\hsize = 6.5 true in %THE FIRST 14 LINES ARE TYPESETTING CODE. \input fontmac %delete to download, except on Univ. Mich. (MTS) equipment. \setpointsize{12}{9}{8} \parskip=3pt \baselineskip=14 pt \mathsurround=1pt \headline = {\ifnum\pageno=1 \hfil \else {\ifodd\pageno\righthead \else\lefthead\fi}\fi} \def\righthead{\sl\hfil SOLSTICE } \def\lefthead{\sl Summer, 1990 \hfil} \def\ref{\noindent\hang} \font\big = cmbx17 \font\tn = cmr10 \font\nn = cmr9 %The code has been kept simple to facilitate reading as e-mail \centerline{\big SOLSTICE:} \vskip.5cm \centerline{\bf AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS} \vskip5cm \centerline{\bf SUMMER, 1990} \vskip12cm \centerline{\bf Volume I, Number 1} \smallskip \centerline{\bf Institute of Mathematical Geography} \vskip.1cm \centerline{\bf Ann Arbor, Michigan} \vfill\eject \hrule \smallskip \centerline{\bf SOLSTICE} \line{Founding Editor--in--Chief: {\bf Sandra Lach Arlinghaus}. \hfil} \smallskip \centerline{\bf EDITORIAL BOARD} \smallskip \line{{\bf Geography} \hfil} \line{{\bf Michael Goodchild}, University of California, Santa Barbara. \hfil} \line{{\bf Daniel A. Griffith}, Syracuse University. \hfil} \line{{\bf Jonathan D. Mayer}, University of Washington; joint appointment in School of Medicine.\hfil} \line{{\bf John D. Nystuen}, University of Michigan (College of Architecture and Urban Planning).} \smallskip \line{{\bf Mathematics} \hfil} \line{{\bf William C. Arlinghaus}, Lawrence Technological University. \hfil} \line{{\bf Neal Brand}, University of North Texas. \hfil} \line{{\bf Kenneth H. Rosen}, A. T. \& T. Information Systems Laboratory. \hfil} \smallskip \line{{\bf Business} \hfil} \line{{\bf Robert F. Austin}, Director, Automated Mapping and Facilities Management, CDI. \hfil} \smallskip \hrule \smallskip The purpose of {\sl Solstice\/} is to promote interaction between geography and mathematics. Articles in which elements of one discipline are used to shed light on the other are particularly sought. Also welcome, are original contributions that are purely geographical or purely mathematical. These may be prefaced (by editor or author) with commentary suggesting directions that might lead toward the desired interaction. Individuals wishing to submit articles, either short or full-- length, as well as contributions for regular features, should send them, in triplicate, directly to the Editor--in--Chief. Contributed articles will be refereed by geographers and/or mathematicians. Invited articles will be screened by suitable members of the editorial board. IMaGe is open to having authors suggest, and furnish material for, new regular features. \vskip2in \noindent {\bf Send all correspondence to:} \vskip.1cm \centerline{\bf Institute of Mathematical Geography} \centerline{\bf 2790 Briarcliff} \centerline{\bf Ann Arbor, MI 48105-1429} \vskip.1cm \centerline{\bf (313) 761-1231} \centerline{\bf IMaGe@UMICHUM} \vfill\eject This document is produced using the typesetting program, {\TeX}, of Donald Knuth and the American Mathematical Society. Notation in the electronic file is in accordance with that of Knuth's {\sl The {\TeX}book}. The program is downloaded for hard copy for on The University of Michigan's Xerox 9700 laser-- printing Xerox machine, using IMaGe's commercial account with that University. Unless otherwise noted, all regular features are written by the Editor--in--Chief. \smallskip {\nn Upon final acceptance, authors will work with IMaGe to get manuscripts into a format well--suited to the requirements of {\sl Solstice\/}. Typically, this would mean that authors would submit a clean ASCII file of the manuscript, as well as hard copy, figures, and so forth (in camera--ready form). Depending on the nature of the document and on the changing technology used to produce {\sl Solstice\/}, there may be other requirements as well. Currently, the text is typeset using {\TeX}; in that way, mathematical formul{\ae} can be transmitted as ASCII files and downloaded faithfully and printed out. The reader inexperienced in the use of {\TeX} should note that this is not a ``what--you--see--is--what--you--get" display; however, we hope that such readers find {\TeX} easier to learn after exposure to {\sl Solstice\/}'s e-files written using {\TeX}!} {\nn Copyright will be taken out in the name of the Institute of Mathematical Geography, and authors are required to transfer copyright to IMaGe as a condition of publication. There are no page charges; authors will be given permission to make reprints from the electronic file, or to have IMaGe make a single master reprint for a nominal fee dependent on manuscript length. Hard copy of {\sl Solstice\/} will be sold (contact IMaGe for price--{\sl Solstice\/} will be priced to cover expenses of journal production); it is the desire of IMaGe to offer electronic copies to interested parties for free--as a kind of academic newsstand at which one might browse, prior to making purchasing decisions. Whether or not it will be feasible to continue distributing complimentary electronic files remains to be seen.} \vskip.5cm Copyright, August, 1990, Institute of Mathematical Geography. All rights reserved. \vskip1cm ISBN: 1-877751-51-0 \vfill\eject \centerline{\bf SUMMARY OF CONTENT} \smallskip Note: in this first issue, there is one of each type of article--this need not be the case in the future. \vskip.2cm 1. REPRINT. \smallskip William Kingdon Clifford, {\sl Postulates of the Science of Space\/}. This reprint of a portion of Clifford's lectures to the Royal Institution in the 1870's suggests many geographic topics of concern in the last half of the twentieth century. Look for connections to boundary issues, to scale problems, to self-- similarity and fractals, and to non--Euclidean geometries (from those based on denial of Euclid's parallel postulate to those based on a sort of mechanical ``polishing"). What else did, or might, this classic essay foreshadow? \smallskip 2. FULL--LENGTH ARTICLE. \smallskip Sandra L. Arlinghaus, {\sl Beyond the Fractal\/}. Figures are transmitted in this e-file only for the half of the article described in the first paragraph below. An original article. The fractal notion of self-- similarity is useful for characterizing change in scale; the reason fractals are effective in the geometry of central place theory is because that geometry is hierarchical in nature. Thus, a natural place to look for other connections of this sort is to other geographical concepts that are also hierarchical. Within this fractal context, this article examines the case of spatial diffusion. When the idea of diffusion is extended to see ``adopters" of an innovation as ``attractors" of new adopters, a Julia set is introduced as a possible axis against which to measure one class of geographic phenomena. Beyond the fractal context, fractal concepts, such as ``compression" and ``space--filling" are considered in a broader graph--theoretic context. \smallskip 3. SHORT ARTICLE. \smallskip William C. Arlinghaus, {\sl Groups, graphs, and God\/} An original article based on a talk given before a MIdwest GrapH TheorY (MIGHTY) meeting. The author, an algebraic graph theorist, ties his research interests to a broader philosophical realm, suggesting the breadth of range to which algebraic structure might be applied. The fact that almost all graphs are rigid (have trivial automorphism groups) is exploited to argue probabilistically for the existence of God. This is presented in the context that applications of mathematics need not be limited to scientific ones. \smallskip Note: In this first issue, there is one of each type of article--this need not be the case in the future. \smallskip 4. REGULAR FEATURES \smallskip \item{i.} {\bf Theorem Museum} --- Desargues's Two Triangle Theorem of projective geometry. \item{ii.} {\bf Construction Zone} --- a centrally symmetric hexagon is derived from an arbitrary convex hexagon. \item{iii.} {\bf Reference Corner} --- Point set theory and topology. \item{iv.} {\bf Games and other educational features} --- Crossword puzzle focused on spices. \item{v.} {\bf Coming attractions} --- Indication of topics for the ``REGULAR FEATURES" section in forthcoming issues. \smallskip \item{vi.}{\bf Solution to puzzle} \smallskip 5. SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE \smallskip This section shows the exact set of commands that work to download this file on The University of Michigan's Xerox 9700. Because different universities will have different installations of {\TeX}, this is only a rough guideline which {\sl might\/} be of use to the reader or to the reader's computing center. \vfill\eject \noindent 1. REPRINT \smallskip \centerline{\bf THE POSTULATES OF THE SCIENCE OF SPACE} \vskip.2cm \centerline{William Kingdon Clifford} \smallskip From a set of lectures given before the Royal Institution, 1873 -- ``The Philosophy of the Pure Sciences." Reprinted excerpt longer than this one appears in {\sl The World of Mathematics\/}, edited by James R. Newman, New York: Simon and Schuster, 1956. \smallskip \noindent In my first lecture I said that, out of the pictures which are all that we can really see, we imagine a world of solid things; and that this world is constructed so as to fulfil a certain code of rules, some called axioms, and some called definitions, and some called postulates, and some assumed in the course of demonstration, but all laid down in one form or another in Euclid's Elements of Geometry. It is this code of rules that we have to consider to--day. I do not, however, propose to take this book that I have mentioned, and to examine one after another the rules as Euclid has laid them down or unconsciously assumed them; notwithstanding that many things might be said in favour of such a course. This book has been for nearly twenty--two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state. The encouragement; for it contained a body of knowledge that was really known and could be relied on, and that moreover was growing in extent and application. For even at the time this book was written---shortly after the foundation of the Alexandrian Museum--Mathematic was no longer the merely ideal science of the Platonic school, but had started on her career of conquest over the whole world of Phenomena. The guide; for the aim of every scientific student of every subject was to bring his knowledge of that subject into a form as perfect as that which geometry had attained. Far up on the great mountain of Truth, which all the sciences hope to scale, the foremost of that sacred sisterhood was seen, beckoning to the rest to follow her. And hence she was called, in the dialect of the Pythagoreans, `the purifier of the reasonable soul.' Being thus in itself at once the inspiration and the aspiration of scientific thought, this Book of Euclid's has had a history as chequered as that of human progress itself. [Deleted text.] The geometer of to--day knows nothing about the nature of actually existing space at an infinite distance; he knows nothing about the properties of this present space in a past or a future eternity. He knows, indeed, that the laws assumed by Euclid are true with an accuracy that no direct experiment can approach, not only in this place where we are, but in places at a distance from us that no astronomer has conceived; but he knows this as of Here and Now; beyond his range is a There and Then of which he knows nothing at present, but may ultimately come to know more. So, you see, there is a real parallel between the work of Copernicus and his successors on the one hand, and the work of Lobatchewsky and his successors on the other. In both of these the knowledge of Immensity and Eternity is replaced by knowledge of Here and Now. And in virtue of these two revolutions the idea of the Universe, the Macrocosm, the All, as subject of human knowledge, and therefore of human interest, has fallen to pieces. It will now, I think, be clear to you why it will not do to take for our present consideration the postulates of geometry as Euclid has laid them down. While they were all certainly true, there might be substituted for them some other group of equivalent propositions; and the choice of the particular set of statements that should be used as the groundwork of the science was to a certain extent arbitrary, being only guided by convenience of exposition. But from the moment that the actual truth of these assumptions becomes doubtful, they fall of themselves into a necessary order and classification; for we then begin to see which of them may be true independently of the others. And for the purpose of criticizing the evidence for them, it is essential that this natural order should be taken; for I think you will see presently that any other order would bring hopeless confusion into the discussion. Space is divided into parts in many ways. If we consider any material thing, space is at once divided into the part where that thing is and the part where it is not. The water in this glass, for example, makes a distinction between the space where it is and the space where it is not. Now, in order to get from one of these to the other you must cross the {\it surface\/} of the water; this surface is the boundary of the space where the water is which separates it from the space where it is not. Every {\it thing\/}, considered as occupying a portion of space, has a surface which separates that space where it is from the space where it is not. But, again, a surface may be divided into parts in various ways. Part of the surface of this water is against the air, and part is against the glass. If you travel over the surface from one of these parts to the other, you have to cross the {\it line\/} which divides them; it is this circular edge where water, air, and glass meet. Every part of a surface is separated from the other parts by a line which bounds it. But now suppose, further, that this glass had been so constructed that the part towards you was blue and the part towards me was white, as it is now. Then this line, dividing two parts of the surface of the water, would itself be divided into two parts; there would be a part where it was against the blue glass, and a part where it was against the white glass. If you travel in thought along that line, so as to get from one of these two parts to the other, you have to cross a {\it point\/} which separates them, and is the boundary between them. Every part of a line is separated from the other parts by points which bound it. So we may say altogether --- \vskip.1cm The boundary of a solid ({\it i.e.