SOLSTICE:  AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS.
Volume III, Number 2.  Winter, 1992.
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\centerline{\big SOLSTICE:}
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\centerline{\bf  AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS}
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\centerline{\bf WINTER, 1992}
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\centerline{\bf Volume III, Number 2}
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\centerline{\bf Institute of Mathematical Geography}
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\centerline{\bf Ann Arbor, Michigan}
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\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. Bell Laboratories.
\hfil}
\smallskip
\line{{\bf Engineering Applications} \hfil}
\line{{\bf William D. Drake}, University of Michigan, \hfil}
\smallskip
\line{{\bf Robert F. Austin, Ph.D.} \hfil}
\line{President, Austin Communications Education Services \hfil}
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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.

The opinions expressed are those of the authors, alone, and the
authors alone are responsible for the accuracy of the facts in
the articles.
\smallskip
\noindent {\bf Send all correspondence to:
Institute of Mathematical Geography, 2790 Briarcliff,
Ann Arbor, MI 48105-1429, (313) 761-1231, IMaGe@UMICHUM,
Solstice@UMICHUM}
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Suggested form for citation.  If standard referencing to the hardcopy
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{\nn  Upon final acceptance,  authors will work with IMaGe
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{\nn  Copyright  will  be taken out in  the  name  of  the
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There are no page charges; authors will be given  permission  to
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length.   Hard  copy of {\sl Solstice\/} is  available at a cost
of \$15.95 per year (plus shipping and handling; hard copy is issued once yearly, in the Monograph series of the Institute of Mathematical Geography. Order directly from IMaGe. It is the desire of IMaGe to offer electronic copies to interested parties for free. Whether or not it will be feasible to continue distributing complimentary electronic files remains to be seen. Presently {\sl Solstice\/} is funded by IMaGe and by a generous donation of computer time from a member of the Editorial Board. Thank you for participating in this project focusing on environmentally-sensitive publishing.} \vskip.5cm \copyright Copyright, December, 1992 by the Institute of Mathematical Geography. All rights reserved. \vskip1cm {\bf ISBN: 1-877751-54-5} {\bf ISSN: 1059-5325} \vfill\eject \centerline{\bf SUMMARY OF CONTENT} \smallskip \noindent{\bf 1. A WORD OF WELCOME FROM A TO U} \smallskip \noindent{\bf 2. PRESS CLIPPINGS---SUMMARY} \smallskip \noindent{\bf 3. REPRINTS.} \smallskip\noindent A. {\bf What are mathematical models and what should they be ?} \smallskip\noindent {\bf by Frank Harary}. \smallskip \noindent Reprinted, with permission, from {\sl Biometrie-Praximetrie\/}; full citation at the front of the article. \smallskip\noindent 1. What Are They? 2. Two Worlds: Abstract and Empirical 3. Two Worlds: Two Levels 4. Two Levels: Derivation and Selection. 5. Research Schema. 6. Sketches of Discovery. 7. What Should They Be? \smallskip \smallskip\noindent B. {\bf Where are we? Comments on the concept of the center of population"} \smallskip\noindent {\bf by Frank E. Barmore} \smallskip\noindent Reprinted, with permission, from {\sl The Wisconsin Geographer\/}; full citation at the front of the article. \smallskip\noindent 1. Introduction 2. Preliminary Remarks 3. Census Bureau Center of Population Formul{\ae} 4. Census Bureau Center of Population Description. 5. Agreement between Description and Formul{\ae}. 6. Proposed Definition of the Center of Population. 7. Summary. 8. Appendix A. 9. Appendix B. 10. References. \smallskip \noindent{\bf 4. ARTICLE } \smallskip\noindent {\bf The pelt of the Earth: An essay on reactive diffusion} \smallskip\noindent {\bf by Sandra L. Arlinghaus and John D. Nystuen}. \smallskip\noindent 1. Pattern formation: Global views. 2. Pattern formation: Local views. 3. References cited. 4. Literature of apparent related interest. \smallskip \noindent{\bf 5. FEATURE} \smallskip\noindent {\bf Meet New Solstice Board Member, William D. Drake} \smallskip \noindent {\bf 6. DOWNLOADING OF SOLSTICE} \smallskip \noindent{\bf 7. INDEX to Volumes I (1990), II (1991), and III (1992, Number 1) of {\sl Solstice}.} \smallskip \noindent{\bf 8. OTHER PUBLICATIONS OF IMaGe } \vfill\eject \centerline{\bf 1. A WORD OF WELCOME--FROM A TO U!} Welcome to new subscribers from Alice Springs, Australia, to Ulm, Germany and to all those in between (alphabetically or otherwise)! We hope you enjoy participating in this means of journal distribution. Instructions for downloading the typesetting have been repeated in this issue, at the end. They are specific to the {\TeX} installation at The University of Michigan, but apparently they have been helpful in suggesting to others the sorts of commands that might be used on their own particular mainframe installation of {\TeX}. New subscribers might wish to note that the electronic files are typeset files---the mathematical notation will print out as typeset notation. For example, $$\Sigma_{i=1}^n$$ when properly downloaded, will print out a typeset summation as i goes from one to n symbol, as a centered display on the page. Complex notation is no barrier to this form of journal production. Many thanks to the members of the Editorial Board of {\sl Solstice\/}; with the publication of this issue we welcome the addition to that Board of William D. Drake, Ph.D. Engineering (Operations Research), and Professor in various departments of The University of Michigan. Bill has a brief note later in this issue of {\sl Solstice\/} in which he introduces himself, some of his recent interests, and some of his students' interests. \vfill\eject \noindent{\bf 2. PRESS CLIPPINGS---SUMMARY} \noindent Brief write-ups about {\sl Solstice\/} have appeared in the following publications: \noindent 1. {\bf Science}, Online Journals" Briefings. [by Joseph Palca] 29 November 1991. Vol. 254. \smallskip \smallskip \noindent 2. {\bf Science News}, Math for all seasons" by Ivars Peterson, January 25, 1992, Vol. 141, No. 4. \smallskip \smallskip \noindent 3. {\bf American Mathematical Monthly}, Telegraphic Reviews" --- mentioned as one of the World's first electronic journals using {\TeX}," September, 1992. \vfill\eject \centerline{\bf 3A. WHAT ARE MATHEMATICAL MODELS AND WHAT SHOULD THEY BE?} \smallskip \smallskip \centerline{\bf Frank Harary} \centerline{Distinguished Professor of Computer Science} \centerline{New Mexico State University} \smallskip \centerline{Professor Emeritus of Mathematics} \centerline{University of Michigan} \smallskip \centerline{Reprinted with permission from } \centerline{\sl Biometrie-Praximetrie } \centerline{Vol. XII, 1-4, 1971, pp. 3-18} \centerline{Published by Soci\'et\'e Adolphe Quetelet} \centerline{Belgian Region of Biometric Society} \centerline{Av. de la Facult\'e, 22} \centerline{B-5030 Gembloux, Belgium} \smallskip \centerline{At the time this paper was written,} \centerline{Frank Harary} \centerline{was also a member of the} \centerline{Research Center for Group Dynamics} \centerline{Institute for Social Research} \centerline{The University of Michigan} \smallskip \noindent No matter what the area of scientific research, whether social or physical, mathematical thinking is involved, explicitly or implicitly. At the least, the precise formulation of a problem entails some aspect of set theory and logic. Generally speaking, the working scientist uses the term mathematical model' for whatever branch of mathematics he may be applying to his present problem. On the other hand, the purist mathematician-logician insists strictly on the use of model' to mean a certain interpretation of an abstract axiom system in the real world. We begin with a self-contained development of the concepts needed for the discussion of research processes. This leads to the distinction between the real and abstract world, and the interaction between them by interpretation and abstraction. A similar, but conceptually different bifurcation is proposed for the two levels of research: digging into the foundations versus extending the horizons of knowledge. These considerations are assembled into a comprehensive Research Schema which enables a concise analysis of scientific discovery. Classical illustrations are provided, including true stories about Newton, Darwin, Freud, and Einstein. We conclude with some subjective evaluations of acceptability of mathematical models. \noindent{\bf 1. What Are They?} We have just noted that the word model' has different meanings for the mathematician and the scientist. When a mathematician uses the word, he is referring to the physical or social realization of his theory. On the other hand, when a scientist speaks of a mathematical model, he means the area of mathematics which applies to his work. Thus one (following Abraham Kaplan, oral communication) could say as a mnemonic aid that a model is always the other fellow's system. Contrariwise it also appears to be customary by usage to refer to research'' as what goes on in your own domain. In order to define a model rigorously, it is convenient to develop (as in Wilder [4] or in a more elementary presentation, Richardson [3]) several notions in the foundations of mathematics. Recall from high school geometry that Euclid's axioms are about as follows (depending on which book you read). The words point'' and line'' are undefined terms.$A_1$(Axiom 1) Every line is a collection of points.$A_2$There exist at least two points.$A_3$If$u$and$v$are points, then there exists one and only one line containing$u$and$v$.$A_4$If$L$is a line, then there exists a point not on$L$.$A_5$If$L$is a line, and$v$is a point not on$L$, then there exists one and only one line$L'$containing$v$which is parallel to$L$, i.e.,$L \cap L' = \emptyset $. Axiom 5 is the celebrated Parallel Postulate'' of Euclid. An {\sl axiom system\/}$\Sigma = (P, A)$consists of two sets: a set$P$of primitives and a set$A$of axioms. {\sl Primitives} are the deliberately undefined terms upon which all definitions in the system are based. {\sl Axioms\/} are statements which are assumed to be true, and from which other statements called {\sl theorems\/}, can be derived. Primitives and axioms serve to avoid so-called circular definitions and circular reasoning. Each axiom in the system is an assertion about the primitives. Euclid's axiom system consists of two primitives, point' and line', and five axioms. When Euclid developed geometry, he made a distinction between axioms and postulates. Both were statements whose truth was assumed, but axioms were considered self-evident while postulates were not! This distinction eventually proved unnecessary and even undesirable, and today axiom and postulate are synonyms. We shall denote by$T$or$T(\Sigma )$the set of all theorems derivable from an axiom system$\Sigma $. Then a {\sl mathematical system\/}$(P, A, T)$is an axiom system together with all theorems derivable from it. An {\sl independent axiom\/}$A$of$\Sigma $is one which cannot be derived from the remaining axioms. An {\sl axiom system\/} is {\sl independent\/} if every axiom is independent. In is called {\sl consistent\/} if there are no two contradictory statements in$T(\Sigma )$. One of the classical problems in 19th Century mathematics was to determine whether or not Euclid's Parallel Postulate,$A_5$, was independent. The consensus of opinion was that$A_5$was dependent, that is, it could be derived from$A_1$---$A_4$. Unsuccessful attempts to derive$A_5$led to the discovery instead of non-euclidean geometry. The two types of non-euclidean geometry are now respectively called {\sl hyperbolic geometry\/} (Bolyai-Lobachewski independently) in which there can be many parallels to a line through a point, and { \sl elliptic geometry\/} (Riemann) in which there can be no such parallel. An {\sl interpretation\/} of an axiom system is an assignment of meanings to its primitives which makes the axioms become true statements. The results of an interpretation of$\Sigma $is called a {\sl model\/} for$\Sigma $. This is the strict use of model' mentioned earlier. An axiom system is called {\sl satisfiable\/} if it has at least one model. Two models,$M_1$and$M_2$of$\Sigma $are {\sl isomorphic\/} if there is a 1-1 correspondence between the elements of$M_1$and those of$M_2$which preserves every$\Sigma$-statement. In a {\sl categorical\/} axiom system, any two models are isomorphic. To illustrate, consider an axiom system with primitives$P=\{S,
\circ \}$, where$S$is a set of integers, and$\circ $, is chosen as an undefined term for a binary operation denoted$a
\circ b$, in order to avoid preconceived notions that a familiar symbol like$a+b$would bring to mind. The following statements$A_1$---$A_4$are called {\sl group axioms\/}, and any set$S$on which they hold under the operation${\circ }$is called a {\sl group\/}.$A_1$(Closure Law)$S$is closed under${\circ }$, that is, if$a$and$b$are in$S$,$a {\circ } \, b$is in$S$.$A_2$(Associative Law) Operation$\circ $is associative, that is,$a \circ (b\circ c) = (a \circ b) \circ c $for all$a$,$b$, and$c$in$S$.$A_3$(Identity Law) There is a unique element$i$in$S$, called the identity element, such that$a \circ i = i \circ a = a$for all$a$in$S$.$A_4$(Inverse Law) For every$a$in$S$, there is a unique element, written$a^{-1}$and called the {\sl inverse\/} of$a$, such that$a \circ \, a^{-1}  =  a^{-1} \circ \, a  = i$. Each of the four group axioms is independent, and so this is an independent axiom system. To verify that this axiom system is satisfiable, we now display a model. One model for this system is the set$S_1 = \{1,-1\}$under multiplication$\times $. Thus this is called a group of {\sl order\/} 2, i.e., having just two elements. The identity element is 1, each element has itself as an inverse, and$S$is obviously closed and associative, as can be seen from the following multiplication table: \smallskip \line{\hfil \phantom{$-1$}$\times$$\vert\phantom{-}1\quad-1\hfil} \line{\hfil -----------------------\hfil} \line{\hfil \phantom{\times}\phantom{-}1$$\vert$\phantom{$-$}$1$\quad$-1$\hfil} \line{\hfil \phantom{$\times$}$-1$$\vert$$-1$\quad\phantom{$-$}$1$\hfil} Another model for this axiom system is the set$S_2=\{0,1\}$under addition modulo 2. We define the sum of$a$and$b$mod 2 to be the remainder of$a+b$after division by 2. Under this operation, we see at once from the next table that$S_2$is closed and associative, 0 is the identity, and each element is again its own inverse. Thus$S_2$is also a group of order 2. \smallskip \line{\hfil$+$mod 2\phantom{1}$\vert0$\quad$1$\hfil} \line{\hfil ----------------------- \hfil} \line{\hfil \phantom{$+$mod 2}$0$$\vert$$0$\quad$1$\hfil} \line{\hfil \phantom{$+$mod 2}$1$$\vert$$1$\quad$0$\hfil} More generally, one can take$S$to be the set$\{0, 1, 2,
