the actual
truth of these assumptions becomes doubtful, they fall of
themselves into a necessary order and classification; for we then
begin to see which of them may be true independently of the
others. And for the purpose of criticizing the evidence for
them, it is essential that this natural order should be taken;
for I think you will see presently that any other order would
bring hopeless confusion into the discussion.
Space is divided into parts in many ways. If we consider
any material thing, space is at once divided into the part where
that thing is and the part where it is not. The water in this
glass, for example, makes a distinction between the space where
it is and the space where it is not. Now, in order to get from
one of these to the other you must cross the {\it surface\/} of
the water; this surface is the boundary of the space where the
water is which separates it from the space where it is not.
Every {\it thing\/}, considered as occupying a portion of space,
has a surface which separates that space where it is from the
space where it is not. But, again, a surface may be divided into
parts in various ways. Part of the surface of this water is
against the air, and part is against the glass. If you travel
over the surface from one of these parts to the other, you have
to cross the {\it line\/} which divides them; it is this circular
edge where water, air, and glass meet. Every part of a surface
is separated from the other parts by a line which bounds it. But
now suppose, further, that this glass had been so constructed
that the part towards you was blue and the part towards me was
white, as it is now. Then this line, dividing two parts of the
surface of the water, would itself be divided into two parts;
there would be a part where it was against the blue glass, and a
part where it was against the white glass. If you travel in
thought along that line, so as to get from one of these two parts
to the other, you have to cross a {\it point\/} which separates
them, and is the boundary between them. Every part of a line is
separated from the other parts by points which bound it. So we
may say altogether ---
\vskip.1cm
The boundary of a solid ({\it i.e.\/}, of a part of space) is a
surface.
\vskip.1cm
The boundary of a part of a surface is a line.
\vskip.1cm
The boundaries of a part of a line are points.
And we are only settling the meanings in which words are to
be used. But here we may make an observation which is true of
all space that we are acquainted with: it is that the process
ends here. There are no parts of a point which are separated
from one another by the next link in the series. This is also
indicated by the reverse process.
For I shall now suppose this point --- the last thing that we
got to --- to move round the tumbler so as to trace out the
line, or edge, where air, water, and glass meet. In this way I
get a series of points, one after another; a series of such a
nature that, starting from any one of them, only two changes are
possible that will keep it within the series: it must go
forwards or it must go backwards, and each of these if perfectly
definite. The line may then be regarded as an aggregate of
points. Now let us imagine, further, a change to take place in
this line, which is nearly a circle. Let us suppose it to
contract towards the centre of the circle, until it becomes
indefinitely small, and disappears. In so doing it will trace
out the upper surface of the water, the part of the surface where
it is in contact with the air. In this way we shall get a
series of circles one after another --- a series of such a nature
that, starting from any one of them, only two changes are
possible that will keep it within the series: it must expand or
it must contract. This series, therefore, of circles, is just
similar to the series of points that make one circle; and just as
the line is regarded as an aggregate of points, so we may regard
this surface as an aggregate of lines. But this surface is also
in another sense an aggregate of point, in being an aggregate of
aggregates of points. But, starting from a point in the surface,
more than two changes are possible that will keep it within the
surface, for it may move in any direction. The surface, then, is
an aggregate of points of a different kind from the line. We
speak of the line as a point--aggregate of one dimension,
because, starting from one point, there are only two possible
directions of change; so that the line can be traced out in one
motion. In the same way, a surface is a line--aggregate of one
dimension, because it can be traced out by one motion of the
line; but it is a point--aggregate of two dimensions, because, in
order to build it up of points, we have first to aggregate points
into a line, and then lines into a surface. It requires two
motions of a point to trace it out.
Lastly, let us suppose this upper surface of the water to
move downwards, remaining always horizontal till it becomes the
under surface. In so doing it will trace out the part of space
occupied by the water. We shall thus get a series of surfaces
one after another, precisely analogous to the series of points
which make a line, and the series of lines which make a surface.
The piece of solid space is an aggregate of surfaces, and an
aggregate of the same kind as the line is of points; it is a
surface--aggregate of one dimension. But at the same time it is
a line--aggregate of two dimensions, and a point--aggregate of
three dimensions. For if you consider a particular line which
has gone to make this solid, a circle partly contracted and part
of the way down, there are more than two opposite changes which
it can undergo. For it can ascend or descend, or expand or
contract, or do both together in any proportion. It has just as
great a variety of changes as a point in a surface. And the
piece of space is called a point--aggregate of three dimensions,
because it takes three distinct motions to get it from a point.
We must first aggregate points into a line, then lines into a
surface, then surfaces into a solid.
At this step it is clear, again, that the process must stop
in all the space we know of. For it is not possible to move that
piece of space in such a way as to change every point in it.
When we moved our line or our surface, the new line or surface
contained no point whatever that was in the old one; we started
with one aggregate of points, and by moving it we got an entirely
new aggregate, all the points of which were new. But this cannot
be done with the solid; so that the process is at an end. We
arrive, then, at the result that {\it space is of three
dimensions\/}.
Is this, then, one of the postulates of the science of
space? No; it is not. The science of space, as we have it,
deals with relations of distance existing in a certain space of
three dimensions, but it does not at all require us to assume
that no relations of distance are possible in aggregates of more
than three dimensions. The fact that there are only three
dimensions does regulate the number of books that we write, and
the parts of the subject that we study: but it is not itself a
postulate of the science. We investigate a certain space of
three dimensions, on the hypothesis that it has certain
elementary properties; and it is the assumptions of these
elementary properties that are the real postulates of the science
of space. To these I now proceed.
The first of them is concerned with {\it points\/}, and with
the relation of space to them. We spoke of a line as an
aggregate of points. Now there are two kinds of aggregates,
which are called respectively continuous and discrete. If you
consider this line, the boundary of part of the surface of the
water, you will find yourself believing that between any two
points of it you can put more points of division, and between any
two of these more again, and so on; and you do not believe there
can be any end to the process. We may express that by saying you
believe that between any two points of the line there is an
infinite number of other points. But now here is an aggregate of
marbles, which, regarded as an aggregate, has many characters of
resemblance with the aggregate of points. It is a series of
marbles, one after another; and if we take into account the
relations of nextness or contiguity which they possess, then
there are only two changes possible from one of them as we travel
along the series: we must go to the next in front, or to the
next behind. But yet it is not true that between any two of them
here is an infinite number of other marbles; between these two,
for example, there are only three. There, then, is a distinction
at once between the two kinds of aggregates. But there is
another, which was pointed out by Aristotle in his Physics and
made the basis of a definition of continuity. I have here a row
of two different kinds of marbles, some white and some black.