\/}, of a part of space) is a surface. \vskip.1cm The boundary of a part of a surface is a line. \vskip.1cm The boundaries of a part of a line are points. And we are only settling the meanings in which words are to be used. But here we may make an observation which is true of all space that we are acquainted with: it is that the process ends here. There are no parts of a point which are separated from one another by the next link in the series. This is also indicated by the reverse process. For I shall now suppose this point --- the last thing that we got to --- to move round the tumbler so as to trace out the line, or edge, where air, water, and glass meet. In this way I get a series of points, one after another; a series of such a nature that, starting from any one of them, only two changes are possible that will keep it within the series: it must go forwards or it must go backwards, and each of these if perfectly definite. The line may then be regarded as an aggregate of points. Now let us imagine, further, a change to take place in this line, which is nearly a circle. Let us suppose it to contract towards the centre of the circle, until it becomes indefinitely small, and disappears. In so doing it will trace out the upper surface of the water, the part of the surface where it is in contact with the air. In this way we shall get a series of circles one after another --- a series of such a nature that, starting from any one of them, only two changes are possible that will keep it within the series: it must expand or it must contract. This series, therefore, of circles, is just similar to the series of points that make one circle; and just as the line is regarded as an aggregate of points, so we may regard this surface as an aggregate of lines. But this surface is also in another sense an aggregate of point, in being an aggregate of aggregates of points. But, starting from a point in the surface, more than two changes are possible that will keep it within the surface, for it may move in any direction. The surface, then, is an aggregate of points of a different kind from the line. We speak of the line as a point--aggregate of one dimension, because, starting from one point, there are only two possible directions of change; so that the line can be traced out in one motion. In the same way, a surface is a line--aggregate of one dimension, because it can be traced out by one motion of the line; but it is a point--aggregate of two dimensions, because, in order to build it up of points, we have first to aggregate points into a line, and then lines into a surface. It requires two motions of a point to trace it out. Lastly, let us suppose this upper surface of the water to move downwards, remaining always horizontal till it becomes the under surface. In so doing it will trace out the part of space occupied by the water. We shall thus get a series of surfaces one after another, precisely analogous to the series of points which make a line, and the series of lines which make a surface. The piece of solid space is an aggregate of surfaces, and an aggregate of the same kind as the line is of points; it is a surface--aggregate of one dimension. But at the same time it is a line--aggregate of two dimensions, and a point--aggregate of three dimensions. For if you consider a particular line which has gone to make this solid, a circle partly contracted and part of the way down, there are more than two opposite changes which it can undergo. For it can ascend or descend, or expand or contract, or do both together in any proportion. It has just as great a variety of changes as a point in a surface. And the piece of space is called a point--aggregate of three dimensions, because it takes three distinct motions to get it from a point. We must first aggregate points into a line, then lines into a surface, then surfaces into a solid. At this step it is clear, again, that the process must stop in all the space we know of. For it is not possible to move that piece of space in such a way as to change every point in it. When we moved our line or our surface, the new line or surface contained no point whatever that was in the old one; we started with one aggregate of points, and by moving it we got an entirely new aggregate, all the points of which were new. But this cannot be done with the solid; so that the process is at an end. We arrive, then, at the result that {\it space is of three dimensions\/}. Is this, then, one of the postulates of the science of space? No; it is not. The science of space, as we have it, deals with relations of distance existing in a certain space of three dimensions, but it does not at all require us to assume that no relations of distance are possible in aggregates of more than three dimensions. The fact that there are only three dimensions does regulate the number of books that we write, and the parts of the subject that we study: but it is not itself a postulate of the science. We investigate a certain space of three dimensions, on the hypothesis that it has certain elementary properties; and it is the assumptions of these elementary properties that are the real postulates of the science of space. To these I now proceed. The first of them is concerned with {\it points\/}, and with the relation of space to them. We spoke of a line as an aggregate of points. Now there are two kinds of aggregates, which are called respectively continuous and discrete. If you consider this line, the boundary of part of the surface of the water, you will find yourself believing that between any two points of it you can put more points of division, and between any two of these more again, and so on; and you do not believe there can be any end to the process. We may express that by saying you believe that between any two points of the line there is an infinite number of other points. But now here is an aggregate of marbles, which, regarded as an aggregate, has many characters of resemblance with the aggregate of points. It is a series of marbles, one after another; and if we take into account the relations of nextness or contiguity which they possess, then there are only two changes possible from one of them as we travel along the series: we must go to the next in front, or to the next behind. But yet it is not true that between any two of them here is an infinite number of other marbles; between these two, for example, there are only three. There, then, is a distinction at once between the two kinds of aggregates. But there is another, which was pointed out by Aristotle in his Physics and made the basis of a definition of continuity. I have here a row of two different kinds of marbles, some white and some black. This aggregate is divided into two parts, as we formerly supposed the line to be. In the case of the line the boundary between the two parts is a point which is the element of which the line is an aggregate. In this case before us, a marble is the element; but here we cannot say that the boundary between the two parts is a marble. The boundary of the white parts is a white marble, and the boundary of the black parts is a black marble; these two adjacent parts have different boundaries. Similarly, if instead of arranging my marbles in a series, I spread them out on a surface, I may have this aggregate divided into two portions --- a white portion and a black portion; but the boundary of the white portion is a row of white marbles, and the boundary of the black portion is a row of black marbles. And lastly, if I made a heap of white marbles, and put black marbles on the top of them, I should have a discrete aggregate of three dimensions divided into two parts: the boundary of the white part would be a layer of white marbles, and the boundary of the black part would be a layer of black marbles. In all these cases of discrete aggregates, when they are divided into two parts, the two adjacent parts have different boundaries. But if you come to consider an aggregate that you believe to be continuous, you will see that you think of two adjacent parts as having the {\it same\/} boundary. What is the boundary between water and air here? Is it water? No; for there would still have to be a boundary to divide that water from the air. For the same reason it cannot be air. I do not want you at present to think of the actual physical facts by the aid of any molecular theories; I want you only to think of what appears to be, in order to understand clearly a conception that we all have. Suppose the things actual in contact. If, however much we magnified them, they still appeared to be thoroughly homogeneous, the water filling up a certain space, the air an adjacent space; if this held good indefinitely through all degrees of conceivable magnifying, then we could not say that the surface of the water was a layer of water and the surface of air a layer of air; we should have to say that the same surface was the surface of both of them, and was itself neither one nor the other---that this surface occupied {\it no\/} space at all. Accordingly, Aristotle defined the continuous as that of which two adjacent parts have the same boundary; and the discontinuous or discrete as that of which two adjacent parts have direct boundaries. Now the first postulate of the science of space is that space in a continuous aggregate of points, and not a discrete aggregate. And this postulate---which I shall call the postulate of continuity---is really involved in those three of the six postulates of Euclid for which Robert Simson has retained the name of postulate. You will see, on a little reflection, that a discrete aggregate of points could not be so arranged that any two of them should be relatively situated to one another in exactly the same manner, so that any two points might be joined by a straight line which should always bear the same definite relation to them. And the same difficulty occurs in regard to the other two postulates. But perhaps the most conclusive way of showing that this postulate is really assumed by Euclid is to adduce the proposition he probes, that every finite straight line may be bisected. Now this could not be the case if it consisted of an odd number of separate points. As the first of the postulates of the science of space, the, we must reckon this postulate of Continuity; according to which two adjacent portions of space, or of a surface, or of a line, have the {\it same\/} boundary, {\it viz\/}.--- a surface, a line, or a point; and between every two points on a line there is an infinite number of intermediate points. The next postulate is that of Elementary Flatness. You know that if you get hold of a small piece of a very large circle, it seems to you nearly straight. So, if you were to take any curved line, and magnify it very much, confining your attention to a small piece of it, that piece would seem straighter to you than the curve did before it was magnified. At least, you can easily conceive a curve possessing this property, that the more you magnify it, the straighter it gets. Such a curve would possess the property of elementary flatness. In the same way, if you perceive a portion of the surface of a very large sphere, such as the earth, it appears to you to be flat. If, then, you take a sphere of say a foot diameter, and magnify it more and more, you will find that the more you magnify it the flatter it gets. And you may easily suppose that this process would go on indefinitely; that the curvature would become less and less the more the surface was magnified. Any curved surface which is such that the more you magnify it the flatter it gets, is said to possess the property of elementary flatness. But if every succeeding power of our imaginary microscope disclosed new wrinkles and inequalities without end, then we should say that the surface did not possess the property of elementary flatness. But how am I to explain how solid space can have this property of elementary flatness? Shall I leave it as a mere analogy, and say that it is the same kind of property as this of the curve and surface, only in three dimensions instead of one or two? I think I can get a little nearer to it than that; at all events I will try. If we start to go out from a point on a surface, there is a certain choice of directions in which we may go. These directions make certain angles with one another. We may suppose a certain direction to start with, and then gradually alter that by turning it round the point: we find thus a single series of directions in which we may start from the point. According to our first postulate, it is a continuous series of directions. Now when I speak of a direction from the point, I mean a direction of starting; I say nothing about the subsequent path. Two different paths may have the same direction at starting; in this case they will touch at the point; and there is an obvious difference between two paths which touch and two paths which meet and form an angle. Here, then, is an aggregate of directions, and they