\ldots ,  n-1\}$and$a \circ b$to mean$a+b\,\hbox{mod}\,n$. Then for each positive integer$n$, we get a distinct group of order$n$. Thus the above axiom system for groups is not categorical, since it has many non-isomorphic models. These two groups,$S_1$and$S_2$, are isomorphic since we can let operation$\times $correspond with$+$mod 2 and set$\{1, -1\}$to correspond with$\{0, 1\}$. All statements derivable from the axioms still hold. That the two models are isomorphic is also shown in the fact that their tables both have the following form: \smallskip \smallskip \line{\hfil$\circ$\phantom{a}$\verta$\quad$b$\hfil} \line{\hfil ----------------------- \hfil} \line{\hfil \phantom{$\circ$}$a\verta$\quad$b$\hfil} \line{\hfil \phantom{$\circ$}$b\vertb$\quad$a$\hfil} In fact, any pair of groups with two elements are isomorphic, so it is customary to speak of the group of order two." The study of group theory was originally motivated by properties which are possessed by the symmetries of a configuration, whether it be geometric, algebraic, architectural, physical, or chemical. It is readily verified that symmetries satisfy the four group axioms. For example, the inverse of a symmetry of a configuration is the corresponding symmetry mapping done in reverse. \noindent{\bf 2. Two Worlds: Abstract and Empirical} The realm of research activity is naturally divided into two worlds: the abstract and the empirical. The abstract world is generally regarded as the domain of the mathematician, logician, or purely theoretical physicist, while the empirical world is inhabited by experimental scientists of many varieties: physical, social, and others. [It has been established empirically that the less scientific a subject, the more likely it is that its practitioners call it a science. Outstanding examples include (in alphabetical order): divinity science, library science, military science, political science, and secretarial science.] There is a growing tendency, however, for people to live in both worlds in these interdisciplinary times. Those who work entirely in the abstract world are engaged in deriving new theorems either from axioms or from an existing theory or coherent body of theorems. Such results are usually expressed in symbols rather than numbers, and rarely touch upon the real world. On the other hand, the inhabitants of the empirical world work for a living." Some live in laboratories and perform experiments in order to collect meaningful data leading to a scientific theory. The two worlds are shown in Figure 1. The two loops, called theory building and experimentation, represent purely theoretical and purely experimental research. Figure 1 exhibits a symmetric pair of directed links between the worlds, the first of which can be called interpretation in accordance with the use of this word in the preceding section. In a confrontation between these two worlds, the mathematician's theorems become predictions about the real world, which can be tested by the scientist. If a prediction is verified by an appropriate experiment, the scientist feels that the theorem really works, and the mathematician has found a {\sl realization\/}. \smallskip \midinsert\vskip2in \noindent{\bf Figure 1 --- Two worlds}. [Two rectangles, representing the two worlds, are linked by a left arrow and a right arrow. The left rectangle is labelled Abstract"; the right rectangle is labelled Empirical." The right arrow, from the abstract world to the empirical world is labelled interpretation." The left arrow, from the empirical world to the abstract world is labelled abstraction. A loop, labelled theory building," is attached to the upper left of the abstract world. A loop, labelled experimentation," is attached to the upper right of the empirical world. Inserted by Ed.] \endinsert If the predictions are entirely incorrect, the model cannot be used. However, in cases where the predictions are not verified, yet are rather close" to correct, further abstraction is in order to construct a working model. This abstraction in the light of the experiment may suggest alternate hypotheses which should result in new theorems. These theorems hopefully will lead to better predictions than previously, and to a working model. \noindent{\bf 3. Two Worlds: Two Levels} Each of our two worlds may be divided into two levels. As we have indicated, the upper level of the abstract world deals with the development of mathematical systems by the derivation of theorems. We have discussed interaction between worlds at this level by means of interpretation and abstraction. In this section we shall observe that this same type of interaction can occur at the lower level. The lower level of the abstract world deals with the foundations of mathematics, axioms, and logic. The research activities might involve trying to prove consistency or independence of an axiom system. A rather esoteric and dramatic recent example of an important discovery at this level is given by the definitive work of Paul Cohen [1]. It is known (see Wilder [4], for example) that a$1-1$correspondence can be constructed between the natural numbers$1$,$2$,$\ldots$, and all the integers,$\ldots-3$,$-2$,$-1$,$0$,$1$,$2$,$\ldots$, and between the integers and the rational numbers. These three sets of numbers are all said to have the same (infinite) {\sl cardinality\/} which is conventionally denoted$\aleph_0$. It is also known that there are more real numbers than integers. The real line is sometimes called the {\sl continuum\/}, and so${\bf c}$is written for the number of reals. The {\sl continuum hypothesis\/} states that there is no infinite set with cardinality between$\aleph_0$and${\bf c}$. Cohen proved that the continuum hypothesis (as well as its negation) is consistent with the usual axioms of set theory. As a consequence, it is independent and can neither be proved nor disproved in that axiom system. Analogous to the development of non-euclidean geometry, two entirely different axiom systems have been created; one by assuming the continuum hypothesis, and the other by taking its negation. Cohen also proved the independence of the axiom of choice." On the other hand, the lower level of the empirical world also deals with foundations, but in the form of the basic laws of science. Kepler's Laws of Planetary Motion, Darwin's Law of Natural Selection, Newton's Laws of Motion, Kirchhoff's Laws of Electricity, and Einstein's Law of Special Relativity are all there. The link between the two worlds at this lower level is quite analogous to that at the upper level. Thus interpretation of an axiom leads to a basic law about the real world, while an abstraction, a coherent set of scientific laws becomes an axiom system. The schematic representation of interaction between the two worlds is shown in Figure 2. \smallskip \midinsert \vskip 3in \noindent{\bf Figure 2 --- Interaction between the two worlds}. [There are four rectangles in this figure, arranged at the upper left, upper right, lower left, and lower right. The two uppers have a left arrow and right arrow linking them, as do the two lowers. The upper left rectangle is labelled Theorems"; the upper right, Data"; the lower left, Axioms"; and, the lower right Laws." The right arrow in each case is labelled interpretation." The left arrow in each case is labelled abstraction." The left hand side of the figure is labelled Abstract"; the right, Empirical." There is a loop attached to each of the four rectangles. Ed.] \endinsert \noindent{\bf 4. Two Levels: Derivation and Selection} Having discussed interaction between the two worlds, we shall now establish links between their upper and lower levels. The process of climbing from the lower level to the upper in the abstract world can be regarded as {\sl derivation\/}. For we begin with an axiom system and then, sometimes painfully, derive progressively complicated theorems to obtain a mathematical system. Now consider how one goes from the upper level to the lower. From an existing body of theorems, an axiom system is to be built. To accomplish this, we select a body of particularly appropriate and fruitful theorems to use as axioms. This process of {\sl selection\/} yields a small, more manageable and often more powerful system, which is conducive to the derivation of new theorems. Selection in the empirical world involves collecting and studying vast amounts of data, and observing a pattern which may suggest a general law. Thus it is actually the {\sl induction\/} process. There appears to be no direct link in the empirical world from the lower level to the upper. Derivation does occur, and in fact uses the deduction process, but again and again we find that it takes the long way around," as shown in Figure 3. One begins with several scientific laws (lower right), and abstracts them to formulas (lower left) from which theorems can be derived (upper left) which make predictions about the real world (upper right). It is convenient, however, to draw the link representing derivation directly as well, as we do later. \smallskip \midinsert \vskip 2in \noindent{\bf Figure 3 --- Derivation in the empirical world} [The four rectangles of Figure 2, are linked with three arrows from Laws to Axioms to Theorems to Data. Ed.] \endinsert In general, innovative research is initiated in the upper level, and particularly in the upper right quadrant. This is due to the fact that the great majority of natural and fundamental questions arise from an attempt to observe or explain empirical phenomena. In fact, most research is done at the upper level, both right and left, while almost no one continuously remains at the lower level. For example, in ancient Egypt, the discovery of geometric formulas was necessitated by the search for improved techniques in measuring and surveying. Problems in geometry were solved long before Euclid organized the subject in an axiomatic formulation. \noindent{\bf 5. Research Schema} We contend that the above Research Schema represents all the types of interaction between the abstract and empirical worlds during the processes of research and discovery. Its two diagonal links are shortcuts which represent research processes that go directly to opposite" quadrants. There do not seem to be any directly ascending diagonal links. It is rarely but definitely possible to predict scientific laws from a body of theorems without actually working with experimental data. This is represented by the diagonal from upper left to lower right in the Research Schema. We shall see that Einstein took this route in his formulation of the theory of special relatively. \smallskip \midinsert \vskip 3in \noindent{\bf Figure 4. Research schema}. [Draw Figure 2. Label the loop on Theorems" as theory building"; that on Data" as experimentation"; that on Axioms" as axiomatic archaeology"; and, that on Laws" as empirical archaeology." Add up and down vertical arrows joining the rectangles; label the downward arrow in each case as selection"; the upward as derivation." Draw the two diagonals -- one with an arrow to suggest going from Theorems" to Laws" and the other from Data" to Axioms." Ed.] \endinsert The shortcut from experimental data to axioms, skipping the formulation of laws, occasionally occurs in the social sciences when a careful analysis of data patterns produces a set of formulas that can be taken as axioms. These are then interpreted, and hopefully suggest an empirical law, without the selection process. When considering routes between the two worlds, one must also allow for traversing loops at any quadrant one or more times. The upper right loop, for example, when traversed several times, indicates repeated efforts in observation and collection of data, before attempting to select corresponding laws. One must also note that the most direct route is not often taken in research. This will become evident in the next section when we take a closer look at particular cases of discovery. \noindent{\bf 6. Sketches of Discovery} We shall illustrate the Research Schema with the work of several men who represent varied branches of science and mathematics. We begin with Euclid, whose work in the axiomatization and derivation of what we now call euclidean geometry is represented schematically in Figure 5. [It has been said that the ultimate recognition of a man's contribution is conferred when his name is made an adjective and not capitalized.] \smallskip \midinsert \vskip2in \noindent{\bf Figure 5 --- Euclid's Research Schema}. [Draw three rectangles: upper left, upper right, lower left. Label them, respectively, Theorems of Geometry," Egyptian observations on measure," Axioms of Geometry." Add a loop to the two rectangles on the left. Join the upper left and lower left rectangles by an up arrow and a down arrow. Draw an arrow from the upper right to the upper left rectangle. Ed.] \endinsert {\sl Euclid\/}: Although Euclid is the acknowledged father of geometry, his main contribution was to its organization rather than to its derivation. The early Egyptians already knew the rudiments of geometry, including a form of the pythagorean theorem, and formulas for the area and volume of many geometric figures. Thus we attribute the upper right quadrant in Figure 5 to the Egyptians. The emphasis on proof, however, was introduced by the early Greeks and Euclid's contemporaries developed many of the theorems of geometry. Euclid selected the five axioms above from existing results. He then proved from these all the theorems of geometry then known and a few new ones, and presented a logical organization of the material in an exhaustive text. By today's standards, Euclid's axiomatic work is not rigorous, but it was an outstanding accomplishment for its time. \smallskip \midinsert\vskip2in \noindent{\bf Figure 6 --- Newton's Research Schema}. [Draw four rectangles: upper left -- Theorems of Calculus"; upper right -- left half labelled Verification" right half labelled Collection of Data"; lower left -- Abstraction of Laws of Motion; lower right -- Laws of Motion." Join the rectangles with arrows forming a rectangular cycle oriented in a clockwise direction. Add a loop to the lower left rectangle; label the loop Formalization of Calculus by Cauchy." Add a loop to the upper right rectangle; label the loop Galileo." Link the Galileo" loop to the down arrow as a dashed line separating Verification" from Collection of Data" in the upper right hand box. Ed.] \endinsert {\sl Newton\/}: Unlike Euclid, Newton occupied every quadrant of the Research Schema. His first work was on the upper level of the empirical world, where he experimented in chemistry and optics while still a student. Newton's most important results, however, were not derived from his own data, but from the work of those before him. His formulation of the Laws of Motion was induced from Galileo's extensive experimentation. Hence we credit the upper right loop in Newton's Research Schema to Galileo. Newton's Laws of Motion have been stated as follows: \item{1.