This aggregate is divided into two parts, as we formerly supposed
the line to be. In the case of the line the boundary between the
two parts is a point which is the element of which the line is an
aggregate. In this case before us, a marble is the element; but
here we cannot say that the boundary between the two parts is a
marble. The boundary of the white parts is a white marble, and
the boundary of the black parts is a black marble; these two
adjacent parts have different boundaries. Similarly, if instead
of arranging my marbles in a series, I spread them out on a
surface, I may have this aggregate divided into two portions ---
a white portion and a black portion; but the boundary of the
white portion is a row of white marbles, and the boundary of the
black portion is a row of black marbles. And lastly, if I made a
heap of white marbles, and put black marbles on the top of them,
I should have a discrete aggregate of three dimensions divided
into two parts: the boundary of the white part would be a layer
of white marbles, and the boundary of the black part would be a
layer of black marbles. In all these cases of discrete
aggregates, when they are divided into two parts, the two
adjacent parts have different boundaries. But if you come to
consider an aggregate that you believe to be continuous, you will
see that you think of two adjacent parts as having the {\it
same\/} boundary. What is the boundary between water and air
here? Is it water? No; for there would still have to be a
boundary to divide that water from the air. For the same reason
it cannot be air. I do not want you at present to think of the
actual physical facts by the aid of any molecular theories; I
want you only to think of what appears to be, in order to
understand clearly a conception that we all have. Suppose the
things actual in contact. If, however much we magnified them,
they still appeared to be thoroughly homogeneous, the water
filling up a certain space, the air an adjacent space; if this
held good indefinitely through all degrees of conceivable
magnifying, then we could not say that the surface of the water
was a layer of water and the surface of air a layer of air; we
should have to say that the same surface was the surface of both
of them, and was itself neither one nor the other---that this
surface occupied {\it no\/} space at all. Accordingly, Aristotle
defined the continuous as that of which two adjacent parts have
the same boundary; and the discontinuous or discrete as that of
which two adjacent parts have direct boundaries.
Now the first postulate of the science of space is that
space in a continuous aggregate of points, and not a discrete
aggregate. And this postulate---which I shall call the postulate
of continuity---is really involved in those three of the six
postulates of Euclid for which Robert Simson has retained the
name of postulate. You will see, on a little reflection, that a
discrete aggregate of points could not be so arranged that any
two of them should be relatively situated to one another in
exactly the same manner, so that any two points might be joined
by a straight line which should always bear the same definite
relation to them. And the same difficulty occurs in regard to
the other two postulates. But perhaps the most conclusive way of
showing that this postulate is really assumed by Euclid is to
adduce the proposition he probes, that every finite straight line
may be bisected. Now this could not be the case if it consisted
of an odd number of separate points. As the first of the
postulates of the science of space, the, we must reckon this
postulate of Continuity; according to which two adjacent portions of
space, or of a surface, or of a line, have the {\it same\/}
boundary, {\it viz\/}.--- a surface, a line, or a point; and
between every two points on a line there is an infinite number of
intermediate points.
The next postulate is that of Elementary Flatness. You
know that if you get hold of a small piece of a very large
circle, it seems to you nearly straight. So, if you were to take
any curved line, and magnify it very much, confining your
attention to a small piece of it, that piece would seem
straighter to you than the curve did before it was magnified. At
least, you can easily conceive a curve possessing this property,
that the more you magnify it, the straighter it gets. Such a
curve would possess the property of elementary flatness. In the
same way, if you perceive a portion of the surface of a very
large sphere, such as the earth, it appears to you to be flat.
If, then, you take a sphere of say a foot diameter, and magnify
it more and more, you will find that the more you magnify it the
flatter it gets. And you may easily suppose that this process
would go on indefinitely; that the curvature would become less
and less the more the surface was magnified. Any curved surface
which is such that the more you magnify it the flatter it gets,
is said to possess the property of elementary flatness. But if
every succeeding power of our imaginary microscope disclosed new
wrinkles and inequalities without end, then we should say that
the surface did not possess the property of elementary flatness.
But how am I to explain how solid space can have this
property of elementary flatness? Shall I leave it as a mere
analogy, and say that it is the same kind of property as this of
the curve and surface, only in three dimensions instead of one or
two? I think I can get a little nearer to it than that; at all
events I will try.
If we start to go out from a point on a surface, there is a
certain choice of directions in which we may go. These
directions make certain angles with one another. We may suppose
a certain direction to start with, and then gradually alter that
by turning it round the point: we find thus a single series of
directions in which we may start from the point. According to
our first postulate, it is a continuous series of directions.
Now when I speak of a direction from the point, I mean a
direction of starting; I say nothing about the subsequent path.
Two different paths may have the same direction at starting; in
this case they will touch at the point; and there is an obvious
difference between two paths which touch and two paths which meet
and form an angle. Here, then, is an aggregate of directions,
and they can be changed into one another. Moreover, the changes
by which they pass into one another have magnitude, they
constitute distance--relations; and the amount of change
necessary to turn one of them into another is called the angle
between them. It is involved in this postulate that we are
considering, that angles can be compared in respect of
magnitude. But this is not all. If we go on changing a
direction of start, it will, after a certain amount of turning, come
round into itself again, and be the same direction. On
every surface which has the property of elementary flatness, the
amount of turning necessary to take a direction all round into
its first position is the same for all points of the surface. I
will now show you a surface which at one point of it has not this
property. I take this circle of paper from which a sector has
been cut out, and bend it round so as to join the edges; in this
way I form a surface which is called a {\it cone\/}. Now on all
points of this surface but one, the law of elementary flatness
holds good. At the vertex of the cone, however, notwithstanding
that there is an aggregate of directions in which you may start,
such that by continuously changing one of them you may get it
round into its original position, yet the whole amount of change
necessary to effect this is not the same at the vertex as it is
at any other point of the surface. And this you can see at once
when I unroll it; for only part of the directions in the plane
have been included in the cone. At this point of the cone, then,
it does not possess the property of elementary flatness; and no
amount of magnifying would ever make a cone seem flat at its
vertex.