can be changed into one another. Moreover, the changes by which they pass into one another have magnitude, they constitute distance--relations; and the amount of change necessary to turn one of them into another is called the angle between them. It is involved in this postulate that we are considering, that angles can be compared in respect of magnitude. But this is not all. If we go on changing a direction of start, it will, after a certain amount of turning, come round into itself again, and be the same direction. On every surface which has the property of elementary flatness, the amount of turning necessary to take a direction all round into its first position is the same for all points of the surface. I will now show you a surface which at one point of it has not this property. I take this circle of paper from which a sector has been cut out, and bend it round so as to join the edges; in this way I form a surface which is called a {\it cone\/}. Now on all points of this surface but one, the law of elementary flatness holds good. At the vertex of the cone, however, notwithstanding that there is an aggregate of directions in which you may start, such that by continuously changing one of them you may get it round into its original position, yet the whole amount of change necessary to effect this is not the same at the vertex as it is at any other point of the surface. And this you can see at once when I unroll it; for only part of the directions in the plane have been included in the cone. At this point of the cone, then, it does not possess the property of elementary flatness; and no amount of magnifying would ever make a cone seem flat at its vertex. To apply this to solid space, we must notice that here also there is a choice of directions in which you may go out from any point; but it is a much greater choice than a surface gives you. Whereas in a surface the aggregate of directions is only of one dimension, in solid space it is of two dimensions. But here also there are distance--relations, and the aggregate of directions may be divided into parts which have quantity. For example, the directions which start from the vertex of this cone are divided into those which go inside the cone, and those which go outside the cone. The part of the aggregate which is inside the cone is called a solid angle. Now in those spaces of three dimensions which have the property of elementary flatness, the whole amount of solid angle round one point is equal to the whole amount round another point. Although the space need not be exactly similar to itself in all parts, yet the aggregate of directions round one point is exactly similar to the aggregate of directions round another point, if the space has the property of elementary flatness. How does Euclid assume this postulate of Elementary Flatness? In his fourth postulate he has expressed it so simply and clearly that you will wonder how anybody could make all this fuss. He says, `All right angles are equal.' Why could I not have adopted this at once, and saved a great deal of trouble? Because it assumes the knowledge of a surface possessing the property of elementary flatness in all its points. Unless such a surface is first made out to exist, and the definition of a right angle is restricted to lines drawn upon it---for there is no necessity for the word {\it straight\/} in that definition---the postulate in Euclid's form is obviously not true. I can make two lines cross at the vertex of a cone so that the four adjacent angles shall be equal, and yet not one of them equal to a right angle. I pass on to the third postulate of the science of space--- the postulate of Superposition. According to this postulate a body can be moved about in space without altering its size or shape. This seems obvious enough, but it is worth while to examine a little more closely into the meaning of it. We must define what we mean by size and by shape. When we say that a body can be moved about without altering its size, we mean that it can be so moved as to keep unaltered the length of all the lines in it. This postulate therefore involves that lines can be compared in respect of magnitude, or that they have a length independent of position; precisely as the former one involved the comparison of angular magnitudes. And when we say that a body can be moved about without altering its shape, we mean that it can be so moved as to keep unaltered all the angles in it. It is not necessary to make mention of the motion of a body, although that is the easiest way of expressing and of conceiving this postulate; but we may, if we like, express it entirely in terms which belong to space, and that we should do in this way. Suppose a figure to have been constructed in some portion of space; say that a triangle has been drawn whose sides are the shortest distances between its angular points. Then if in any other portion of space two points are taken whose shortest distance is equal to a side of the triangle, and at one of them an angle is made equal to one of the angles adjacent to that side, and a line of shortest distance drawn equal to the corresponding side of the original triangle, the distance from the extremity of this to the other of the two points will be equal to the third side of the original triangle, and the two will be equal in all respects; or generally, if a figure has been constructed anywhere, another figure, with all its lines and all its angles equal to the corresponding lines and angles of the first, can be constructed anywhere else. Now this is exactly what is meant by the principle of superposition employed by Euclid to prove the proposition that I have just mentioned. And we may state it again in this short form---All parts of space are exactly alike. But this postulate carries with it a most important consequence. In enables us to make a pair of most fundamental definitions---those of the plane and of the straight line. In order to explain how these come out of it when it is granted, and how they cannot be made when it is not granted, I must here say something more about the nature of the postulate itself, which might otherwise have been left until we come to criticize it. We have stated the postulate as referring to solid space. But a similar property may exist in surfaces. Here, for instance, is part of the surface of a sphere. If I draw any figure I like upon this, I can suppose it to be moved about in any way upon the sphere, without alteration of its size or shape. If a figure has been drawn on any part of the surface of a sphere, a figure equal to it in all respects may be drawn on any other part of the surface. Now I say that this property belongs to the surface itself, is a part of its own internal economy, and does not depend in any way upon its relation to space of three dimensions. For I can pull it about and bend it in all manner of ways, so as altogether to alter its relation to solid space; and yet, if I do not stretch it or tear it, I make no difference whatever in the length of any lines upon it, or in the size of any angles upon it. I do not in any way alter the figures drawn upon it, or the possibility of drawing figures upon it, {\it so far as their relations with the surface itself are concerned\/}. This property of the surface, then, could be ascertained by people who lived entirely in it, and were absolutely ignorant of a third dimension. As a point--aggregate of two dimensions, it has in itself properties determining the distance--relations of the points upon it, which are absolutely independent of the existence of any points which are not upon it. Now here is a surface which has not that property. You observe that it is not of the same shape all over, and that some parts of it are more curved than other parts. If you drew a figure upon this surface, and then tried to move it about, you would find that it was impossible to do so without altering the size and shape of the figure. Some parts of it would have to expand, some to contract, the lengths of the lines could not all be kept the same, the angles would not hit off together. And this property of the surface---that its parts are different from one another---is a property of the surface itself, a part of its internal economy, absolutely independent of any relations it may have with space outside of it. For, as with the other one, I can pull it about in all sorts of ways, and, so long as I do not stretch it or tear it, I make no alteration in the length of lines drawn upon it or in the size of the angles. Here, then, is an intrinsic difference between these two surfaces, as surfaces. They are both point--aggregates of two dimensions; but the points in them have certain relations of distance (distance measured always {\it on\/} the surface), and these relations of distance are not the same in one case as they are in the other. The supposed people living in the surface and having no idea of a third dimension might, without suspecting that third dimension at all, make a very accurate determination of the nature of their {\it locus in quo\/}. If the people who lived on the surface of the sphere were to measure the angles of a triangle, they would find them to exceed two right angles by a quantity proportional to the area of the triangle. This excess of the angles above two right angles, being divided by the area of the triangle, would be found to give exactly the same quotient at all parts of the sphere. That quotient is called the curvature of the surface; and we say that a sphere is a surface of uniform curvature. But if the people living on this irregular surface were to do the same thing, they would not find quite the same result. The sum of the angles would, indeed, differ from two right angles, but sometimes in excess, and sometimes in defect, according to the part of the surface where they were. And though for small triangles in any on neighbourhood the excess or defect would be nearly proportional to the area of the triangle, yet the quotient obtained by dividing this excess or defect by the area of the triangle would vary from one part of the surface to another. In other words, the curvature of this surface varies from point to point; it is sometimes positive, sometimes negative, sometimes nothing at all. But now comes the important difference. When I speak of a triangle, what do I suppose the sides of that triangle to be? If I take two points near enough together upon a surface, and stretch a string between them, that string will take up a certain definite position upon the surface, marking the line of shortest distance from one point to the other. Such a line is called a geodesic line. It is a line determined by the intrinsic properties of the surface, and not by its relations with external space. The line would still be the shortest line, however the surface were pulled about without stretching or tearing. A geodesic line may be {\it produced\/}, when a piece of it is given; for we may take one of the points, and, keeping the string stretched, make it go round in a sort of circle until the other end has turned through two right angles. The new position will then be a prolongation of the same geodesic line. In speaking of a triangle, then, I meant a triangle whose sides are geodesic lines. But in the case of a spherical surface---or, more generally, of a surface of constant curvature---these geodesic lines have another and most important property. They are {\it straight\/}, so far as the surface is concerned. On this surface a figure may be moved about without altering its size or shape. It is possible, therefore, to draw a line which shall be of the same shape all along and on both sides. That is to say, if you take a piece of the surface on one side of such a line, you may slide it all along the line and it will fit; and you may turn it round and apply it to the other side, and it will fit there also. This is Leibniz's definition of a straight line, and, you see, it has no meaning except in the case of a surface of constant curvature, a surface all parts of which are alike. Now let us consider the corresponding things in solid space. In this also we may have geodesic lines; namely, lines formed by stretching a string between two points. But we may also have geodesic surfaces; and they are produced in this manner. Suppose we have a point on a surface, and this surface possesses the property of elementary flatness. Then among all the directions of starting from the point, there are some which start {\it in the surface\/}, and do not make an angle with it. Let all these be prolonged into geodesics; then we may imagine one of these geodesics to travel round and coincide with all the others in turn. In so doing it will trace out a surface which is called a geodesic surface. Now in the particular case where a space of three dimensions has the property of superpositoin, or is all over alike, these geodesic surfaces are {\it planes\/}. That is to say, since the space is all over alike, these surfaces are also of the same shape all over and on both sides; which is Leibniz's definition of a plane. If you take a piece of space on one side of such a plane, partly bounded by the plane, you may slide it all over the plane, and it will fit; and you may turn it round and apply it to the other side, and it will fit there also. Now it is clear that this definition will have no meaning unless the third postulate be granted. So we may say that when the postulate of Superposition is true, then there are planes and straight lines; and they are defined as being of the same shape throughout and on both sides. It is found that the whole geometry of a space of three dimensions is known when we know the curvature of three geodesic surfaces at every point. The third postulate requires that the curvature