}Every body will continue in its state of rest or uniform motion in a straight line unless it is compelled to change that state by impressed force. \item{2.} The rate of change of momentum is proportional to the impressed force and takes place in the line in which the force acts. \item{3.} For every action, there is an equal and opposite reaction. Newton left the empirical world and entered the abstract by expressing his laws symbolically as equations. His work with these resulted in the discovery of both differential and integral calculus. Others independently discovered these concepts, but it is believed that only Newton and Leibnitz (who discovered calculus independently) realized that differentiation and integration were inverse processes. Calculus did not become mathematically precise until the next century when Cauchy introduced the necessary concepts of limit and infinite sequence. We draw a loop in the lower left quadrant of Figure 6 to represent Cauchy's work in the foundations of calculus. This new branch of mathematics readily produced an abundant supply of theorems. The predictions which resulted were tested in the laboratory, and found to be entirely correct within the range of current measuring instruments. \smallskip \midinsert \vskip2in \noindent{\bf Figure 7 --- Einstein's Research Schema}. [Draw four rectangles. Label upper left: Theorems for Special Relativity"; upper right -- Michelson-Morley and others"; lower left -- Formulas"; and, lower right is split (by a dashed line) -- top half Laws of Light Motion," bottom half Special Relativity Theory." Arrows from upper left to upper right -- prediction"; from upper right to lower right -- selection"; from lower right to lower left; from lower left to upper left -- derivation." Ed.] \endinsert {\sl Einstein\/}: Eventually, more accurate measuring devices revealed that Newton's Laws of Motion could not explain the behavior of light on either the microscopic or astronomical level. Furthermore, the Michelson-Morley experiment proved conclusively that ether" did not exist. These discoveries led to a period of great activity in physics pioneered by Albert Einstein. Like Newton, Einstein's major work resulted from data collected by scientists before him. Einstein was a purely theoretical physicist, and never worked in the upper right quadrant of the Research Schema himself. But he certainly stimulated an enormous number of experiments there. He proposed the following empirical axiom system as laws of light motion: \item{1.} No physical object can travel faster than the speed of light. \item{2.} The speed of light depends not at all on the relative positions of the source of light and the observer, or their relative speeds. \item{3.} The mass at a velocity$v$of a particle equals its mass at velocity 0 divided by$\sqrt{1-v^2/c^2}$, where$c$is the speed of light. Einstein abstracted these three laws to an axiom system, from which he derived the body of theorems interpreted as the theory of special relativity. He found that in particular, his distance formulas for relativity theory were related to those of hyperbolic non-euclidean geometry; thus relativity theory provides a physical model for hyperbolic geometry. The Research Schema for this discovery is shown in Figure 7. We begin in the upper right with the Michelson-Morley experiment, and then go to the Laws of Motion of Light in the lower right, and their abstractions in the lower left. From there we go to the theorems of special relativity in the upper left, and finally to the experimental verification in the upper right where this cycle started. Einstein then went around this cycle again with his more refined theory of general relativity, which led to more precise predictions of physical measurements. \smallskip \midinsert\vskip2in \noindent{\bf Figure 8 --- Darwin's Research Schema}. [Draw two rectangles, one above and one below. The top one is labelled Data"; the bottom one is labelled Theory of Evolution." There are three loops attached to the top one. There is a line linking the two rectangles. Ed.] \endinsert {\sl Darwin\/}: Charles Darwin spent most of his life doing research in only one quadrant of the Research Schema, the upper right. His research career began when he became the official naturalist on the good ship Beagle, and embarked upon a five-year voyage. He made observations on all species of animals he could find, and took voluminous notes. During the remainder of his life, Darwin analyzed and classified these notes and all other available information. The climax of his work was the formulation of his Law of Natural Selection and his Theory of Evolution. Darwin's theory asserts that all animal species have descended from a common origin. The variety of species results from natural selection," in which those animals which are best adapted to their environment survive. Due to occasional mutations, certain animals in a species are better able to survive than others. These mutations may be passed on to their offspring who in turn will tend to survive and reproduce, eventually resulting in a new species which has been naturally selected. \smallskip \midinsert \vskip2in \noindent{\bf Figure 9 --- Freud's Research Schema}. [Draw two rectangles, one above and one below. Label the top one Medical Practice." Label the bottom one Psychoanalytic Theory." Join the two rectangles with an up arrow and a down arrow. There is a loop attached to the top rectangle. Ed.] \endinsert {\sl Freud\/}: Sigmund Freud, like Darwin, stayed in the empirical world. In fact, their Research Schemata are quite alike, as seen in Figures 8 and 9. He began with a medical degree and turned from general practice to specialization. Freud (in collaboration with J. Breuer initially) did research in the treatment of hysterical" patients who had physical symptoms for which no physical cause could be found. He inferred from the study of many cases that the symptoms could be traced back to some repressed childhood trauma, and went on to develop the concept of the subconscious together with the id, ego, and superego. First through hypnosis, and later through free association," Freud was able to induce himself and his patients to recall these forgotten experiences, and alleviate their symptoms. Much of the psychoanalytic theory which Freud developed is still highly controversial today, although it has made a lasting impact on the development of many modern theories in psychology. There has been a highly publicized report of the proof of a deep and important theorem by a mathematician while boarding a bus in Paris. It may be just as true as the anecdote about Newton's finding his law of gravitational attraction when an apple fell off its tree and landed on his head. This sort of phenomenon does occur, but fortunately is not an intrinsic part of the discovery procedure. In the words of Hans Zinsser, {\narrower\smallskip\noindent It is an erroneous impression, fostered by sensational popular biography, that scientific discovery is often made by inspiration$\ldots$. This is rarely the case. Even Archimedes' sudden inspiration in the bathtub; Descartes' geometrical discoveries in his bed; Darwin's flash of lucidity on reading a passage in Malthus; Kekule's vision of the closed carbon ring came to him on top of a London bus; and Einstein's brilliant solution of the Michelson puzzle in the patent office in Bern, were not messages out of the blue. They were the final co-ordinations, by minds of genius, of innumerable accumulated facts and impressions which lesser men could grasp only in their uncorrelated isolation, which --- by them --- were seen in entirety and integrated into general principles. The scientist takes off from the manifold observations of predecessors, and shows his intelligence, if any, by his ability to discriminate between the important and the negligible, by selecting here and there the significant steppingstones that will lead across the difficulties to new understanding. The one who places the last stone and steps across to the {\sl terra firma\/} of accomplished discovery gets all the credit. Only the initiated know and honor those whose patient integrity and devotion to exact observation have made the last step possible. \smallskip} When a researcher has become sufficiently steeped in his problem, he has amassed enough meaningful data (mathematicians also accumulate data via thought-experiments") to perceive the proper pattern and conceive the correct conjecture. This is a necessary but not sufficient step toward establishing a theorem. A proof, which is valid, must be supplied; otherwise, the assertion remains a conjecture. The two talents of conjecture and proof appear to be quite separable. \noindent{\bf 7. What Should They Be?} It is becoming more fashionable to use mathematical models as a powerful analytic device for advancing scientific research in a remarkable variety of disciplines. This usage is certainly not unwarranted, since models, when used with care and discretion, can and should be of great value in the clarification of existing problems and the formulation of important new ones. Unfortunately, it seems that models are misused all too often. The word model' is sometimes bandied about by people with little conception of its real meaning simply because it is {\sl au courant\/}. They don't even define model', but use the word to suit their own purposes. A model need not be impressively confusing in order to be valuable. In fact, one of the main contributions of a model lies in its ability to simplify a problem, and so it should be no more complicated than necessary. Neither should a model be symbol-rich but idea-poor. Models which hide miniscule content behind a mass of symbolic formulas tend to look impressive, but add nothing. Mystery is no criterion of knowledge." For example, a recent paper in a leading psychological journal had only one abstract idea: the number of elements in the union of two sets is the sum of the number of elements in each minus the number they have in common. Alas, the author apparently did not recognize it as the simplest special case of the Principle of Inclusion and Exclusion. Another unfortunate use of mathematical models occurred in a published paper in sociology in which there were ten axioms and zero theorems. However, an interpretation was then given which resulted in ten empirical theorems," one for each axiom. This 1-1 correspondence between axioms and empirical theorems simply involves the preparation of axioms which will yield desired empirical assertions. Furthermore, an axiom system should not be constructed for the artificial purpose of deriving just one theorem which has already been verified statistically. Clearly such a model only clutters the literature and does not involve genuine derivation. We do not wish to lay all the blame for the misuse of mathematical models on scholars in the empirical world; it occurs in the abstract world as well. The following passage by the eminent linguist Gustave Herdan [2] shows the dual roles the two worlds can play in the misuse of models. {\narrower\smallskip\noindent Without going into details, I will only mention a certain quantitative relation known to linguists as the Zipf law'. Mathematicians believe in it as a law, because they think that linguists have established it as a relation of linguistic facts, and linguists believe in it because they, on their part, think that mathematicians have established it to be a mathematical law. As can be shown in five minutes, it is not a law at all in the sense in which we speak of natural laws. \smallskip} Loosely stated, this law of Zipf proposes a high correlation between the frequency of use of words and their brevity. Another typical superficial use of mathematical models involves the bland assumption that the most elementary parts of an existing branch of mathematics apply unchanged to a problem in social science. Typical examples include high school algebra, coordinate geometry, matrix manipulation, graph theory, and the probabilistic theory of Markov chains. In such models, the typical procedure is to assign empirical terms to the mathematical variables by way of interpretation at the lower level. Then the existing theorems and methods of calculation are translated at the upper level into statements which are claimed to be new empirical findings. What, then, should mathematical models be? We have suggested that they should lead to new theorems, but this is not always necessary. The precise thinking involved in the careful formulation of an axiom system will lead to an improved conceptualization of the empirical phenomena at hand. This in turn can suggest the proper variables to measure, and perhaps an approach to the measurement problem. Sometimes an existing area of mathematics can be quite useful as a mathematical model provided it is augmented by one or more new axioms suggested by the real world. The most productive models, however, have involved derivation. For it is only after the derivation of new theorems that unexpected and far-reaching predictions can be made. From a mathematician's viewpoint, it is best if derivation leads to nontrivial theorems, which actually qualify for publication in the mathematical literature. To summarize, it is our personal and perhaps controversial contention that mathematical models will lead to significant and natural growth in both the abstract and empirical worlds. \smallskip \noindent{\sl Acknowledgment.} Research supported in part by Grant MH22743 from the National Institute of Mental Health. \vfill\eject \noindent{\bf References} \ref [1] P. J. Cohen, {\sl Set Theory and the Continuum Hypothesis\/}, Benjamin, New York, 1966. \ref [2] G. Herdan, {\sl Sonderdruck aus Zeichen und System der Sprache\/}, Vol. 2, Akademie-Verlag, Berlin, 1962, p. 108. \ref [3] M. Richardson, {\sl Fundamentals of Mathematics\/}, 3rd ed., Macmillan, New York, 1966. \ref [4] R. L. Wilder, {\sl Introduction to the Foundations of Mathematics\/}, 2nd ed., Wiley, New York, 1965. \vfill\eject \centerline{\bf 3B. WHERE ARE WE?