To apply this to solid space, we must notice that here also
there is a choice of directions in which you may go out from any
point; but it is a much greater choice than a surface gives you.
Whereas in a surface the aggregate of directions is only of one
dimension, in solid space it is of two dimensions. But here also
there are distance--relations, and the aggregate of directions
may be divided into parts which have quantity. For example, the
directions which start from the vertex of this cone are divided
into those which go inside the cone, and those which go outside
the cone. The part of the aggregate which is inside the cone is
called a solid angle. Now in those spaces of three dimensions
which have the property of elementary flatness, the whole amount
of solid angle round one point is equal to the whole amount round
another point. Although the space need not be exactly similar to
itself in all parts, yet the aggregate of directions round one
point is exactly similar to the aggregate of directions round
another point, if the space has the property of elementary
flatness.
How does Euclid assume this postulate of Elementary
Flatness? In his fourth postulate he has expressed it so simply
and clearly that you will wonder how anybody could make all this
fuss. He says, `All right angles are equal.'
Why could I not have adopted this at once, and saved a
great deal of trouble? Because it assumes the knowledge of a
surface possessing the property of elementary flatness in all
its points. Unless such a surface is first made out to exist,
and the definition of a right angle is restricted to lines drawn
upon it---for there is no necessity for the word {\it straight\/}
in that definition---the postulate in Euclid's form is obviously
not true. I can make two lines cross at the vertex of a cone so
that the four adjacent angles shall be equal, and yet not one of
them equal to a right angle.
I pass on to the third postulate of the science of space---
the postulate of Superposition. According to this postulate a
body can be moved about in space without altering its size or
shape. This seems obvious enough, but it is worth while to
examine a little more closely into the meaning of it. We must
define what we mean by size and by shape. When we say that a
body can be moved about without altering its size, we mean that
it can be so moved as to keep unaltered the length of all the
lines in it. This postulate therefore involves that lines can be
compared in respect of magnitude, or that they have a length
independent of position; precisely as the former one involved the
comparison of angular magnitudes. And when we say that a body
can be moved about without altering its shape, we mean that it
can be so moved as to keep unaltered all the angles in it. It is
not necessary to make mention of the motion of a body, although
that is the easiest way of expressing and of conceiving this
postulate; but we may, if we like, express it entirely in terms
which belong to space, and that we should do in this way.
Suppose a figure to have been constructed in some portion of
space; say that a triangle has been drawn whose sides are the
shortest distances between its angular points. Then if in any
other portion of space two points are taken whose shortest
distance is equal to a side of the triangle, and at one of them
an angle is made equal to one of the angles adjacent to that
side, and a line of shortest distance drawn equal to the
corresponding side of the original triangle, the distance from
the extremity of this to the other of the two points will be
equal to the third side of the original triangle, and the two
will be equal in all respects; or generally, if a figure has been
constructed anywhere, another figure, with all its lines and all
its angles equal to the corresponding lines and angles of the
first, can be constructed anywhere else. Now this is exactly
what is meant by the principle of superposition employed by
Euclid to prove the proposition that I have just mentioned. And
we may state it again in this short form---All parts of space are
exactly alike.
But this postulate carries with it a most important
consequence. In enables us to make a pair of most fundamental
definitions---those of the plane and of the straight line. In
order to explain how these come out of it when it is granted, and
how they cannot be made when it is not granted, I must here say
something more about the nature of the postulate itself, which
might otherwise have been left until we come to criticize it.
We have stated the postulate as referring to solid space.
But a similar property may exist in surfaces. Here, for
instance, is part of the surface of a sphere. If I draw any
figure I like upon this, I can suppose it to be moved about in
any way upon the sphere, without alteration of its size or shape.
If a figure has been drawn on any part of the surface of a
sphere, a figure equal to it in all respects may be drawn on any
other part of the surface. Now I say that this property belongs
to the surface itself, is a part of its own internal economy, and does
not depend in any way upon its relation to space of three
dimensions. For I can pull it about and bend it in all manner
of ways, so as altogether to alter its relation to solid space;
and yet, if I do not stretch it or tear it, I make no difference
whatever in the length of any lines upon it, or in the size of
any angles upon it. I do not in any way alter the figures drawn
upon it, or the possibility of drawing figures upon it, {\it so
far as their relations with the surface itself are concerned\/}.
This property of the surface, then, could be ascertained by
people who lived entirely in it, and were absolutely ignorant of
a third dimension. As a point--aggregate of two dimensions, it
has in itself properties determining the distance--relations of
the points upon it, which are absolutely independent of the
existence of any points which are not upon it.
Now here is a surface which has not that property. You
observe that it is not of the same shape all over, and that some
parts of it are more curved than other parts. If you drew a
figure upon this surface, and then tried to move it about, you
would find that it was impossible to do so without altering the
size and shape of the figure. Some parts of it would have to
expand, some to contract, the lengths of the lines could not all
be kept the same, the angles would not hit off together. And
this property of the surface---that its parts are different from
one another---is a property of the surface itself, a part of its
internal economy, absolutely independent of any relations it may
have with space outside of it. For, as with the other one, I can
pull it about in all sorts of ways, and, so long as I do not
stretch it or tear it, I make no alteration in the length of
lines drawn upon it or in the size of the angles.
Here, then, is an intrinsic difference between these two
surfaces, as surfaces. They are both point--aggregates of two
dimensions; but the points in them have certain relations of
distance (distance measured always {\it on\/} the surface), and
these relations of distance are not the same in one case as they
are in the other.