of all geodesic surfaces should be everywhere equal to the same quantity. I pass to the fourth postulate, which I call the postulate of Similarity. According to this postulate, any figure may be magnified or diminished in any degree without altering its shape. If any figure has been constructed in one part of space, it may be reconstructed to any scale whatever in any other part of space, so that no one of the angles shall be altered through all the lengths of lines will of course be altered. This seems to be a sufficiently obvious induction from experience; for we have all frequently seen different sixes of the same shape; and it has the advantage of embodying the fifth and sixth of Euclid's postulates in a single principle, which bears a great resemblance in form to that of Superposition, and may be used in the same manner. It is easy to show that it involves the two postulates of Euclid: `Two straight lines cannot enclose a space,' and `Lines in one plane which never meet make equal angles with every other line.' This fourth postulate is equivalent to the assumption that the constant curvature of the geodesic surfaces is zero; or the third and fourth may be put together, and we shall then say that the three curvatures of space are all of them zero at every point. The supposition made by Lobatchewsky was, that the three first postulates were true, but not the fourth. Of the two Euclidean postulates included in this, he admitted one, {\it viz\/}., that two straight lines cannot enclose a space, or that two lines which once diverge go on diverging for ever. But he left out the postulate about parallels, which may be stated in this form. If through a point outside of a straight line there be drawn another, indefinitely produced both ways; and if we turn this second one round so as to make the point of intersection travel along the first line, then at the very instant that this point of intersection disappears at one end it will reappear at the other, and there is only one position in which the lines do not intersect. Lobatchewsky supposed, instead, that there was a finite angle through which the second line must be turned after the point of intersection had disappeared at one end, before it reappeared at the other. For all positions of the second line within this angle there is then no intersection. In the two limiting positions, when the lines have just done meeting at one end, and when they are just going to meet at the other, they are called parallel; so that two lines can be drawn through a fixed point parallel to a given straight line. The angle between these two depends in a certain way upon the distance of the point from the line. The sum of the angles of a triangle is less than two right angles by a quantity proportional to the area of the triangle. The whole of this geometry is worked out in the style of Euclid, and the most interesting conclusions are arrived at; particularly in the theory of solid space, in which a surface turns up which is not plane relatively to that space, but which, for purposes of drawing figures upon it, is identical with the Euclidean plane. It was Riemann, however, who first accomplished the task of analysing all the assumptions of geometry, and showing which of them were independent. This very disentangling and separation of them is sufficient to deprive them for the geometer of their exactness and necessity; for the process by which it is effected consists in showing the possibility of conceiving these suppositions one by one to be untrue; whereby it is clearly made out how much is supposed. But it may be worth while to state formally the case for and against them. When it is maintained that we know these postulates to be universally true, in virtue of certain deliverances of our consciousness, it is implied that these deliverances could not exist, except upon the supposition that the postulates are true. If it can be shown, then, from experience that our consciousness would tell us exactly the same things if the postulates are not true, the ground of their validity will be taken away. But this is a very easy thing to show. That same faculty which tells you that space is continuous tells you that this water is continuous, and that the motion perceived in a wheel of life is continuous. Now we happen to know that if we could magnify this water as much again as the best microscopes can magnify it, we should perceive its granular structure. And what happens in a wheel of life is discovered by stopping the machine. Even apart, then, from our knowledge of the way nerves act in carrying messages, it appears that we have no means of knowing anything more about an aggregate than that it is too fine--grained for us to perceive its discontinuity, if it has any. Nor can we, in general, receive a conception as positive knowledge which is itself founded merely upon inaction. For the conception of a continuous thing is of that which looks just the same however much you magnify it. We may conceive the magnifying to go on to a certain extent without change, and then, as it were, leave it going on, without taking the trouble to doubt about the changes that may ensue. In regard to the second postulate, we have merely to point to the example of polished surfaces. The smoothest surface that can be made is the one most completely covered with the minutest ruts and furrows. Yet geometrical constructions can be made with extreme accuracy upon such a surface, on the supposition that it is an exact plane. If, therefore, the sharp points, edges, and furrows of space are only small enough, there will be nothing to hinder our conviction of its elementary flatness. It has even been remarked by Riemann that we must not shrink from this supposition if it is found useful in explaining physical phenomena. The first two postulates may therefore be doubted on the side of the very small. We may put the third and fourth together, and doubt them on the side of the very great. For if the property of elementary flatness exist on the average, the deviations from it being, as we have supposed, too small to be perceived, then, whatever were the true nature of space, we should have exactly the conceptions of it which we now have, if only the regions we can get at were small in comparison with the areas of curvature. If we suppose the curvature to vary in an irregular manner, the effect of it might be very considerable in a triangle formed by the nearest fixed stars; but if we suppose it approximately uniform to the limit of telescopic reach, it will be restricted to very much narrower limits. I cannot perhaps do better than conclude by describing to you as well as I can what is the nature of things on the supposition that the curvature of all space is nearly uniform and positive. In this case the Universe, as known, becomes again a valid conception; for the extent of space is a finite number of cubic miles. And this comes about in a curious way. If you were to start in any direction whatever, and move in that direction in a perfect straight line according to the definition of Leibniz; after travelling a most prodigious distance, to which the parallactic unit---200,000 times the diameter of the earth's orbit---would be only a few steps, you would arrive at---this place. Only, if you had started upwards, you would appear from below. Now, one of two things would be true. Either, when you had got half--way on your journey, you came to a place that is opposite to this, and which you must have gone through, whatever direction you started in; or else all paths you could have taken diverge entirely from each other till they meet again at this place. In the former case, every two straight lines in a plane meet in two points, in the latter they meet only in one. Upon this supposition of a positive curvature, the whole of geometry is far more complete and interesting; the principle of duality, instead of half breaking down over metric relations, applies to all propositions without exception. In fact, I do no mind confessing that I personally have often found relief from the dreary infinities of homaloidal space in the consoling hope that, after all, this other may be the true state of things. \vfill\eject \noindent 2. FULL--LENGTH ARTICLE \vskip.5cm \centerline{\bf BEYOND THE FRACTAL} \vskip.1cm \centerline{\sl Sandra Lach Arlinghaus} \vskip.5cm \centerline{``I never saw a moor,} \centerline{I never saw the sea;} \centerline{Yet know I how the heather looks,} \centerline{And what a wave must be."} \vskip.1cm {\sl Emily Dickinson, ``Chartless."} \vskip.5cm \centerline{\bf Abstract.} {\nn The fractal notion of self--similarity is useful for characterizing change in scale; the reason fractals are effective in the geometry of central place theory is because that geometry is hierarchical in nature. Thus, a natural place to look for other connections of this sort is to other geographical concepts that are hierarchical in nature. Within this fractal context, this chapter examines the case of spatial diffusion. When the idea of diffusion is extended to see ``adopters" of an innovation as ``attractors" of new adopters, a Julia set is introduced as a possible axis against which to measure one class of geographic phenomena. Beyond the fractal context, fractal concepts, such as ``compression" and ``space-- filling" are considered in a broader graph--theoretic context.} \centerline{\bf Introduction.} Because a fractal may be considered as a randomly generated statistical image (Mandelbrot, 1983), one place to look for geometric fractals tailored to fit geographic concepts is within the set of ideas behind spatial configurations traditionally characterized using randomness. The spatial diffusion of an innovation is one such case; H\"agerstrand characterized it using probabilistic simulation techniques (H\"agerstrand, 1967). This chapter builds directly on H\"agerstrand's work in order to demonstrate, in some detail, how fractals might arise in spatial diffusion. From there, and with a view of an adopter of an innovation as an ``attractor" of other adopters, the connected Julia set $z = z^2-1$ is examined, only broadly, for its potential to serve as an axis from which to measure spatial ``attraction." More generally, it is not necessary to consider fractal-- like concepts such as ``attraction," ``space--filling," or ``compression" relative to any metric, as in the diffusion example, or relative to any axis, as in the Julia set case. These broad fractal notions are examined, in some detail, in a graph--theoretic realm, free from metric/axis encumbrance, as one step beyond the fractal. An effort has been made to explain key geographical and mathematical concepts so that much of the material, and the flow of ideas, is self-- contained and accessible to readers from various disciplines. \centerline{\bf A fractal connection to spatial diffusion} The diffusion of the knowledge of an innovation across geographic space may be simulated numerically using Monte Carlo techniques based in probability theory (H\"agerstrand, 1967). A simple example illustrates the basic mechanics of H\"agerstrand's procedure. Consider a geographic region and cover it with a grid of uniform cell size suited to the scale of the available empirical information about the innovation. Enter the number of initial adopters of the innovation in the grid: an entry of ``$1$" means one person (household, or other set of people) knows of the innovation. Over time, this person will tell others. Assume that the spread of the news, from this person to others, decays with distance. To simulate this spread, probabilities of the likelihood of contact will be assigned to each cell surrounding each initial adopter. A table of random numbers is used in conjunction with the probabilities, as follows. Given a gridded geographic region and a distribution of three initial adopters of an innovation (Figure 1). Assume that an initial telling occurs no more than two cells away from the initial adopters' cells. This assumption creates a five--by--five grid in which interchange can occur between an initial adopter in the central cell and others. Assign probabilities of contact to each of these twenty--five cells as a percentage likelihood that a randomly chosen four digit number falls within a given interval of numbers assigned to each cell (Figure 2). Because the intervals in Figure 2 partition the set of four digit numbers, the percentage probabilities assigned to each cell add to 100\%. Pick up the five--by--five grid and center it on the original adopter in cell H3 (Figure 1). Choose the first number, 6248, in the list of random numbers (Figure 2). It falls in the interval of numbers in the central cell. Enter a ``$+1$" in the associated cell, H3, to represent this new adopter. Move the five--by-- five grid across the distribution of original adopters, stopping it and repeating this procedure with the next random number in the list each time a new original adopter is encountered. Center the five--by--five grid on H4; the next random number is 0925 which falls in the interval in the cell immediately northwest of center (Figure 2). Enter a ``$+1$" in cell G3 (Figure 1), the cell immediately northwest of H4. Finally, center the moving grid on H5. The next random number, 4997, falls in the center cell; therefore, enter a ``$+1$" in cell H5. Once this procedure has been applied to all original adopters, the population of adopters doubles and a ``first generation" of adopters, comprising original adopters and newer adopters represented as ``$+1$'s", emerges (Figure 1). Any number of additional generations of adopters of the innovation may be simulated by iteration of this procedure. \topinsert\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.Figure 1.\noindent{\bf Figure 1}.