} \centerline{\bf COMMENTS ON THE CONCEPT OF THE CENTER OF POPULATION''} \smallskip \centerline{\bf Frank E. Barmore} \smallskip \centerline{Associate Professor of Physics} \centerline{University of Wisconsin, La Crosse} \smallskip \centerline{Reprinted, with permission, from} \centerline{\sl The Wisconsin Geographer} \centerline{Vol. 7, pp. 40-50, 1991} \centerline{A publication of The Wisconsin Geographical Society} \smallskip \smallskip \noindent {\bf 1. Introduction} \noindent I was recently flabbergasted when I discovered how the Bureau of the Census calculates the location of the center of population of the United States following each decennial census. I found that: {\narrower\smallskip\noindent The center of population is the point at which an imaginary, flat, weightless, and rigid map of the United Sates would balance if weights of identical value were placed on it so that each weight represented the location of one person on April 1, 1980. Located at latitude 38 degrees, 8 minutes, 13 seconds north, and longitude 90 degrees, 34 minutes, 26 seconds west,$\ldots $. The computation of the center of population in 1980 was based on the 1980 population counts and the 1970 centers of population for counties. County population centers have not been determined for 1980. The center is the point whose latitude (LAT) and longitude (LONG) satisfy the equations $$\hbox{LAT}={{\Sigma w_i \times \hbox{lat}_i} \over {\Sigma w_i}},$$ $$\hbox{LONG}={{\Sigma w_i \times \hbox{long}_i \times \hbox{Cos} (\hbox{lat}_i)} \over {\Sigma w_i \times \hbox{Cos}(\hbox{lat}_i)}},$$ where$\hbox{lat}_i$,$\hbox{long}_i$,$w_i$are the latitude, longitude, and population, respectively, of the counties. (U. S. Bureau of the Census, 1983, Appendix A, p. A-5.) \smallskip} The statements were surprising for a number of reasons. First, as every good introductory physical geography text book points out, a flat map of the earth's curved surface is a {\sl distorted\/} map. Though distortions can be reduced by a careful choice of projection, appropriately centered, some distortion always remains and cannot be eliminated. Of course, one may define something any way one wishes, but at least it should be stated which method of projection is used to create this flat map." Second, the formul{\ae} given suggest that east-west distances are measured (reasonably, though arbitrarily) along parallels of latitude, which are small circles. But distances are usually measured along great circles and this would produce different results. Third, the formul{\ae} yield results that differ from results of other accepted definitions of the center of population. Fourth, the formul{\ae} can produce some rather peculiar results. These will be discussed below. The purpose of this paper is to discuss some of the difficulties in the concept of the center of population" when applied to populations spread over enough of the earth's surface that its curvature is noticeable. A more satisfactory definition of the center of population is proposed. \vfill\eject \noindent{\bf 2. Preliminary Remarks} When considering the characteristics of a large group of anything distributed over a region it is often useful to concentrate on only the most basic characteristics, the first three moments of the distribution: 1) the population, 2) the location, and 3) the spatial dispersion of the group. This paper concentrates on the second moment, the location. There are many ways location could be specified. Almost a century ago Hayford (1902) convincingly argued that the most appropriate measure of location of an area or population is a statistic called the average (arithmetic mean). Abler, {\it et al.\/} (1971) agree: When we ask {\sl where\/} questions about distributions we almost always desire an average location which represents the entire set." When averaging the location, each location is weighted'' according to the specific characteristic of interest and the result is the average location of the weighting character. The result is a center of mass'' if the weighting character is mass, a center of area'' (or geographic center) if the weighting character is area, a center of population'' if the weighting character is population, {\it etc\/}. Sviatlovsky and Eells (1937) have discussed in some detail the use and significance of the concept of the center'' in geographical regional analysis. Locations can be described as vectors whose magnitudes and directions are taken as the distances and directions of the items whose center is to be calculated. Then the power and convenience of vector algebra can be used to calculate the center in one, two, three or even higher dimensional spaces. If the space is flat" (Euclidean) then the process is quite straightforward, though tedious. Also, by using vectors in a Euclidean space to represent the distances and directions of individuals in a population, there exist several interesting and useful concepts. First, when the center of the coordinate system used is {\sl at\/} the center of population then the vector sum of the distances of all the people is zero and the sum of the squares of the distances of all the individuals is at a minimum. The minimum sum of the distances squared when measured from the average is not an accident, but rather, the result of a fundamental mathematical relation between the two quantities. Sufficiently fundamental is this relation that Warntz and Neft (1960), for example, define the mean as the place from which the sum of squares of the distances to each member of the population is minimum. Second, when the center of the coordinate system used is {\sl not at\/} the center of population then the vector sum of the distances of all the individuals provides the distance and direction of the center of population from the center of the coordinate system. I will use these characteristics later. However, the surface of the earth is not flat,'' but curved,'' and though finite, is without a boundary. On such a surface one can get into difficulty with the concept of average location. Where, for example, is the center of area'' (geographic center) of the earth's entire surface? If one chooses to preserve an earth surface provincialism it is not clear how one can modify the vector representation'' of distance and direction for the locations of individuals in a population. Some criteria are needed. I suggest that any reasonable definition of center of population'' should meet at least the following standards: (1) population distributions which are symmetric about some central point should have their center of population at this central point and (2) distances should be measured as true distances, either on the surface along great circles or in three dimensions along straight lines. It would also be desirable to have any definition satisfy additional restrictions: a) it should correspond to one's common understanding of center of population," b) it should reduce to the usual definition of center of population'' when there is no curvature and c) it should be easily extended to nonspherical surfaces --- for example, the International Ellipsoid or the geoid. In the examples and discussions which follow, all angles and great circle arc lengths are given in degrees and decimal degrees. Azimuths of places from any point are measured from North toward East. Latitudes and longitudes are designated as North or South and East or West respectively. I have ignored the differences between the shape of the earth and a sphere, as does the Bureau of the Census, when calculating centers of population (U. S. Bureau of the Census, 1973). When calculating the 1980 center of population of the United States in the various examples, I have used the original published 1980 populations (U. S. Bureau of the Census, 1983) and the unpublished 1980 centers of population for the fifty states and the District of Columbia (see Appendix A). When calculating the 1910 and 1880 centers of population of the United States I have used the published populations and centers of population of the various states and the District of Columbia (U.S. Bureau of the Census, 1913 and 1914). \noindent{\bf 3. Census Bureau Center of Population Formul{\ae}} Imagine a circumpolar population uniformly distributed along, say, the 70th parallel of latitude north (see Fig. 1). If longitude is measured from$180^{\circ}$W through$0^{\circ}$to$180^{\circ}$E then the Bureau of the Census formul{\ae} put the center of population at$70^{\circ}$N on the Greenwich meridian. Yet, surely the center of this population is at the North Pole. The Bureau of the Census formul{\ae} fail to meet the suggested standard. \topinsert \vskip 4in \noindent{\bf Figure 1.} Example population centers and distributions used in Part 3 of the text. Figure available in hard copy only; content should be clear from text and from caption. A sphere is drawn containing parallels and meridians. Three figures are highlighted on this sphere. I: a circumpolar population distributed along the 70th parallel of latitude north. II. a symmetric population distribution centered at latitude$38^{\circ}$N and longitude$90^{\circ}$W. This population is spaced at eight locations around the perimeter of a circle. III: a simple symmetric population distribution centered at$38^{\circ}$N and longitude$30^{\circ}$W. This population is spaced at either end of a line segment centered at III. The precise locations of places in distributions II and III are listed in Tables 1 and 2, respectively. \endinsert This failure is not due to the choice of a circumpolar population. Even at mid-latitudes the formul{\ae} fail to meet the suggested standard. Consider a second example, a collection of eight equally populated places located on a circle of 15 degrees of arc radius. Center the circle at$38^{\circ}$N and$90^{\circ}$W. Choose the position of the first place so that its azimuth from the center point is 15 degrees and each succeeding place has its azimuth 45 degrees greater than the preceding place (see Fig. 1). The latitude and longitude of each of the eight places can be calculated with spherical trigonometry. The results are shown in Table 1. When the Bureau of the Census formul{\ae} are used to calculate the center of population of the odd numbered places, then of the even numbered places and finally for all eight places the results differ. Specifically, for the odd numbered places one finds: $$\hbox{LAT} = 37.2351\,\hbox{N}, \qquad \hbox{LONG} = 90.0146\,\hbox{W}.$$ For the even numbered places one finds: $$\hbox{LAT} = 37.2155\,\hbox{N}, \qquad \hbox{LONG} = 89.9855\,\hbox{W}.$$ While for all eight places one finds: $$\hbox{LAT} = 37.2253\,\hbox{N}, \qquad \hbox{LONG} = 90.0000\,\hbox{W}.$$ Yet, surely the different symmetric distributions centered on the same point should have their center of population at the same place and surely that place (in this example) should be$38^{\circ}$N and$90^{\circ}$W ! Once more, the Bureau of the Census formul{\ae} do not meet the suggested standard. \smallskip \midinsert \hrule \smallskip \hrule \smallskip \centerline{\bf Table 1. Locations of places, Example II} \smallskip \hrule \smallskip \settabs\+\indent\qquad\qquad\qquad&Place\qquad&N. Latitude\qquad &W. Longitude\qquad&\cr \+&Place&N. Latitude&W. Longitude\cr \smallskip \hrule \smallskip \+&1&52.3434&\phantom{1}83.7050\cr \+&2&44.1596&\phantom{1}71.7938\cr \+&3&32.8128&\phantom{1}72.6948\cr \+&4&24.7119&\phantom{1}81.8100\cr \+&5&23.4333&\phantom{1}94.1868\cr \+&6&29.5187&104.9264\cr \+&7&40.3511&109.1501\cr \+&8&50.4718&101.7316\cr \smallskip \hrule \smallskip \endinsert For a third example, consider the simplest possible case: two equally populated places equidistant and in opposite directions from a central place. Specifically, place the center at$38^{\circ}$N and$30^{\circ}$W and the two places 15 degrees of arc from the center to the northeast (Az = 45) and to the southwest (Az = 225). (See Fig. 1). The latitude and longitude of the two places can be calculated and are shown in Table 2. When the center of population is calculated for this very simple population distribution of two places using the Bureau of the Census formul{\ae} the result is: $$\hbox{LAT} = 37.2057\,\hbox{N}, \qquad \hbox{LONG} = 29.9626\,\hbox{W}.$$ Yet, surely the true center is at the central point:$38^{\circ}$N,$30^{\circ}$W ! And yet again, the Bureau of the Census formul{\ae} do not meet the suggested standard. \smallskip \hrule \smallskip \hrule \smallskip \centerline{\bf Table 2. Locations of places, Example III} \smallskip \hrule \smallskip \settabs\+\indent\qquad\qquad\qquad&Place\qquad&N. Latitude\qquad &W. Longitude\qquad&\cr \+&Place&N. Latitude&W. Longitude\cr \smallskip \hrule \smallskip \+&\phantom{1}9&47.6377&14.2401\cr \+&10&26.7737&41.8289\cr \smallskip \hrule \smallskip What the Bureau of the Census formul{\ae} calculate is not the latitude and longitude of the center of population, but rather, two different and separate statistics: 1) the average latitude of the population and 2) the longitude of the average east-west distance of the population {\sl on a specific map projection\/}. This longitude is {\sl neither\/} the average longitude {\sl nor\/} the longitude of the center of population. The formul{\ae} calculate the location of a place that differs from other common measures of the center, such as the median or mean location (as defined by Warntz and Neft, 1960, and used by Haggett, {\it et al.\/}, 1977). The result of the Bureau of the Census formul{\ae} is the latitude and longitude of a place that cannot justifiably be named the center of population'' as the examples above clearly demonstrate. For comparison with later examples and further discussion, I have calculated the 1980 center of population of the United States with the Bureau of the Census formul{\ae}. The result is: $$\hbox{LAT} = 38.1376\,\hbox{N}, \qquad \hbox{LONG} = 90.5737\,\hbox{W}.