The supposed people living in the surface and having no
idea of a third dimension might, without suspecting that third
dimension at all, make a very accurate determination of the
nature of their {\it locus in quo\/}. If the people who lived on
the surface of the sphere were to measure the angles of a
triangle, they would find them to exceed two right angles by a
quantity proportional to the area of the triangle. This excess
of the angles above two right angles, being divided by the area
of the triangle, would be found to give exactly the same quotient
at all parts of the sphere. That quotient is called the curvature
of the surface; and we say that a sphere is a surface of uniform
curvature. But if the people living on this irregular surface
were to do the same thing, they would not find quite the same
result. The sum of the angles would, indeed, differ from two
right angles, but sometimes in excess, and sometimes in defect,
according to the part of the surface where they were. And
though for small triangles in any on neighbourhood the excess or defect
would be nearly proportional to the area of the triangle,
yet the quotient obtained by dividing this excess or defect by
the area of the triangle would vary from one part of the surface
to another. In other words, the curvature of this surface varies
from point to point; it is sometimes positive, sometimes
negative, sometimes nothing at all.
But now comes the important difference. When I speak of a
triangle, what do I suppose the sides of that triangle to be?
If I take two points near enough together upon a surface,
and stretch a string between them, that string will take up a
certain definite position upon the surface, marking the line of
shortest distance from one point to the other. Such a line is
called a geodesic line. It is a line determined by the intrinsic
properties of the surface, and not by its relations with external
space. The line would still be the shortest line, however the
surface were pulled about without stretching or tearing. A
geodesic line may be {\it produced\/}, when a piece of it is
given; for we may take one of the points, and, keeping the string
stretched, make it go round in a sort of circle until the other
end has turned through two right angles. The new position will
then be a prolongation of the same geodesic line.
In speaking of a triangle, then, I meant a triangle whose
sides are geodesic lines. But in the case of a spherical
surface---or, more generally, of a surface of constant
curvature---these geodesic lines have another and most important
property. They are {\it straight\/}, so far as the surface is
concerned. On this surface a figure may be moved about without
altering its size or shape. It is possible, therefore, to draw
a line which shall be of the same shape all along and on both
sides. That is to say, if you take a piece of the surface on one
side of such a line, you may slide it all along the line and it
will fit; and you may turn it round and apply it to the other
side, and it will fit there also. This is Leibniz's definition
of a straight line, and, you see, it has no meaning except in the
case of a surface of constant curvature, a surface all parts of
which are alike.
Now let us consider the corresponding things in solid space.
In this also we may have geodesic lines; namely, lines formed by
stretching a string between two points. But we may also have
geodesic surfaces; and they are produced in this manner. Suppose
we have a point on a surface, and this surface possesses the
property of elementary flatness. Then among all the directions
of starting from the point, there are some which start {\it in
the surface\/}, and do not make an angle with it. Let all these
be prolonged into geodesics; then we may imagine one of these
geodesics to travel round and coincide with all the others in
turn. In so doing it will trace out a surface which is called a
geodesic surface. Now in the particular case where a space of
three dimensions has the property of superpositoin, or is all
over alike, these geodesic surfaces are {\it planes\/}. That is
to say, since the space is all over alike, these surfaces are also
of the same shape all over and on both sides; which is
Leibniz's definition of a plane. If you take a piece of space
on one side of such a plane, partly bounded by the plane, you may
slide it all over the plane, and it will fit; and you may turn it
round and apply it to the other side, and it will fit there also.
Now it is clear that this definition will have no meaning unless
the third postulate be granted. So we may say that when the
postulate of Superposition is true, then there are planes and
straight lines; and they are defined as being of the same shape
throughout and on both sides.
It is found that the whole geometry of a space of three
dimensions is known when we know the curvature of three geodesic
surfaces at every point. The third postulate requires that the
curvature of all geodesic surfaces should be everywhere equal to
the same quantity.
I pass to the fourth postulate, which I call the postulate
of Similarity. According to this postulate, any figure may be
magnified or diminished in any degree without altering its shape.
If any figure has been constructed in one part of space, it may
be reconstructed to any scale whatever in any other part of
space, so that no one of the angles shall be altered through all
the lengths of lines will of course be altered. This seems to be
a sufficiently obvious induction from experience; for we have all
frequently seen different sixes of the same shape; and it has the
advantage of embodying the fifth and sixth of Euclid's postulates
in a single principle, which bears a great resemblance in form to
that of Superposition, and may be used in the same manner. It is
easy to show that it involves the two postulates of Euclid: `Two
straight lines cannot enclose a space,' and `Lines in one plane
which never meet make equal angles with every other line.'
This fourth postulate is equivalent to the assumption that
the constant curvature of the geodesic surfaces is zero; or the
third and fourth may be put together, and we shall then say that
the three curvatures of space are all of them zero at every
point.
The supposition made by Lobatchewsky was, that the three
first postulates were true, but not the fourth. Of the two
Euclidean postulates included in this, he admitted one, {\it
viz\/}., that two straight lines cannot enclose a space, or that
two lines which once diverge go on diverging for ever. But he
left out the postulate about parallels, which may be stated in
this form. If through a point outside of a straight line there
be drawn another, indefinitely produced both ways; and if we turn
this second one round so as to make the point of intersection
travel along the first line, then at the very instant that this
point of intersection disappears at one end it will reappear at
the other, and there is only one position in which the lines do
not intersect. Lobatchewsky supposed, instead, that there was a
finite angle through which the second line must be turned after
the point of intersection had disappeared at one end, before it
reappeared at the other. For all positions of the second line
within this angle there is then no intersection. In the two
limiting positions, when the lines have just done meeting at one
end, and when they are just going to meet at the other, they are
called parallel; so that two lines can be drawn through a fixed
point parallel to a given straight line. The angle between these
two depends in a certain way upon the distance of the point from
the line. The sum of the angles of a triangle is less than two
right angles by a quantity proportional to the area of the
triangle. The whole of this geometry is worked out in the style
of Euclid, and the most interesting conclusions are arrived at;
particularly in the theory of solid space, in which a surface
turns up which is not plane relatively to that space, but which,
for purposes of drawing figures upon it, is identical with the
Euclidean plane.
It was Riemann, however, who first accomplished the task of
analysing all the assumptions of geometry, and showing which of
them were independent. This very disentangling and separation of
them is sufficient to deprive them for the geometer of their
exactness and necessity; for the process by which it is effected
consists in showing the possibility of conceiving these
suppositions one by one to be untrue; whereby it is clearly made
out how much is supposed. But it may be worth while to state
formally the case for and against them.