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.

TYPESETTING USING TeX OF THE FIGURE. $$ \matrix{ {\phantom{+0} \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {1 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {2 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {3 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {4 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {5 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {6 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {7 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {8 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {Totals}\cr A&&&&&&&&&\cr B&&&&&&&&&\cr C&&&&&&&&&\cr D&&&&&&&&&\cr E&&&&&&&&&\cr F&{\phantom{0}0.96} &{\phantom{0}2.36} &{\phantom{0}4.04} &{\phantom{0}4.48} &{\phantom{0}4.04} &{\phantom{0}2.36} &{\phantom{0}0.96} & &\phantom{0}19.20\cr G&{\phantom{0}1.40} &{\phantom{0}4.41} &{\phantom{0}9.88} &{11.49} &{\phantom{0}9.88} &{\phantom{0}4.41} &{\phantom{0}1.40} & &\phantom{0}42.87\cr H&{\phantom{0}1.68} &{\phantom{0}7.15} &{51.46} &{55.25} &{51.46} &{\phantom{0}7.15} &{\phantom{0}1.68} & &175.83\cr I&{\phantom{0}1.40} &{\phantom{0}4.41} &{\phantom{0}9.88} &{11.49} &{\phantom{0}9.88} &{\phantom{0}4.41} &{\phantom{0}1.40} & &\phantom{0}42.87\cr J&{\phantom{0}0.96} &{\phantom{0}2.36} &{\phantom{0}4.04} &{\phantom{0}4.48} &{\phantom{0}4.04} &{\phantom{0}2.36} &{\phantom{0}0.96} & &\phantom{0}19.20\cr K&&&&&&&&&\cr &&&&&&&&&&\cr &{\phantom{0}6.40} &{20.69} &{79.30} &{87.19} &{79.31} &{20.70} &{\phantom{0}6.41} & &{300\phantom{.00}}\cr } $$ \endinsert