$$ This differs from the location published by the Bureau of the Census (latitude of$38^{\circ}08'13''$or 38.1369 N and longitude of$90^{\circ}34'26''$or 90.5739 W) by one to three seconds of arc. This very small difference results from my using the populations and centers of the fifty states and the District of Columbia rather than the much larger full list of populations and centers of all the individual counties or enumeration districts used by the Census Bureau. As the discussion and all the examples that follow are based on the same set of data this small difference is unimportant --- the examples approximate and represent more extensive computations and their outcome well enough. The concerns in this paper are the methods used rather than the data on which they operate. \noindent{\bf 4. Census Bureau Center of Population Description} The Bureau of the Census description of the center of population does not give the map projection used. If the center of population (the balance point'' mentioned in the description) is calculated on a flat map constructed using various map projections the results vary. In order to demonstrate this I have calculated the 1980 centers of population (the balance point of the population distribution) for the U.S. using several different flat map projections. The projections used are all well known, having been developed in the 18th century, the 16th century and much earlier. For the projections chosen, descriptions given in numerous texts were sufficient for the derivation of the relevant formul{\ae} for laying out the projections. Alternately, one may refer to detailed monographs, such as the one by Snyder (1987), for the appropriate formul{\ae}. The results for each of the selected projections are listed in Table 3 and displayed in Fig. 2. \smallskip \hrule \smallskip \hrule \smallskip \centerline{\bf Table 3. The 1980 center of population of the United States} \centerline{\bf when using the Bureau of the Census prose definition} \centerline{\bf and various different map projections} \smallskip \hrule \smallskip \settabs\+&No.$\,\,$&Azimuthal Equal-Area (centered at 0N, 0W)$\,\,$&N. Latitude$\,\,$&W. Longitude\quad&\cr \+&&&\quad Center of Population&\cr \+&No.&Projection and Comments&N. Latitude&W. Longitude\cr \smallskip \hrule \smallskip \+&\phantom{1}1&Cylindrical Equal-Area&37.9818&90.4237\cr \+&\phantom{1}2&Equidistant Cylindrical (Plate Carr\'ee)&38.1376&90.4237\cr \+&\phantom{1}3&Sinusoidal (centered at 0 W)&38.1376&90.2532\cr \+&\phantom{1}4&Sinusoidal (centered at 60 W)&38.1376&90.4655\cr \+&\phantom{1}5&Sinusoidal (centered at 120 W)&38.1376&90.6778\cr \+&\phantom{1}6&Sinusoidal (centered at 180 W)&38.1376&90.8901\cr \+&\phantom{1}7&Equatorial Mercator&38.2945&90.4237\cr \+&\phantom{1}8&Transverse Mercator (centered at 90 W)&39.2344&90.6732\cr \+&\phantom{1}9&Azimuthal Equal-Area (centered at 0N,0W)&39.2583&89.0197\cr \+&10&Stereographic (centered at N. Pole)&39.7137&90.4888\cr \smallskip \hrule \smallskip \topinsert \vskip 5in \noindent{\bf Figure 2.} The Centers of population" for 1980 for the United States calculated according to various definitions. The center shown by an asterisk (*) and labeled COP was determined by the method proposed in this paper which takes the curvature of the earth's surface into account in an appropriate manner. The place shown as an open circle ($\circ$) and labelled BC is that published by the Bureau of the Census as the location of the center of population. As discussed in the text, this location should not be called the center of population. Places shown as solid circles and numbered are those which result when the center of population is calculated on various map projections using the Bureau of the Census prose definition of the center. The prose definition does not specify which projection should be used. The numbers refer to the list of projections given in Table 3. The Illinois-Missouri boundary, shown dashed, was taken from The National Atlas (U. S. Geological Survey, 1970). This figure is available in hard copy only; its content should be clear from Table 3 when one also notes that the Census calculated Center" lies in Missouri as do centers 1, 2, 4, 5, 6, and 7 (from Table 3); the COP Center lies in Illinois as do centers 8, 9, and 10 (from Table 3); Center 3 appears to lie on the border between Illinois and Missouri. \endinsert Note that the calculated centers depend not only on the projection chosen but also on the center and the orientation selected for the projection. The results differ as little as they do from one another because the projections chosen leave the United States in those portions of the resulting maps where the distortions are not extreme. Indeed, using the Bureau of the Census descriptive definition of the center of population, I believe, that, given sufficient time and mischievousness, one could choose projections of various orientations and center that would place the center of population {\sl any place one wished\/} ! \noindent{\bf 5. Agreement between Description and Formul{\ae}} As it happens, the description and the formul{\ae} currently given by the Bureau of the Census agree for one map projection --- a normal Sinusoidal (Sanson-Flamsteed) {\sl with the central meridian of the map the same as the meridian of the center of population\/}. Indeed, the Bureau of the Census formula for the longitude of the center of population can be derived by answering the following question: What must the central meridian for a normal Sinusoidal projection be in order for the center of population (that is, the balance point of the population distribution) to lie on the central meridian? But, why is the Sinusoidal projection (and associated formul{\ae}) preferred? If this projection holds special appeal, why isn't the latitude of the center of population determined in a similar way? One could determine the longitude {\sl and\/} latitude by answering the following question: What must the center for a Sinusoidal projection be in order for the center of population (the balance point of the population distribution) to lie at the center of the map projection? The result, for 1980, is: $$\hbox{LAT} = 39.1825\,\hbox{N}, \qquad \hbox{LONG} = 90.4934\,\hbox{W}.$$ The Bureau of the Census first calculated the center of population'' of the United States following the census of 1870 (Walker, 1874). Then, as now, the concept of a balance point was stated as underlying the computation of the center of population. The description of the calculation method (the formul{\ae} are not displayed) indicates the method was very similar to that currently used. East-west locations were taken as the distance from the 67th meridian west, measured along parallels of latitude. North-south locations were taken as the latitude above the 24th parallel north. Thus, the calculations of the center of population for the 1879 census are equivalent to calculating the balance point on a Sinusoidal projection with its central meridian at$67^{\circ}$W. Since 1870, east-west distances were measured from other meridians, chosen to be near the estimated center of population. In 1910 for example, the 86th meridian was chosen (U. S. Bureau of the Census, 1913) and thus the calculations of the center of population for the 1910 census are equivalent to calculating the balance point on a Sinusoidal projection with its central meridian at$86^{\circ}$W. \noindent{\bf 6. Proposed Definition of the Center of Population} One could avoid some of the problems discussed above by avoiding the use of the statistic," the average. There are other measures of the center," such as the median. But as Hayford (1902) pointed out long ago, there are fundamental difficulties with the concept of the median for two dimensional distributions. There is no unique point that divides" two dimensional distributions in half. Another possible measure is the point of minimum aggregate travel" --- the point for which the sum of all the distances to the various individuals would be minimum. But, as Eells (1930) demonstrates, this point has some peculiar characteristics. Also, as Court (1964) makes clear, the point of minimum aggregate travel is very difficult to find. There are now elegant, very powerful and very general techniques for solving such problems (Kirkpatrick, {\it et al.\/}, 1983). But, this is beside the point. The statistic that we want is one that reflects where people {\sl are\/}, not where they might congregate with the least total travel. The appropriate statistic is the mean. Whether calculating the mean location or the point of minimum aggregate travel, an arbitrary decision must be made: are the calculations to be done on the curved two dimensional surface of the globe (or appropriate flat map) or are they to be carried out in three dimensions? If the point of minimum aggregate travel were calculated in three dimensions the paths to be traveled and the resulting point would lie below the earth's surface. What value would there be in finding a point of least cumulative travel when the place to congregate and the paths to be traveled are inaccessible? I conclude that if one is interested in the point of minimum aggregate travel, it should be calculated {\sl on\/} the earth's surface. If the mean (average) location of a population is calculated in three dimensions, the resulting point is located below the surface. In the case of the United States in 1980, I find this mean location at: $$\hbox{LAT} = 39.1823\,\hbox{N},$$ $$\hbox{LONG}=90.3477\,W\,\, \hbox{and}\,\,\hbox{Depth}=0.0259 \times R$$ Where$R$is the radius of the sphere representing the earth. Taking$R=6371$km, the depth is about 165 km. But calculating the average location of simple surface distributions in three dimensions can yield some peculiar results. Consider three different equatorial population distributions, each consisting of four equally populated places with longitudes as follows: \smallskip \centerline{Example IV: places at 50.00 W, 15.00 W, 15.00 E, 50.00 E.} \centerline{Example \phantom{I}V: places at 50.00 W, 15.00 W, 15.00 E, 130.00 E.} \centerline{Example VI: places at 62.23 W, 60.00 W, 60.00 E, 62.23 E.} In all three examples the center of population (average location) ends up at the same place --- on the equator at the Greenwich meridian --- and differ only in their depth below the surface, if at all. (Examples V and VI have centers at the same depth.) But, we are largely confined to the surface of the earth and from this provincial point of view the center of population in example V should be far to the east of the centers in examples IV and VI. I find it unsatisfactory for populations of such different East-West distribution to have centers" which differ only in depth or not at all. I conclude that average locations should be calculated {\sl on\/} the earth's surface. To insist that calculation of the average location or point of minimum aggregate travel must be done in three dimensions is no more (or less) reasonable than to insist that the only proper map projection is on a globe. I leave to others the task of championing the computations of two dimensional population distribution statistics in three dimensions. I believe it is legitimate to consider the population distribution a two dimensional distribution and display its characteristics on the two dimensional surface of a globe or appropriate flat map. I suggest that the appropriate definition of the center of population is one similar to the descriptive definition given by the Bureau of the Census but with one addition. Specifically, the center of population is the point at which an imaginary, flat, weightless, and rigid map of the United States {\sl constructed by a specific method of projection\/} would balance if weights of identical value were placed on it so that each weight represented the location of one person$\ldots $. It only remains for one to choose the specific type of projection. It is distance and direction which are central to any calculation of the center of any population distribution. Therefore, I suggest that the only map projection (on which to find the balance point of the population distribution) is one where distances and directions of the individuals in the population are undistorted. If one chooses to measure the distances and directions from the center of an Azimuthal Equidistant map, or on the surface of a globe, they will be undistorted. Then one can create vectors whose magnitudes and directions are the true distances and directions of the various populated places. A vector sum can be done and the result is {\sl an estimate\/} of the distance and direction of the center of population. It is only an estimate because, although a map is flat, the earth's surface is not. However, it is a very good estimate, {\sl and the closer the map projection's center is to the center of population the better is the estimate\/}. Whether one chooses to carry out the computation on an Azimuthal Equidistant map or on the surface of a sphere is immaterial, since the process is algebraically identical and the results are numerically identical. Because the calculating of the center only produces an estimate, the procedure must be an iterative one, with the center of the projection in each iteration being the estimate of the center of population from the previous iteration. But the estimate is an excellent estimate, so the process converges rapidly. The iteration continues until the center of population is as close to the center of the map as one wishes. When calculating the U. S. center of population in this manner I find: \smallskip \centerline{Latitude of the 1980 U. S. center of population = 39.1980 N.} \centerline{Longitude of the 1980 U. S. center of population=90.4978 W.} This point lies about 125 km from the center given by the Bureau of the Census and is in Greene County, Illinois, about 14 km southwest by south of Carrollton, the county seat (see Figs. 2 and 3). The iterative calculation is not the computational nightmare that one might imagine (see Appendix B). Even when choosing the initial starting point at latitude zero and longitude zero or at latitude$20^{\circ}$S and longitude$20^{\circ}$E, the process rapidly converges to within 0.000001 degrees of the answer in four or five iterations. But one knows that the U.S. center of population is not in or near Africa --- there is no point in beginning the computations there. As the approximate center of population can be guessed, only two or three iterations are necessary to calculate the center of population to ample accuracy. I have also tested this procedure on the three example distributions used in Part 3 above and find that in all three cases the process rapidly converges on the expected central point. Therefore, I suggest that the proper definition of the center of population of the United States (or for any population distributed over a substantial portion of the earth's surface) is: {\narrower\smallskip\noindent The center of population is the point at which an imaginary, flat, weightless, and rigid map of the United States would balance if weights of identical value were placed on it so that each weight represented the location of one person on a specific date. The map in question is an azimuthal equidistant map whose center is at the center of population which must be calculated by successive approximation.