When it is maintained that we know these postulates to be
universally true, in virtue of certain deliverances of our
consciousness, it is implied that these deliverances could not
exist, except upon the supposition that the postulates are true.
If it can be shown, then, from experience that our consciousness
would tell us exactly the same things if the postulates are not
true, the ground of their validity will be taken away. But this
is a very easy thing to show.
That same faculty which tells you that space is continuous
tells you that this water is continuous, and that the motion
perceived in a wheel of life is continuous. Now we happen to
know that if we could magnify this water as much again as the
best microscopes can magnify it, we should perceive its granular
structure. And what happens in a wheel of life is discovered by
stopping the machine. Even apart, then, from our knowledge of
the way nerves act in carrying messages, it appears that we have
no means of knowing anything more about an aggregate than that it
is too fine--grained for us to perceive its discontinuity, if it
has any.
Nor can we, in general, receive a conception as positive
knowledge which is itself founded merely upon inaction. For the
conception of a continuous thing is of that which looks just the
same however much you magnify it. We may conceive the magnifying
to go on to a certain extent without change, and then, as it
were, leave it going on, without taking the trouble to doubt
about the changes that may ensue.
In regard to the second postulate, we have merely to point to
the example of polished surfaces. The smoothest surface that
can be made is the one most completely covered with the minutest
ruts and furrows. Yet geometrical constructions can be made with
extreme accuracy upon such a surface, on the supposition that it
is an exact plane. If, therefore, the sharp points, edges, and
furrows of space are only small enough, there will be nothing to
hinder our conviction of its elementary flatness. It has even
been remarked by Riemann that we must not shrink from this
supposition if it is found useful in explaining physical
phenomena.
The first two postulates may therefore be doubted on the
side of the very small. We may put the third and fourth
together, and doubt them on the side of the very great. For if
the property of elementary flatness exist on the average, the
deviations from it being, as we have supposed, too small to be
perceived, then, whatever were the true nature of space, we
should have exactly the conceptions of it which we now have, if
only the regions we can get at were small in comparison with the
areas of curvature. If we suppose the curvature to vary in an
irregular manner, the effect of it might be very considerable in
a triangle formed by the nearest fixed stars; but if we suppose
it approximately uniform to the limit of telescopic reach, it
will be restricted to very much narrower limits. I cannot
perhaps do better than conclude by describing to you as well as I
can what is the nature of things on the supposition that the
curvature of all space is nearly uniform and positive.
In this case the Universe, as known, becomes again a valid
conception; for the extent of space is a finite number of cubic
miles. And this comes about in a curious way. If you were to
start in any direction whatever, and move in that direction in a
perfect straight line according to the definition of Leibniz;
after travelling a most prodigious distance, to which the
parallactic unit---200,000 times the diameter of the earth's
orbit---would be only a few steps, you would arrive at---this
place. Only, if you had started upwards, you would appear from
below. Now, one of two things would be true. Either, when you
had got half--way on your journey, you came to a place that is
opposite to this, and which you must have gone through, whatever
direction you started in; or else all paths you could have taken
diverge entirely from each other till they meet again at this
place. In the former case, every two straight lines in a plane
meet in two points, in the latter they meet only in one. Upon
this supposition of a positive curvature, the whole of geometry
is far more complete and interesting; the principle of duality,
instead of half breaking down over metric relations, applies to
all propositions without exception. In fact, I do no mind
confessing that I personally have often found relief from the
dreary infinities of homaloidal space in the consoling hope that,
after all, this other may be the true state of things.
\vfill\eject
\noindent 2. FULL--LENGTH ARTICLE
\vskip.5cm
\centerline{\bf BEYOND THE FRACTAL} \vskip.1cm
\centerline{\sl Sandra Lach Arlinghaus}
\vskip.5cm
\centerline{``I never saw a moor,}
\centerline{I never saw the sea;}
\centerline{Yet know I how the heather looks,}
\centerline{And what a wave must be."}
\vskip.1cm
{\sl Emily Dickinson, ``Chartless."}
\vskip.5cm
\centerline{\bf Abstract.}
{\nn The fractal notion of self--similarity is
useful for characterizing change in scale; the reason fractals
are effective in the geometry of central place theory is
because that geometry is hierarchical in nature. Thus, a
natural place to look for other connections of this sort is
to other geographical concepts that are hierarchical in
nature. Within this fractal context, this chapter examines
the case of spatial diffusion.
When the idea of diffusion is extended to see ``adopters"
of an innovation as ``attractors" of new adopters, a Julia
set is introduced as a possible axis against which to measure
one class of geographic phenomena. Beyond the fractal
context, fractal concepts, such as ``compression" and ``space--
filling" are considered in a broader graph--theoretic context.}
\centerline{\bf Introduction.}
Because a fractal may be considered as a randomly
generated statistical image (Mandelbrot, 1983), one place
to look for geometric fractals tailored to fit geographic
concepts is within the set of ideas behind spatial
configurations traditionally characterized using randomness.
The spatial diffusion of an innovation is one such case;
H\"agerstrand characterized it using probabilistic simulation
techniques (H\"agerstrand, 1967). This chapter builds
directly on H\"agerstrand's work in order to demonstrate,
in some detail, how fractals might arise in spatial diffusion.
From there, and with a view of an adopter of an
innovation as an ``attractor" of other adopters, the
connected Julia set $z = z^2-1$ is examined, only broadly, for
its potential to serve as an axis from which to measure spatial
``attraction."
More generally, it is not necessary to consider fractal--
like concepts such as ``attraction," ``space--filling," or
``compression" relative to any metric, as in the diffusion
example, or relative to any axis, as in the Julia set
case. These broad fractal notions are examined, in some
detail, in a graph--theoretic realm, free from metric/axis
encumbrance, as one step beyond the fractal. An effort has
been made to explain key geographical and mathematical concepts
so that much of the material, and the flow of ideas, is self--
contained and accessible to readers from various disciplines.