It is easy to see that the values in the cells of Figure 3.A must add to a total of 300 if one views them as derived from each of three five--by--five grids centered on each original adopter. A ``zone of interaction" of entries from two or more five--by--five grids is outlined by a heavy line; 25 cells are enclosed in it in Figure 3.A. The pattern of numbers exhibits bilateral symmetry, insofar as is possible (allowing for the ``appendix" of .01 required to make the numerical partition associated with Figure 2 complete) with respect to both North--South and East--West axes (with the origin in cell H4). Sum and column totals are calculated; as the shape of the distribution of initial adopters is altered (below), these totals will tag sets of cells to demonstrate how changes in the zone of interaction are occurring. Next consider a distribution of three initial adopters derived from the linear one by moving the middle adopter one unit to the North (Figure 3.B). When interaction values are calculated as they were for the initial distribution in Figure 3.A, a comparable, but different numerical pattern emerges (Figure 3.B). Here, the column totals are the same as those in Figure 3.A, but the row totals are different. The zone of interaction contains 23 cells; the highest individual cell value of 50.33 is less than that of the highest cell value, 55.25, in Figure 3.A. Because both sets of values are partitions of the number 300, and because there are more cells with potential for contact in Figure 3.B than in Figure 3.A, the concentration of entries in Figure 3.B is not as compressed as in Figure 3.A. This is reflected in the row totals; a visual device useful for tracking this compression is to think of the five--by--five grid centered on 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 $$ \matrix{ {\phantom{+0} \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {1 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {2 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {3 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {4 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {5 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {6 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {7 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {8 \atop \phantom{+0}}{\phantom{0} \atop \phantom{0}}& {Totals}\cr A&&&&&&&&&\cr B&&&&&&&&&\cr C&&&&&&&&&\cr D&&&&&&&&&\cr E& &{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}1.68} &{\phantom{0}1.40} &{\phantom{0}0.96} & & &{\phantom{0}6.40}\cr F&{\phantom{0}0.96} &{\phantom{0}2.80} &{\phantom{0}5.65} &{\phantom{0}8.27} &{\phantom{0}5.65} &{\phantom{0}2.80} &{\phantom{0}0.96} & &\phantom{0}27.09\cr G&{\phantom{0}1.40} &{\phantom{0}4.69} &{12.34} &{50.33} &{12.34} &{\phantom{0}4.69} &{\phantom{0}1.40} & &\phantom{0}87.19\cr H&{\phantom{0}1.68} &{\phantom{0}6.87} &{49.00} &{16.41} &{49.00} &{\phantom{0}6.87} &{\phantom{0}1.68} & &131.51\cr I&{\phantom{0}1.40} &{\phantom{0}3.97} &{\phantom{0}8.27} &{\phantom{0}7.70} &{\phantom{0}8.27} &{\phantom{0}3.98} &{\phantom{0}1.40} & &\phantom{0}34.99\cr J&{\phantom{0}0.96} &{\phantom{0}1.40} &{\phantom{0}2.64} &{\phantom{0}2.80} &{\phantom{0}2.65} &{\phantom{0}1.40} &{\phantom{0}0.97} & &\phantom{0}12.82\cr K&&&&&&&&&\cr &&&&&&&&&&\cr &{\phantom{0}6.40} &{20.69} &{79.30} &{87.19} &{79.31} &{20.70} &{\phantom{0}6.41} & &{300\phantom{.00}}\cr } $$ \endinsert