\smallskip} This suggested definition of the center of population has the following advantages: (1) The center of populations symmetric about some central point is at that central point. (2) The true distance of each place is used in the computation. (Thus, this definition satisfies the two suggested standards given in Part 2 above.) (3) The suggested definition of the center of population also satisfies two of the three additional restrictions desired and stated in Part 2 above: (a) it corresponds to one's common understanding of center of population in that it does find the balance point of a distribution --- though one must be very specific about how the distribution is displayed, and (b) mathematically there is a correspondence to the usual definition of the center in the sense of the average --- the vector sum of the distances'' is zero when measured from the center and the sum of the squares of the distances'' is minimum when the distances are measured from the center. In addition, when the center of the map is not at the center of population the vector sum of the distances" points approximately to the center of population. Finally, our definition would reduce to the usual mathematical definition when there is no curvature. It is not clear that the definition suggested can be extended to non-spherical curved surfaces and thus satisfy the additional desired restriction (c) mentioned in Part II above. I believe it would work for the center of population of the United States on an ellipsoid of revolution but there could be difficulties for non-spherical surfaces in general --- the shortest distance between two points may not be uniquely defined and one may end up with several centers of population, all equally legitimate. A widely known use of the decennial centers of population determined by the Bureau of the Census is their display on a map of the United States depicting the historic westward shift of the population. In addition to this westward shift, these centers have slowly moved south since the turn of the century. By 1980, the center determined by the Bureau of the Census was more than a degree of latitude ({\it ca\/}. 110 km) south of where it was located in 1790. In contrast, the center of population calculated by the proposed method has followed a different path, diverging from the other path, and in 1980 was located at about the same latitude as the 1790 center. (The smaller the east-west dispersion of the population, the smaller will be the difference between the center of population calculated by the proposed method and the center the Bureau of the Census calculates. Thus, one would expect the locations calculated by either method to be about the same in 1790, before extensive westward national expansion occurred.) In order to show the increasing divergence of the two paths I have calculated the centers of population for 1910 and 1880 using the proposed method. The results are shown in Fig. 3 and labeled COP. Also shown (and labeled BC) are the locations determined and published by the Bureau of the Census for the same years. One can see that the average latitude of the population (which is what the Bureau of the Census calculates) has moved farther south than has the center of population. \midinsert \vskip 5in \noindent{\bf Figure 3.} Centers of population" for 1880, 1910 and 1980 for the United States calculated according to various definitions. Centers shown by asterisks (*) and labeled COP were determined by the method proposed in this paper which takes the curvature of the earth's surface into account in an appropriate manner. Places shown as open circles ($\circ $) and labeled BC are those published by the Bureau of the Census as the location of the center of population. As discussed in the text, these locations should not be called the centers of population. State boundaries, shown dashed, were taken from {\sl The National Atlas\/} (U. S. Geological Survey, 1970). This map is available in hard copy only; it does not transmit electronically. Its content should be clear from the combination of the text and this caption. \endinsert \noindent{\bf 7. Summary} For more than a century the Bureau of the Census has been calculating and displaying on maps a place designated as the center of population" of the United States. The method used in this computation is equivalent to calculating the average location of the population on a Sinusoidal map projection. As indicated in the previous discussion, such a method does not adequately take into account the curvature of the earth's surface. As a result, what the Bureau of the Census calculates should not be called the center of population." It is, rather, the location of a point that has the population's average latitude and the population's average distance (measured east-west along parallels of latitude) from an arbitrarily chosen meridian. A different method of calculating the center of population has been proposed in this paper. Like the Bureau of the Census method of calculation, the proposed method is based on the concept of the balance point of the population distribution and thus corresponds to one's common understanding of the center. In contrast to the Bureau of the Census method, the proposed method takes into account the curvature of the earth's surface and map projection distortions in an appropriate manner and is based on measuring distances along great circles. When calculated as proposed, the center of population's location differs substantially from that calculated by the Bureau of the Census. Not only is this true for 1980, but also for other census years, and the greater the east-west dispersion of the population, the greater will be the difference. \vfill\eject \noindent{\bf 8. Appendix A} The unpublished 1980 population centers of the fifty states and the District of Columbia used in the various examples were obtained from the Bureau of the Census. As they are unpublished, a complete list of the center latitudes and longitudes that were used in the computations discussed in this paper is supplied below. This is the data used as the representative example data set in Where are we? Comments on the concept of the center of population' " by Frank E. Barmore, published in {\sl The Wisconsin Geographer\/}, Vol. 7, pp. 40-50, (1991), a publication of the Wisconsin Geographical Society. The table below shows the original state populations and also, State Centers of Population supplied by the Bureau of the Census. The first coordinate for the center of population is measured in degrees of longitude west of the prime meridian; the second coordinate is measured in degrees of latitude north of the equator. \smallskip \hrule \smallskip \hrule \smallskip \settabs\+\indent\quad&North Carolina\qquad&Population\qquad &population\qquad&population\quad&\cr \+&Place &Population &Center of &Center of \cr \+& &1980 &population &population\cr \+& & &1980 &1980 \cr \smallskip \hrule \smallskip \+&Alabama &\phantom{2}3,893,888 &\phantom{1}86.7750 &32.9923\cr \+&Alaska &\phantom{23,}401,851 &148.4964 &61.3650\cr \+&Arizona &\phantom{2}2,718,215 &111.7186 &33.3245\cr \+&Arkansas &\phantom{2}2,286,435 &\phantom{1}92.4340 &34.9718\cr \+&California &23,667,902 &119.4380 &35.4746\cr \+&Colorado &\phantom{2}2,889,964 &105.1809 &39.4868\cr \+&Connecticut &\phantom{2}3,107,576 &\phantom{1}72.8760 &41.4906\cr \+&Delaware &\phantom{23,}594,338 &\phantom{1}75.5636 &39.4450\cr \+&D. C. &\phantom{23,}638,333 &\phantom{1}77.0088 &38.9074\cr \+&Florida &\phantom{2}9,746,324 &\phantom{1}81.6735 &27.7948\cr \+&Georgia &\phantom{2}5,463,105 &\phantom{1}83.8100 &33.1866\cr \+&Hawaii &\phantom{23,}964,691 &157.6129 &21.2009\cr \+&Idaho &\phantom{23,}943,935 &114.9358 &44.2072\cr \+&Illinois &11,426,518 &\phantom{1}88.4070 &41.2073\cr \+&Indiana &\phantom{2}5,490,224 &\phantom{1}86.2835 &40.1759\cr \+&Iowa &\phantom{2}2,913,808 &\phantom{1}93.0582 &41.9858\cr \+&Kansas &\phantom{2}2,363,679 &\phantom{1}96.6379 &38.4544\cr \+&Kentucky &\phantom{2}3,660,777 &\phantom{1}85.2228 &37.7918\cr \+&Louisiana &\phantom{2}4,205,900 &\phantom{1}91.4656 &30.7177\cr \+&Maine &\phantom{2}1,124,660 &\phantom{1}69.6408 &44.4125\cr \+&Maryland &\phantom{2}4,216,975 &\phantom{1}76.7904 &39.1598\cr \+&Massachusetts&\phantom{2}5,737,037 &\phantom{1}71.3844 &42.2792\cr \+&Michigan &\phantom{2}9,262,078 &\phantom{1}84.1083 &42.8410\cr \+&Minnesota &\phantom{2}4,075,970 &\phantom{1}93.6489 &45.2543\cr \+&Mississippi &\phantom{2}2,520,638 &\phantom{1}89.6224 &32.5778\cr \+&Missouri &\phantom{2}4,916,686 &\phantom{1}92.0799 &38.4815\cr \+&Montana &\phantom{23,}786,690 &110.9159 &46.8610\cr \+&Nebraska &\phantom{2}1,569,825 &\phantom{1}97.5697 &41.1991\cr \+&Nevada &\phantom{23,}800,493 &116.7563 &37.5535\cr \+&New Hampshire&\phantom{23,}920,610 &\phantom{1}71.4735 &43.1783\cr \+&New Jersey &\phantom{2}7,364,823 &\phantom{1}74.4172 &40.4640\cr \+&New Mexico &\phantom{2}1,302,894 &106.2391 &34.6202\cr \+&New York &17,558,072 &\phantom{1}74.7181 &41.5458\cr \+&North Carolina&\phantom{2}5,881,766 &\phantom{1}79.6756 &35.5676\cr \+&North Dakota &\phantom{23,}652,717 &\phantom{1}99.5101&47.4277\cr \+&Ohio &10,797,630 &\phantom{1}82.7006 &40.5199\cr \+&Oklahoma &\phantom{2}3,025,290 &\phantom{1}96.8876 &35.5880\cr \+&Oregon &\phantom{2}2,633,105 &122.5648 &44.6942\cr \+&Pennsylvania &11,863,895 &\phantom{1}77.2024 &40.4699\cr \+&Rhode Island &\phantom{23,}947,154 &\phantom{1}71.4419 &41.7595\cr \+&South Carolina&\phantom{2}3,121,820 &\phantom{1}81.0355 &34.0472\cr \+&South Dakota &\phantom{23,}690,768 &\phantom{1}99.0563 &44.1116\cr \+&Tennessee &\phantom{2}4,591,120 &\phantom{1}86.4217 &35.7793\cr \+&Texas &14,229,191 &\phantom{1}97.4571 &30.9925\cr \+&Utah &\phantom{2}1,461,037 &111.8261 &40.5165\cr \+&Vermont &\phantom{23,}511,456 &\phantom{1}72.8055 &44.0566\cr \+&Virginia &\phantom{2}5,346,818 &\phantom{1}78.0021 &37.6381\cr \+&Washington &\phantom{2}4,132,156 &121.5325 &47.3363\cr \+&West Virginia &\phantom{2}1,949,644 &\phantom{1}80.9407 &38.7202\cr \+&Wisconsin &\phantom{2}4,705,767 &\phantom{1}88.9756 &43.7192\cr \+&Wyoming &\phantom{23,}469,557 &106.9348 &42.6568\cr \smallskip \hrule \smallskip \hrule \vfill\eject \noindent{\bf 9. Appendix B} All the computations reported here were done on an Apple IIgs computer using the spread sheet in AppleWorks 3.0. Computation time for the problems varied. When calculating the center of population using the proposed method, each iteration took about 85 seconds. Computations using a more detailed list of populated places would take longer in direct proportion to the number of places used. Computers and software with enormously greater speed and capability are widely available. There is no computational reason for not using the proposed method. \vfill\eject \noindent{\bf 10. References} \ref Abler, Ronald, J. S. Adams and P. Gould. 1971. {\sl Spatial Organization: The Geographer's View of the World\/}. Englewood Cliffs NJ, Prentice-Hall. p. 59. \ref Court, A. 1964. The elusive point of minimum travel. {\sl Annals of the Association of American Geographers\/} 54: 400-403. \ref Eells, W.C. 1930. A mistaken conception of the center of population. {\sl Journal of the American Statistical Association\/}, New Series No. 169 25:33-40. \ref Haggett, Peter, A. D. Cliff and A. Frey. 1977. {\sl Locational Analysis in Human Geography: Vol. II, Locational Methods, 2nd Ed.\/} New York NY, Wiley, p. 312. \ref Hayford, John F. 1902. What is the center of an area, or the center of a population? {\sl Journal of the American Statistical Association\/}, New Series No. 58, 8:47-58. \ref Kirkpatrick, S., C. D. Gelatt, Jr. and M. P. Vecchi. 1983. Optimization by simulated annealing. {\sl Science\/} 220:671-680. \ref Snyder, J. P. 1987. {\sl Map Projections --- A Working Manual. (USGS Professional Paper 1395)\/}. Washington DC, U. S. Government Printing Office. pp. 41, 58, 77, 91, 157, 185, 195, 247. \ref Sviatlovsky, E. E. and W. C. Eells. 1937. The centrographical method and regional analysis. {\sl The Geographical Review\/}, 27:240-254. \ref U. S. Bureau of the Census. 1983. {\sl 1980 Census of Population, Vol. I\/}, Chapter A, Part 1 [PC80-1-A1]. Washington DC, U. S. Dept. of Commerce, Bureau of the Census. Appendix A, p. A-5 and Table 8, pp. 1-43. \ref U. S. Bureau of the Census. 1973. {\sl 1970 Census of Population and Housing: Procedural History\/} [PHC(R)-1]. Washington DC, U. S. Dept. of Commerce, Bureau of the Census. Appendix B (Computation of the 1970 U. S. center of population), pp. 3-50. \ref U. S. Bureau of the Census. 1914. {\sl Statistical Atlas of the United States\/}. Washington DC, Government Printing Office. pp. 29-32. \ref U. S. Bureau of the Census. 1913. {\sl Thirteenth Census of the United States taken in the year 1910. Vol. I: Population, 1910\/}. Washington DC, Government Printing Office. pp. 30, 46. \ref U. S. Geological Survey. 