\centerline{\bf A fractal connection to spatial diffusion}
The diffusion of the knowledge of an innovation
across geographic space may be simulated numerically using
Monte Carlo techniques based in probability theory
(H\"agerstrand, 1967). A simple example illustrates the basic
mechanics of H\"agerstrand's procedure.
Consider a geographic region and cover it with a grid
of uniform cell size suited to the scale of the available
empirical information about the innovation. Enter the number
of initial adopters of the innovation in the grid: an entry of
``$1$" means one person (household, or other set of
people) knows of the innovation. Over time, this person
will tell others. Assume that the spread of the news, from this
person to others, decays with distance. To simulate this
spread, probabilities of the likelihood of contact will be
assigned to each cell surrounding each initial adopter. A
table of random numbers is used in conjunction with the
probabilities, as follows.
Given a gridded geographic region and a distribution of
three initial adopters of an innovation (Figure 1). Assume
that an initial telling occurs no more than two cells away
from the initial adopters' cells. This assumption creates a
five--by--five grid in which interchange can occur between an
initial adopter in the central cell and others. Assign
probabilities of contact to each of these twenty--five cells
as a percentage likelihood that a randomly chosen four digit
number falls within a given interval of numbers assigned to each
cell (Figure 2). Because the intervals in Figure 2
partition the set of four digit numbers, the percentage
probabilities assigned to each cell add to 100\%. Pick up the
five--by--five grid and center it on the original adopter in
cell H3 (Figure 1). Choose the first number, 6248, in the list
of random numbers (Figure 2). It falls in the interval of
numbers in the central cell. Enter a ``$+1$" in the associated
cell, H3, to represent this new adopter. Move the five--by--
five grid across the distribution of original adopters, stopping
it and repeating this procedure with the next random number in
the list each time a new original adopter is encountered.
Center the five--by--five grid on H4; the next random number is
0925 which falls in the interval in the cell immediately
northwest of center (Figure 2). Enter a ``$+1$" in cell G3
(Figure 1), the cell immediately northwest of H4. Finally,
center the moving grid on H5. The next random number, 4997,
falls in the center cell; therefore, enter a ``$+1$" in cell H5.
Once this procedure has been applied to all original adopters,
the population of adopters doubles and a ``first
generation" of adopters, comprising original adopters and
newer adopters represented as ``$+1$'s", emerges (Figure 1).
Any number of additional generations of adopters of the
innovation may be simulated by iteration of this procedure. \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
$$ \matrix{ {\phantom{+0} \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {1 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {2 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {3 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {4 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {5 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {6 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {7 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {8 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}&\cr A&&&&&&&&\cr B&&&&&&&&\cr C&&&&&&&&\cr D&&&&&&&&\cr E&&&&&&&&\cr F&&&&&&&&\cr G&&&{{+1} \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}&&&&&\cr H&&&{{+1} \atop \phantom{+0}}{\phantom{0} \atop 1}& {\phantom{+0} \atop \phantom{+0}}{\phantom{0} \atop 1}& {{+1} \atop \phantom{+0}}{\phantom{0} \atop 1}&&&\cr I&&&&&&&&\cr J&&&&&&&&\cr K&&&&&&&&\cr } $$ \endinsert \topinsert
\noindent {\bf Figure 2}. \smallskip Five--by--five grid overlay. Numerical entries in cells show the percentage of four digit numbers associated with each cell. The given listing of cells shows which cell is associated with which range of four digit numbers. \smallskip North at the top.
TYPESETTING USING TeX OF THE SCANNED IMAGE, ABOVE, APPEARS BELOW
$$ \matrix{ {\phantom{0} \atop \phantom{0}}& {\phantom{0}1\phantom{.00} \atop \phantom{00.00}}& {\phantom{0}2 \atop \phantom{00.00}}& {\phantom{0}3 \atop \phantom{00.00}}& {\phantom{0}4 \atop \phantom{00.00}}& {\phantom{0}5 \atop \phantom{00.00}}&\cr 1&{\phantom{0}0.96 \atop \phantom{00.00}} &{\phantom{0}1.40 \atop \phantom{00.00}} &{\phantom{0}1.68 \atop \phantom{00.00}} &{\phantom{0}1.40 \atop \phantom{00.00}} &{\phantom{0}0.96 \atop \phantom{00.00}}\cr 2&{\phantom{0}1.40 \atop \phantom{00.00}} &{\phantom{0}3.01 \atop \phantom{00.00}} &{\phantom{0}5.47 \atop \phantom{00.00}} &{\phantom{0}3.01 \atop \phantom{00.00}} &{\phantom{0}1.40 \atop \phantom{00.00}}\cr 3&{\phantom{0}1.68 \atop \phantom{00.00}} &{\phantom{0}5.47 \atop \phantom{00.00}} &{44.31 \atop \phantom{00.00}} &{\phantom{0}5.47 \atop \phantom{00.00}} &{\phantom{0}1.68 \atop \phantom{00.00}}\cr 4&{\phantom{0}1.40 \atop \phantom{00.00}} &{\phantom{0}3.01 \atop \phantom{00.00}} &{\phantom{0}5.47 \atop \phantom{00.00}} &{\phantom{0}3.01 \atop \phantom{00.00}} &{\phantom{0}1.40 \atop \phantom{00.00}}\cr 5&{\phantom{0}0.96 \atop \phantom{00.00}} &{\phantom{0}1.40 \atop \phantom{00.00}} &{\phantom{0}1.68 \atop \phantom{00.00}} &{\phantom{0}1.40 \atop \phantom{00.00}} &{\phantom{0}0.97 \atop \phantom{00.00}}\cr } $$