Naturally, as the middle initial adopter is pulled successively one unit to the north in the configuration of original adopters, the middle five--by--five grid is also pulled one unit to the north (Figures 3.C, 3.D, 3.E, and 3.F). The numerical consequence is to reduce the size of the zone of interaction among the initial adopters and to spread the range of cells over which the value of 300 is partitioned. This implies less concentration near the original adopters and less ``filling in" around them as one proceeds from Figure 3.A to Figure 3.F. Thus, in Figure 3.C the zone of interaction shrinks to 21 cells with a largest individual cell entry of 47.39. At the stage shown in Figure 3.D, the largest cell entry is 45.99; because the cells associated with this value are not overlapped by the five--by-- five grid centered on the middle adopter, this largest value will not change as the middle adopter is pulled more to the north. Table 1 shows the sizes of the zones of interaction of the largest individual cell entry for each of Figures 3.A to 3.F. No new information arises from moving the middle cell to the north beyond the position in Figure 3.F; the five-- by--five grid is revealed and no longer intersects the two overlapping grids associated with the other two initial adopters. \topinsert \noindent{\bf Figure 3.C}. \smallskip The simulation is run on three original adopters with positions given below. Numerical entries show the likelihood, out of 300, that a new adopter will fall into a given cell. Zones of interaction between overlapping five--by--five grids are outlined by a heavy line (begin ulhc of F2; horizontally to urhc of F6; vertically to lrhc of H6; horizontally to lrhc of H5; vertically to lrhc of J5; horizontally to llhc of J3; vertically to llhc of H3; horizontally to llhc of H2; vertically to ulhc of F2 --- should be a ``less--fat" T--shaped enclosure that you have added to this figure). {\bf Original adopters are in cells H3, F4, H5.} \smallskip North at the top.