1970. {\sl The National Atlas of the United States of America\/}. Washington DC, U. S. Dept. of the Interior, Geological Survey. \ref Walker, F. A. 1874. The progress of the nation. In {\sl Statistical Atlas of the United States Based on the Results of the Ninth Census, 1870, with ... \/}. Francis A. Walker, compiler. New York NY, J. Bien, Lithographer. List of Memoirs and Discussions, Part II, pp. 5, 6. \ref Warntz, W. and D. Neft. 1960. Contributions to a Statistical Methodology for Areal Distributions. {\sl Journal of Regional Science\/}, 2:47-66. \smallskip \smallskip NOTE: The original article contains several printing errors which might cause misunderstanding. These were corrected in the reprints of the article. The {\sl Solstice\/} copy was prepared from the corrected reprint. The errors in question in the original include: i. the longitudes of places in Examples IV, V and VI were incorrect and ii. near the end of part 6 of the text a date of 1790 was incorrectly given as 1970. Also, the location, about 14 km southwest by south of Carrollton, was incorrectly given as about 7 km southeast of Carrollton. \vfill\eject \centerline{\bf 4. THE PELT OF THE EARTH:} \centerline{\bf AN ESSAY ON REACTIVE DIFFUSION} \smallskip \centerline{\bf Sandra L. Arlinghaus, John D. Nystuen} \smallskip \centerline{Founding Director, Institute of Mathematical Geography} \centerline{Ann Arbor, MI} \smallskip \centerline{Professor of Geography and Urban Planning} \centerline{The University of Michigan} \smallskip \smallskip \noindent Reactive diffusion (see the many references to Murray) is an idea that draws on the concept that boundary shape can influence the spatial pattern of the developing forms and processes interior to that boundary. A second idea is involved. Once a natural diffusive process has been at work, there is reaction to it, altering the shape of the underlying diffusion. Reactive diffusion is thus a dynamic process that is, to some extent, self-adjusting to change. This sort of idea is one that has met with many expressions in the past --- in the biological as well as in the geographical landscape (Arlinghaus, Nystuen, and Woldenberg, 1992). Boundary shape can determine how matter and energy travel within a closed system. Standing waves can be created in this manner, be they standing waves of translation of pigment on animal coats, producing striped animals; or standing waves of oscillation of water, producing seiches as water-stripes in reaction to lake depth and coastline shape of the containing vessel (e.g., Lake Michigan or Lake Geneva). One might even be tempted to speculate on a possible role for seiche-like stripes in the parting of the Red Sea." Xu, Vest, and Murray (1983) created mock animal outlines on laminar plates shaped like two-dimensional pelts --- as maps" of three-dimensional animals; when small adjustments in the outlines were made, vibrational patterns formed in a surface dust placed within these outlines created various spotted and striped patterns as a reaction to the boundary shape. Indeed, a circular drum head boundary offers one way for the roll of the drum wave of noise to interact with the boundary; using a fractal boundary for the drum head can produce a vastly different pattern of resonating pockets of drum roll (Science, 1991). The continuing work of Batty and Longley (1985 and later) in using fractal concepts to track the pattern of the urban fringe might also be (but has not yet been, to our knowledge) cast in the framework of reactive diffusion. Three-dimensional solids, covered with a coat of spots, might also have their spot patterns determined by some underlying vibrational process that causes the substance of the spots on the surface to react with the three-dimensional volume over which the surface is stretched. Thus, the calico cat and the earth might have a great deal in common when land masses, driven by tectonic rather than by biological rhythms, are seen as the calico spots on the pelt of the earth. For example, the burn pattern created by random lightning strikes in a forest, and the reaction of firefighters to these strikes, displays a clear case of reactive diffusion and pattern formation on the earth. For, in the absence of firefighters, the random strikes start fires which coalesce to form an advancing front that may ultimately burn the entire region. When firefighters enter the scene, they work to confine the random strikes; the fire may leap the barriers they create, and when it does, the firefighters talk about it and react by moving to control the new hotspot. Ultimately, the spread of communication among firefighters, in response to the leapfrogging character of forest fires, produces a forest spotted with burnt dark patches. The reaction of firefighters to the diffusion of information about the location of fires produces characteristic, and predictable, patterns on the earth. \smallskip \noindent{\bf 1. Pattern Formation: Global Views} Nystuen noted (1966) that spatial processes depend upon the shape of the partitions created by their boundary patterns. If the boundary shape is changed the process itself is changed. In fact, the very existence of the process may depend on the boundary shape." The biologist Joseph Birdsell noted that coastline shape has affected genetic diversity in the Australian aborigine population (1950); migration patterns forced away from the concave-up portions of the northern coastline were dispersed, while those forced away from the concave-down portions were focused. With dispersal of hunting and gathering came genetic complexity; with focusing came genetic inbreeding. Arlinghaus (1977; 1986) drew on ideas from Birdsell and Nystuen in using boundary shape of a limited access arterial to suggest where new pockets of population concentration and dispersion will appear relative to the concavity of the arterial. In all of these, as with reactive diffusion, there is an adjustment of process (geographical or biological) to boundary, with implications for the spatial organization of associated human activity coming as a reaction to that adjustment. Indeed, even in medi{\ae}val guilds, retail services clustered in pockets across the geographical landscape, as stripes" or spots" of commercial activity, in reaction to the diffusion of information as to type of service available (Vance, 1980). Similar urban patterns are evident in modern developing countries; and this context thus suggests, very generally, an interesting human dimension in exploring global urban change (Drake, 1993; Meadows {\it et al.\/} 1992). In a classical urban context, one might imagine Harris and Ullman's multiple nuclei" model recast within the replicable theoretical framework of reactive diffusion. As diffusion causes change surrounding and within the nuclei, there is a reaction, and the nuclei shift, or new nuclei spring up. The spatial evidence of reactive diffusion might be substituted for the historical evidence on which Harris and Ullman (1945) based their observational model, pulling the multiple nuclei model more in line with the earlier spatial models of Burgess and Hoyt. The multiple nuclei pattern appears as a reaction to incompatible land uses; it arises from an alternative resistance to residential and commercial land uses in which further employment centers leapfrog over existing urban neighborhoods, leading to extensive additional urban growth. Within the Detroit metropolitan region, for example, the complex changing nature of the local political scene coupled with the increasing crime rates associated with downtown Detroit, often encapsulated quite simply in the minds of many Detroiters by the closing of the downtown Detroit Hudson's store, led to the consequent reaction of many businesses to move to the suburbs. Thus, suburban Southfield became an early hub of urban reaction in the Detroit metropolitan region --- here, a new nucleus emerged. Efforts to restore the prominence of the downtown on the Detroit River are typified by the Renaissance Center --- here, the old nucleus shifts toward the River banks. Indeed, the characterization of the collapse of the central city in terms of the failure of the downtown headquarters of Hudson's department store may not be a strictly simple-minded view. Like the stars, the life of the city may take different paths --- at one time a center may be a viable unit, and at another time the relative size and density of the urban area may cause inner city collapse. In a central place context, in which the threshold" of a firm refers to the minimum number of sales which allows the firm to succeed and give an adequate return to its owners, the situation with Hudson's was simply a matter that the buying population at the center was too small to meet the threshold number. Related central place terminology involves the notions of the maximum range of a good and the minimum range of a good. The maximum range is the absolute limit on the demand of a good --- beyond this limit, transportation costs reduce demand for the good to zero. The minimum range is the distance over which the firm must ship its goods to include the threshold populations. A logical consequence, all else being equal, is that the minimum range of a good is less in a densely settled region than it is in a sparsely settled one. Thus, the common sense notion of how can a big store like Hudson's fail in downtown Detroit?" can be translated as follows. Migration of the affluent population to the suburbs reduced the number of potential customers in the center. The minimum range therefore needed to be extended outward from downtown in order to include the threshold number of customers. But, suburban Hudson stores were already in place and also worked to attract those customers that the downtown branch now required to succeed. The three large suburban stores competed with the downtown store for these customers, won them over, and the downtown store failed. Stability in competition (Hotelling, 1929) was restored when the empty" center was divided among the peripheral competitors, in a sort of central place (two-dimensional) Hotelling model. This sort of geometric view is a minimalist approach --- a best-case scenario; when additional social (and other) issues are superimposed, acceleration along the path to collapse is more likely. When one next considers that this pattern will repeat on the periphery of these suburban stores and within the maximum ranges of the various goods, a sort of leapfrogging of circular/hexagonal trade areas occurs and suggests, once again, a conceptual context of reactive diffusion as an alternative, and addition to traditional spatial analysis. Unlike earlier models of urban ecologists (Burgess, 1925; Hoyt, 1939), this sort of urban view of the world is not a generalization of a particular example --- that is why it is important to see reactive diffusion cast in the geographical as well as the biological (or other) realms. The pattern of clusters of urban activity on a regional part of the earth's surface is one that is produced in reaction to the diffusion of urban process. \smallskip \noindent{\bf 2. Pattern Formation: Local Views} Some current urban research strives to develop indices that offer an easy means for replication of experiments and that are sensitive to the role of boundary. Thus, Morrill (1991) proposed an index of segregation, modified by boundary considerations, to quantify urban spatial segregation. Wong (1992) modifies Morrill's indices by arguing that the length of the boundary separating adjacent urban areal units, as well as the shape of these adjacent units, is significant in determining segregation. Indices such as these, that already are sensitive to some boundary considerations, may offer one means to tighten the focus of application of the concept of reactive diffusion in various specific urban situations. Often reactions to incompatible urban land uses are circumscribed by the boundary of the system of local jurisprudence. When these reactions fit reasonably well within the laws, competing commercial and residential land uses are in relative harmony. Laws, such as the apocryphal it is illegal to tie an alligator to a parking meter" suggest a reaction to an unusual situation. When that reaction is passed as law, it diffuses to the population of the surrounding area and may disturb the sensitive balance between incompatible land uses. Perhaps the most difficult situation of this sort is in establishing rules (legal, ethical, or otherwise) to position locally unwanted land uses (lulus"). Human laws permit or forbid institutional boundaries that can influence how process works. Typically, a lulu, such as an adult bookstore or a toxic waste site, causes a strong local reaction around this hotspot." This reaction is confined and suppressed by municipal authorities using the local legal system as their hose" or barrier" to confine the effects of the unwanted activity. As with the forest fire example, the lulu leapfrogs, and yet another hotspot of locally unwanted activity occurs. Reactive diffusion offers an attractive conceptual context in which to examine pattern formation on the pelt of the earth: from local scenarios that mimic the forest fire example to global scenarios that examine entire closed and bounded surfaces. Beyond this essay, the next step is to use this context in specific urban or physical settings. \vfill\eject \noindent{\bf 3. References Cited} \ref Arlinghaus, S. L. 1986. Concavity and human settlement patterns," {\sl Essays on Mathematical Geography\/}, Monograph \#3. Ann Arbor, MI: Institute of Mathematical Geography; 1977. \ref Arlinghaus, S. L., Nystuen, J. D., and Woldenberg, M. J. 1992. An application of graphical analysis to semidesert soils." {\sl Geographical Review\/}, American Geographical Society. July, pp. 244-252. \ref Batty, M. and Longley, P. 1985. The fractal simulation of urban sructure." {\sl Papers in Planning Research\/} 92, Univ. of Wales Institute of Science and Technology, Colum Drive, Cardiff, CF1 3EU. \ref Birdsell, J. B. 1950. Some implications of the genetical concept of race in terms of spatial analysis," {\sl Symposia on Quantitative Biology\/}, Vol. 15, Origin and Evolution of Man. Long Island, New York: The Biological Laboratory, Cold Springs Harbor. \ref Drake, W. D. forthcoming, 1993. Towards building a theory of population - environment dynamics: a family of transitions. In {\sl Population - Environment Dynamics\/}, Ann Arbor: University of Michigan Press. \ref Hotelling, H. 1929. Stability in competition," {\sl Economic Journal\/}, 39: 41-57. \ref Hoyt, H. W. 1939. According to Hoyt (1966), Washington D. C.: Homer Hoyt Associates. \ref Harris, C. D. and Ullman, E. L. 1945. The nature of cities," {\sl Annals of the American Academy of Political and Social Science\/}, CCXLII, Nov. 1945, pp. 7-17. \ref Meadows, D. H., Meadows, D. L., and Randers, J. 1992. {\sl Beyond the Limits\/}. Post Mills, VT: Chelsea Green Publishing Company. \ref Morrill, R. L. 1991. On the measure of geographic segregation." {\sl Geography Research Forum\/} 11, 25-36. \ref Nystuen, J. D. 1966. Effects of boundary shape and the concept of local convexity." Discussion Paper 10. Michigan Inter-University Community of Mathematical Geographers (John D. Nystuen, ed.), Ann Arbor, MI. (Reprinted by the Institute of Mathematical Geography, 1986). \ref Park, R. E. and Burgess, E. W. 1925. {\sl The City\/}. Chicago: University of Chicago Press. \ref Vance, J. E. 1977. {\sl This Scene of Man: The Role and Structure of the City in the Geography of Western Civilization \/}. New York: Harper's College Press. \ref Wong, D. W. S. 1991. Spatial indices of segregation." Preliminary version, National Meetings, Regional Science Association, 1991. Forthcoming in {\sl Urban Geography\/}, 1992. \ref Xu, Youren; Vest, Charles M.; Murray, James D. 1983. Holographic interferometry used to demonstrate a theory of pattern formation in animal coats." {\sl Applied Optics\/} 15 Nov., Vol. 22, No. 22, pp. 3479-3483. \vfill\eject \noindent{\bf 4. Literature of Apparent Related Interest} \ref Bard, Jonathan B. L. 1977. A unity underlying the different zebra striping patterns." {\sl Journal of Zoology\/}, Vol. 183, part 4, pp. 527-539. \ref Boal, F. W. 1972. Close together and far apart: Religious and class divisions in Belfast." {\sl Community Forum\/}, Vol. 3, No. 2, pp. 3-11. \ref Boal, F. W. and Livingstone 1986. Protestants in Belfast: A view from the inside." {\sl Contemporary Review\/}, 248: 169-75. \ref Boyce, R. R., and W. A. V. Clark. 