\smallskip A random set of numbers (source: {\sl CRC Handbook of Standard Mathematical Tables\/}): \smallskip \line{6248, 0925, 4997, 9024, 7754 \hfil} \smallskip \line{7617, 2854, 2077, 9262, 2841 \hfil} \smallskip \line{9904, 9647, \hfil} \smallskip \noindent and so forth. \vskip.5cm Random number assignment to matrix cells, with cell number given as an ordered pair whose first entry refers to the reference number on the left of the matrix in this figure and whose second entry refers to the reference number at the top of that matrix. \vskip.2cm \line{(1,1): 0000-0095; (1,2): 0096-0235; (1,3): 0236-0403 \hfil} \line{(1,4): 0404-0543; (1,5): 0544-0639 \hfil} \smallskip \line{(2,1): 0640-0779; (2,2): 0780-1080; (2,3): 1081-1627 \hfil} \line{(2,4): 1628-1928; (2,5): 1929-2068 \hfil} \smallskip \line{(3,1): 2069-2236; (3,2): 2237-2783; (3,3): 2784-7214 \hfil} \line{(3,4): 7215-7761; (3,5): 7762-7929 \hfil} \smallskip \line{(4,1): 7930-8069; (4,2): 8070-8370; (4,3): 8371-8917 \hfil} \line{(4,4): 8918-9218; (4,5): 9219-9358 \hfil} \smallskip \line{(5,1): 9359-9454; (5,2): 9455-9594; (5,3): 9595-9762 \hfil} \line{(5,4): 9763-9902; (5,5): 9903-9999 \hfil} \endinsert There are numerous side issues, which are important, that may complicate this basic procedure (H\"agerstrand, 1967; Haggett {\it et al.\/}, 1977). How are the percentages for the five--by--five grid chosen? Indeed, how is the dimension of ``five" chosen for a side of this grid? Should the choices of percentages and of dimension be based on empirical data, on other abstract considerations, or on a mix of the two? What sorts of criteria should there be in judging suitability of empirical data? What if a random entry falls outside the given grid; what sorts of boundary/barrier considerations, both in terms of the position of new adopters relative to the regional boundary and of the symmetry of the probabilities within the five--by--five grid, should be taken into account? Independent of how many generations are calculated using this procedure, the pattern of ``filling in" of new adopters is heavily influenced by the shape of the set of original adopters. Indeed, over time, knowledge of the innovation diffuses slowly initially, picks up in speed of transmission, tapers off, and eventually the population becomes saturated with the knowledge. Typically this is characterized as a continuous phenomenon using a differential equation of inhibited growth that has as an initial supposition that the population may not exceed $M$, an upper bound, and that $P(t)$, the population $P$ at time $t$, grows at a rate proportional to the size of itself and proportional to the fraction left to grow (Haggett {\it et al.\/}, 1977; Boyce and DiPrima, 1977). An equation such as $$ {dP(t) \over dt} = k\, P(t)(1- (P(t)/M)) $$ serves as a mathematical model for this sort of growth in which $k >0$ is a growth constant and the fraction $(1-(P(t)/M)$ acts as a damper on the rate of growth (Boyce and DiPrima, 1977). The graph of the equation is an $S$--shaped (sigmoid) logistic curve with horizontal asymptote at $P(t)=M$ and inflection point at $P(t)=M/2$. When $dP/dt > 0$ the population shows growth; when $d^2 P/dt^2 > 0$ (below $P(t)=M/2$) the rate of growth is increasing; when $d^2P/dt^2<0$ (above $P(t)=M/2$) the rate of growth is decreasing. The differential equation model thus yields information concerning the rate of change of the total population and in the rate of change in growth of the total population. It does not show how to determine $M$; the choice of $M$ is given {\it a priori\/}. Iteration of the H\"agerstrand procedure gives a position for $M$ once the procedure has been run for all the generations desired. For, it is a relatively easy matter to accumulate the distributions of adopters and stack them next to each other, creating an empirical sigmoid logistic curve based on the simulation (Haggett {\it et al.\/}, 1977). Finding the position for the asymptote (or for {\it an\/} upper bound close to the asymptotic position) is then straightforward. Neither the H\"agerstrand procedure nor the inhibited growth model provides an estimate of saturation level (horizontal asymptote position) (Haggett, {\it et al.\/}, 1977) that can be calculated early in the measurement of the growth. The fractal approach suggested below offers a means for making such a calculation when self--similar hierarchical data are involved; allometry is a special case of this procedure (Mandelbrot, 1983). The reasons for wanting to make such a calculation might be to determine where to position adopter ``seeds" in order to produce various levels of innovation saturation. As is well--known, not all innovations diffuse in a uniform manner; Paris fashions readily available in major U. S. cities up and down each coast might seldom be seen in rural midwestern towns. To determine how the ideas of fractal ``space--filling" and this sort of diffusion--related ``space-- filling" might be aligned, consider the following example. Given a distribution of three original adopters occupying cells H3, H4, and H5 in a linear pattern (Figure 3.A). The probabilities for positions for new adopters are encoded within each cell surrounding each of these (as determined from the five--by--five grid of Figure 2). Thus, for example, when the grid of Figure 2 is superimposed and centered on the original adopter in cell H3, a probability of 3.01\% is assigned to the likelihood for contact from H3 to G4; when it is superimposed and centered on the original adopter in H4, there is a 5.47\% likelihood for contact from H4 to G4; and, when it is superimposed and centered on the original adopter in H5, there is a 3.01\% likelihood for contact from H5 to G4. Therefore, the percentage likelihood of a new first--generation adopter in cell G4, given this initial configuration of adopters, is the sum of the percentages divided by the number of initial adopters, or 11.49/3. For ease in inserting fractions into the grid, only the numerator, 11.49, is shown as the entry (Figure 3.A). It would be useful, for purposes of comparison of this distribution to those with sets of initial adopters of sizes other than 3, to divide by the number of initial adopters in order to derive a percentage that is independent of the size of the initial distribution ({\it i.e.\/}, to normalize the numerical entries). \topinsert \noindent{\bf Figure 3.A}. \smallskip The simulation is run on three original adopters with positions given below. Numerical entries show the likelihood, out of 300, that a new adopter will fall into a given cell. Zones of interaction between overlapping five--by--five grids are outlined by a heavy line (begin at the upper left--hand corner (ulhc) of cell F2; move horizontally to the upper right--hand corner (urhc) of cell F6; vertically to lower right--hand corner (lrhc) of cell J6; horizontally to lower left--hand corner (llhc) of cell J2; vertically to ulhc of F2 --- should be a rectangular enclosure that you have added to this figure). {\bf Original adopters are in cells H3, H4, H5.} \smallskip North at the top.LINES DESCRIBED ABOVE WERE ADDED TO THE SCANNED IMAGE IN COREL PHOTO-PAINT.