Figure 3C.

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

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

Figure 3D.

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

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

Figure 3E.

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

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

Figure 3F.

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

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

Table 1.

TYPESETTING, USING TeX, THAT PRODUCED TABLE 1. \settabs\+\indent \quad&Figure 3.B: middle cell 2 units north \qquad\quad&Number of cells\qquad&(out of 300) in&\cr \smallskip \+&Figure number: &Number of cells &Largest value \cr \+&Position of three &in interaction &(out of 300) in \cr \+&original adopters. &zone. &individual cell.\cr \smallskip \+&Figure 3.A: linear arrangement &25 &55.25\cr \+&Figure 3.B: middle cell 1 unit north &23 &50.33\cr \+&Figure 3.C: middle cell 2 units north &21 &47.39\cr \+&Figure 3.D: middle cell 3 units north &19 &45.99\cr \+&Figure 3.E: middle cell 4 units north &17 &45.99\cr \+&Figure 3.F: middle cell 5 units north &15 &45.99\cr \smallskip \hrule \endinsert The example depicted in Figure 3 shows that even as early as the first generation, the pattern of the positions of the initial adopters affects significantly the configuration of the later adopters. Figure 3.A with the heaviest possible filling of space using three initial adopters represents a most saturated case, which, taken together with an underlying symmetry that is bilateral relative to mutually perpendicular axes, suggests that an associated space--filling curve should have dimension 2, should have a rectilinear appearance, and should be formed from a generator whose shape is related to the pattern of placement of the original adopters. One space--filling curve that meets these requirements is the rectilinear curve of Figure 4.A. The generator is composed of three nodes hooked together by two edges in a straight path. This is scaled--down, by a factor of 1/2, and hooked to the endpoints of the original generator. Iteration of this procedure leads to a rectilinear tree with the desired properties. The approach of looking for a geometric form to fit a given set of conditions is like the calculus approach of looking for a differential equation to fit a given set of conditions. The difference here is that the shape of the generator and other information from early stages may be used to estimate the relative saturation or space--filling level. The spatial position of the original adopters in Figure 3.B suggests a fractal generator in the shape of a ``V" with an interbranch angle, ${\theta}$, of 90 degrees, while the V in Figure 3.C suggests a generator with $\theta \approx 53^{\circ}$, that of Figure 3.D one with $\theta \approx 37^{\circ}$, that of Figure 3.E one with $\theta \approx 28^{\circ}$, and that of Figure 3.F one with $\theta \approx 23^{\circ}$. Figures 4.B, 4.C, 4.D, 4.E, and 4.F suggest trees that can be generated using these values for $\theta$.

\topinsert \vskip 5in \noindent {\bf Figure 4}. \smallskip THIS FIGURE DOES NOT TRANSMIT---AVAILABLE ONLY AS HARD COPY

(THUS, SCANNED IMAGE ONLY IS AVAILABLE HERE...NO CORRESPONDING TeX CODE.)

\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.

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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 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