Figure 3A.
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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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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.
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&\phantom{0}28.58\cr J&{\phantom{0}0.96}
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&&&&&&&&&&\cr
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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
\qquad\quad&Number of cells\qquad&(out of 300) in&\cr
\smallskip
\+&Figure number: &Number of cells &Largest value \cr
\+&Position of three &in interaction &(out of 300) in \cr
\+&original adopters. &zone. &individual cell.\cr
\smallskip
\+&Figure 3.A: linear arrangement &25 &55.25\cr
\+&Figure 3.B: middle cell 1 unit north &23 &50.33\cr
\+&Figure 3.C: middle cell 2 units north &21 &47.39\cr
\+&Figure 3.D: middle cell 3 units north &19 &45.99\cr
\+&Figure 3.E: middle cell 4 units north &17 &45.99\cr
\+&Figure 3.F: middle cell 5 units north &15 &45.99\cr
\smallskip
\hrule
\endinsert
The example depicted in Figure 3 shows that even as early as
the first generation, the pattern of the positions of the initial
adopters affects significantly the configuration of the later
adopters. Figure 3.A with the heaviest possible filling of space
using three initial adopters represents a most saturated case,
which, taken together with an underlying symmetry that is
bilateral relative to mutually perpendicular axes, suggests that
an associated space--filling curve should have dimension 2, should
have a rectilinear appearance, and should be formed from a
generator whose shape is related to the pattern of placement of
the original adopters. One space--filling curve that meets these
requirements is the rectilinear curve of Figure 4.A. The
generator is composed of three nodes hooked together by two edges
in a straight path. This is scaled--down, by a factor of 1/2, and
hooked to the endpoints of the original generator. Iteration of
this procedure leads to a rectilinear tree with the desired
properties. The approach of looking for a geometric form to fit a
given set of conditions is like the calculus approach of looking
for a differential equation to fit a given set of conditions. The
difference here is that the shape of the generator and other
information from early stages may be used to estimate the relative
saturation or space--filling level.
The spatial position of the original adopters in Figure
3.B suggests a fractal generator in the shape of a ``V"
with an interbranch angle, ${\theta}$, of 90 degrees, while the
V in Figure 3.C suggests a generator with $\theta \approx
53^{\circ}$, that of Figure 3.D one with $\theta \approx
37^{\circ}$, that of Figure 3.E one with $\theta \approx
28^{\circ}$, and that of Figure 3.F one with $\theta \approx
23^{\circ}$. Figures 4.B, 4.C, 4.D, 4.E, and 4.F suggest
trees that can be generated using these values for $\theta$.
\topinsert \vskip 5in
\noindent {\bf Figure 4}.
\smallskip
THIS FIGURE DOES NOT TRANSMIT---AVAILABLE ONLY AS HARD COPY
(THUS, SCANNED IMAGE ONLY IS AVAILABLE HERE...NO CORRESPONDING TeX CODE.)
\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).
Figure 10.
\topinsert \vskip5.5in Figure to accompany construction of centrally symmetric hexagon.
\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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\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
(adjectival form).
\item{62.} This broadleaf ``big onion" is a key ingredient in
Vichyssoise.
\item{63.} Herb used in many pickled cucumbers.
\item{64.} ``Spiced--up" multiplication tables might be called
``---" tables.
\vskip.1cm
\noindent DOWN
\vskip.1cm
\item{1.} This herb supposedly has the power to destroy the scent
of garlic and onion.
\item{2.} East, in French.
\item{3.} Italian city -- home to Fibonacci.
\item{4.} Postal letter (abbreviation)
\item{5.} Orangish powder often association with Hungarian dishes.
\item{6.} East Prussia (abbreviation).
\item{7.} Almost everywhere (mathematical term -- abbreviation).
\item{8.} Railroad (abbreviation).
\item{9.} First initial and last name of former Panamanian leader.
\item{10.} A complimentary copy is a --- one.
\item{11.} Left hand opponent (duplicate bridge term, abbreviation).
\item{12.} Jumble of the word ``neared."
\item{17.} ``---s and bounds."
\item{19.} Spiritual guide in Hinduism.
\item{22.} Poland China is a variety of these.
\item{23.} This herb is often held in vinegar because its leaf veins
stiffen when dried and do not resoften when cooked. ``Estragon" in
French.
\item{24.} ``--- A Clear Day"
\item{25.} ``Though" -- some newspapers spell that word in this way.
\item{29.} This herb loses most of its flavor when dried: ``Pluches de
cerfeuil" refers to sprigs of this herb.
\item{31.} If/``---": typical manner in which a theorem is stated.
\item{33.} Removes from political office.
\item{34.} One variety of this herb, often used in conjunction with fat
fish and lentils, is known as Florence ---.
\item{35.} Tidy.
\item{36.} Paramedic vans are often marked with these three letters.
\item{38.} Uncontrolled anger. \item{42.} Company (abbreviation)
\item{45.} Running --- (Malay word). To be in a violently frenzied
state.
\item{48.} Fine German white wine made from grapes harvested after
frost: ---wein.
\item{49.} Oyster Research Institute of Michigan, might be abbreviated
thus.
\item{51.} Popular description of wok cookery: stir---.
\item{53.} Employ.
\item{56.} Identity element of the integers under multiplication.
\item{58.} Anno Domini (abbreviation)
\item{59.} National income (abbreviation)
\item{60.} Elevated train (abbreviation) -- forms ``Loop" in Chicago.
\item{61.} Prefix meaning ``muscle."
\vfill\eject
\noindent{\bf Coming attractions} ---
\vskip.5cm
\line{Feigenbaum's number \hfil}
\line{Pascal's theorem from projective geometry \hfil}
\line{Braikenridge--MacLaurin construction for a conic in the projective
plane. \hfil}
\vfill\eject
\noindent{\bf Crossword puzzle solution}
Crossword Puzzle Solution.
TYPESETTING, USING TeX, THAT PRODUCED THE CROSSWORD PUZZLE SOLUTION.
\vskip.5cm
\topinsert
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$$
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\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