1964. The concept of shape in geography." {\sl Geographical Review\/} 54, 561-572. \ref Dewdney, A. K. A home computer laboratory in which balls become gases, liquids and critical masses." Computer Recreations, {\sl Scientific American\/} pp. 114-117. \ref Freedman, David H. 1991. A chaotic cat takes a swipe at quantum mechanics." {\sl Science\/}, Vol. 253, p. 626. \ref Gierer, A. 1981. Some physical, mathematical and evolutionary aspects of biological pattern formation." {\sl Philosophical Transactions Royal Society, London\/}, Series B, 295, pp. 429-440. \ref Hagerstrand, T. 1967. {\sl Innovation diffusion as a spatial process.\/} Chicago: University of Chicago Press. \ref Kennedy, S., and W. Tobler. 1983. Geographic interpolation." {\sl Geographical Analysis\/} 15, 151-156. \ref Murray, J. D. 1981. A pre-pattern formation mechanism for animal coat markings." {\sl Journal of Theoretical Biology\/}. Vol. 88, No. 1, pp. 161-199. \ref Murray, J. D. 1988. How the leopard gets its spots. {\sl Scientific American\/} 258:80-87. \ref Murray, J. D. 1981. Introductory remarks" (to an entire volume devoted to pattern formation) {\sl Philosophical Transactions Royal Society, London\/}, Series B 295, pp. 427-428. \ref Murray, J. D. 1989. Mathematical Biology. Springer-Verlag, Heidelberg. \ref Murray, J. D. 1981. On pattern formation mechanisms for lepidopteran wing patterns and mammalian coat patterns." {\sl Philosophical Transactions of the Royal Society, London\/} Series B, Vol. 295, No. 1078, pp. 473-496; Oct. 7. \ref Murray, J. D. and P. K. Maini. 1986. A new approach to the generation of pattern and form in embryology." {\sl Science Progress\/}, Vol. 70, No. 280, part 4, 539-553. \ref Nordbeck, S. 1965. The Law of Allometric Growth." Discussion Paper \#7. Michigan Inter-University Community of Mathematical Geographers (John D. Nystuen, ed.), Ann Arbor, MI. (Reprinted by Institute of Mathematical Geography, 1986). \ref Pool, R. 1991. Did Turing discover how the leopard got its spots? {\sl Science\/} 251:627. \ref Shaw, L. J. and Murray, J. D. 1990. Analysis of a model for complex skin patterns." SIAM Journal of Applied Mathematics, pp. 628-648. \ref Tobler, W. R. 1969. The spectrum of U. S. 40," {\sl Papers of the Regional Science Association\/}, Vol. XXIII, pp. 45-52. \ref Turing, A. M. 1952. The chemical basis of morphogenesis." {\sl Philosophical Transactions of the Royal Society, London\/}, Series B, 237, pp. 37-72. \ref White, M. J. 1983. The measurement of spatial segregation." {\sl American Journal of Sociology\/} 88, 1008-1018. \ref Wolpert, L. 1981. Positional information and pattern formation." {\sl Philosophical Transactions Royal Society of London\/}, Series B 295, pp. 441-450. 1981. \vfill\eject \centerline{\bf 5. FEATURE} \smallskip \centerline{\bf Meet New Solstice Board Member} \centerline{\bf William D. Drake} \centerline{\bf The University of Michigan} \smallskip \centerline{Professor of Resource Policy and Planning} \centerline{School of Natural Resources and the Environment} \smallskip \centerline{Professor of Population Planning and International Health} \centerline{School of Public Health, The University of Michigan} \smallskip \centerline{Professor of Urban, Technological and Environmental Planning} \centerline{College of Architecture and Urban Planning} \smallskip Bill Drake teaches courses on the Global Environment and on Population-Environment Dynamics. Much of his research portfolio is drawn from ongoing projects in the developing world. Many of these projects have been underway for several years and have focused on the problems of rural community development particularly relating to reducing child malnutrition. Recently, he has authored articles and was co-editor for a book on population - en\-vi\-ron\-ment dynamics. During the fall of 1992, Drake and S. Arlinghaus offered a course on the same subject which has resulted in a monograph. The focus of this course is captured in its name {\sl Population-Environment Dynamics: Toward Building a Theory\/}. The effort draws upon recent work carried out as part of the University of Michigan's Population - Environment Dynamics Project. Ten graduate students and two faculty participated formally, and several other students and faculty sat in from time-to-time, with one visitor attending every session. Seminar participants came from many disciplinary backgrounds ranging from population planning, economics, engineering, biology, remote sensing, geography, natural resources, sociology, international health, business administration to mathematics. In addition to U.S. students, the course was enriched by colleagues from Mexico, Nepal, Taiwan, and Nigeria. The monograph serves as a kind of a time capsule"--what do students in 1992 think will be issues of great significance in the current, recently identified, need to study global change"? Here are the sectors of that capsule": \line{\bf Dawn M. Anderson \hfil} \line{The Historical Transition of Forest Stock Depletion in Costa Rica \hfil} \smallskip \line{\bf Katharine A. Duderstadt \hfil} \line{The Energy Sector of Population-Environment Dynamics in China \hfil} \smallskip \line{\bf Eugene A. Fosnight \hfil} \line{Population Transition and Changing Land Cover and Land Use in Senegal \hfil} \smallskip \line{\bf Katharine Hornbarger \hfil} \line{The Energy Crisis in India: Options for a Sound Environment \hfil} \smallskip \line{\bf Deepak Khatry \hfil} \line{An Analysis of the Major Sectoral Transitions in Nepal's Middle Hills \hfil} \line{and their Relationship with Forest Degradation \hfil} \smallskip \line{\bf Catherine MacFarlane \hfil} \line{The Interrelationship Between the Forestry Sector \hfil} \line{and Population-Environment Dynamics in Haiti \hfil} \smallskip \line{\bf Gary Stahl \hfil} \line{Transition to Peace: \hfil} \line{Environmental Impacts of Downsizing the U.S. Nuclear Weapons Complex \hfil } \smallskip \line{\bf Stephen Uche \hfil} \line{Population and Forestry Dynamics: At the Crossroads in Nigeria \hfil} \smallskip \line{\bf Hurng-jyuhn Wang \hfil} \line{The Cultivated Land-Rural Industrialization-Urbanization-Population Dynamics in Taiwan \hfil} \vfill\eject \noindent{\bf 6. SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE} \noindent This section shows the exact set of commands that work to download {\sl Solstice\/} 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. First step is to concatenate the files you received via bitnet/internet. Simply piece them together in your computer, one after another, in the order in which they are numbered, starting with the number, 1." The files you have received are ASCII files; the concatenated file is used to form the .tex file from which the .dvi file (device independent) file is formed. The words percent-sign" and backslash" are written out in the example below; the user should type them symbolically. \noindent ASSUME YOU HAVE SIGNED ON AND ARE AT THE SYSTEM PROMPT, \#. \smallskip \# 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 [there may be some underfull boxes that generally cause no problem; there should be no other error" messages in the typesetting--the files you receive were already tested.] \# run *dvixer par=-t.dvi \# control *print* onesided \# run *pagepr scards=-t.xer, par=paper=plain \vfill\eject \noindent{\bf 7. SUMMARY OF CONTENT} \noindent IMaGe is working to establish a Bulletin Board on which back issues of {\sl Solstice\/} can be posted. Subscribers will be notified when this service is available. \smallskip \noindent {\bf Volume III, Number 1, Summer, 1992} \smallskip \smallskip \noindent{\bf 1. ARTICLES.} \smallskip\noindent {\bf Harry L. Stern}. \smallskip\noindent {\bf Computing Areas of Regions With Discretely Defined Boundaries}. \smallskip\noindent 1. Introduction 2. General Formulation 3. The Plane 4. The Sphere 5. Numerical Example and Remarks. Appendix--Fortran Program. \smallskip \noindent{\bf 2. NOTE } \smallskip\noindent {\bf Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg}. \smallskip\noindent {\bf The Quadratic World of Kinematic Waves} \smallskip \noindent{\bf 3. SOFTWARE REVIEW} \smallskip RangeMapper$^{\hbox{TM}}$--- version 1.4. Created by {\bf Kenelm W. Philip}, Tundra Vole Software, Fairbanks, Alaska. Program and Manual by {\bf Kenelm W. Philip}. \smallskip Reviewed by {\bf Yung-Jaan Lee}, University of Michigan. \smallskip \noindent{\bf 4. PRESS CLIPPINGS} \smallskip \noindent{\bf 5. INDEX to Volumes I (1990) and II (1991) of {\sl Solstice}.} \smallskip \noindent {\bf Volume II, Number 1, Summer, 1991} \smallskip \noindent 1. ARTICLE Sandra L. Arlinghaus, David Barr, John D. Nystuen. {\sl The Spatial Shadow: Light and Dark --- Whole and Part\/} This account of some of the projects of sculptor David Barr attempts to place them in a formal, systematic, spatial setting based on the postulates of the science of space of William Kingdon Clifford (reprinted in {\sl Solstice\/}, Vol. I, No. 1.). \smallskip \smallskip \noindent 2. FEATURES \item{i} Construction Zone --- The logistic curve. \item{ii.} Educational feature --- Lectures on Spatial Theory" \smallskip \noindent {\bf Volume II, Number 2, Winter, 1991} \smallskip \noindent 1. REPRINT Saunders Mac Lane, Proof, Truth, and Confusion." Given as the Nora and Edward Ryerson Lecture at The University of Chicago in 1982. Republished with permission of The University of Chicago and of the author. I. The Fit of Ideas. II. Truth and Proof. III. Ideas and Theorems. IV. Sets and Functions. V. Confusion via Surveys. VI. Cost-benefit and Regression. VII. Projection, Extrapolation, and Risk. VIII. Fuzzy Sets and Fuzzy Thoughts. IX. Compromise is Confusing. \noindent 2. ARTICLE Robert F. Austin. Digital Maps and Data Bases: Aesthetics versus Accuracy." I. Introduction. II. Basic Issues. III. Map Production. IV. Digital Maps. V. Computerized Data Bases. VI. User Community. \noindent 3. FEATURES Press clipping; Word Search Puzzle; Software Briefs. \smallskip \smallskip \smallskip \noindent{\bf INDEX to Volume I (1990) of {\sl Solstice}.} \vskip.5cm \noindent{\bf Volume I, Number 1, Summer, 1990} \noindent 1. REPRINT 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? \noindent 2. ARTICLES. Sandra L. Arlinghaus, {\sl Beyond the Fractal.} 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 setting. \smallskip \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 with the idea that applications of mathematics need not be limited to scientific ones. \smallskip \noindent 3. FEATURES \smallskip \item{i.} Theorem Museum --- Desargues's Two Triangle Theorem from projective geometry. \item{ii.} Construction Zone --- a centrally symmetric hexagon is derived from an arbitrary convex hexagon. \item{iii.} Reference Corner --- Point set theory and topology. \item{iv.} Educational Feature --- Crossword puzzle on spices. \item{v.} Solution to crossword puzzle. \smallskip \noindent 4. SAMPLE OF HOW TO DOWNLOAD THE ELECTRONIC FILE \smallskip \noindent{\bf Volume I, Number 2, Winter, 1990} \smallskip \noindent 1. REPRINT John D. Nystuen (1974), {\sl A City of Strangers: Spatial Aspects of Alienation in the Detroit Metropolitan Region\/}. This paper examines the urban shift from people space" to machine space" (see R. Horvath, {\sl Geographical Review\/}, April, 1974) in the Detroit metropolitan region of 1974. As with Clifford's {\sl Postulates\/}, reprinted in the last issue of {\sl Solstice\/}, note the timely quality of many of the observations. \noindent 2. ARTICLES Sandra Lach Arlinghaus, {\sl Scale and Dimension: Their Logical Harmony\/}. Linkage between scale and dimension is made using the Fallacy of Division and the Fallacy of Composition in a fractal setting. \smallskip \smallskip Sandra Lach Arlinghaus, {\sl Parallels between Parallels\/}. The earth's sun introduces a symmetry in the perception of its trajectory in the sky that naturally partitions the earth's surface into zones of affine and hyperbolic geometry. The affine zones, with single geometric parallels, are located north and south of the geographic parallels. The hyperbolic zone, with multiple geometric parallels, is located between the geographic tropical parallels. Evidence of this geometric partition is suggested in the geographic environment --- in the design of houses and of gameboards. \smallskip \smallskip Sandra L. Arlinghaus, William C. Arlinghaus, and John D. Nystuen. {\sl The Hedetniemi Matrix Sum: A Real-world Application\/}. In a recent paper, we presented an algorithm for finding the shortest distance between any two nodes in a network of$n$nodes when given only distances between adjacent nodes [Arlinghaus, Arlinghaus, Nystuen, {\sl Geographical Analysis\/}, 1990]. In that previous research, we applied the algorithm to the generalized road network graph surrounding San Francisco Bay. Here, we examine consequent changes in matrix entires when the underlying adjacency pattern of the road network was altered by the 1989 earthquake that closed the San Francisco --- Oakland Bay Bridge. \smallskip \smallskip Sandra Lach Arlinghaus, {\sl Fractal Geometry of Infinite Pixel Sequences: Su\-per\--def\-in\-i\-tion" Resolution\/}? Comparison of space-filling qualities of square and hexagonal pixels. \smallskip \noindent 3. FEATURES \item{i.} Construction Zone --- Feigenbaum's number; a triangular coordinatization of the Euclidean plane. \item{ii.} A three-axis coordinatization of the plane. \smallskip \vfill\eject \noindent{\bf 8. OTHER PUBLICATIONS OF IMaGe } \smallskip \centerline{\bf MONOGRAPH SERIES} \centerline{\sl Scholarly Monographs--Original Material, refereed} \centerline{Prices on request, exclusive of shipping and handling;} \centerline{payable in U.S. funds on a U.S. bank, only.} \smallskip Monographs are printed by {\bf Digicopy} on 100\% recycled paper of archival quality; both hard and soft cover is available. \vskip 0.2cm 1. Sandra L. Arlinghaus and John D. Nystuen. {\it Mathematical Geography and Global Art: the Mathematics of David Barr's Four Corners Project\/},'' 1986. \vskip 0.1cm This monograph contains Nystuen's calculations, actually used by Barr to position his abstract tetrahedral sculpture within the earth. Placement of the sculpture vertices in Easter Island, South Africa, Greenland, and Indonesia was chronicled in film by The Archives of American Art for The Smithsonian Institution. In addition to the archival material, this monograph also contains Arlinghaus's solutions to broader theoretical questions--was Barr's choice of a tetrahedron unique within his initial constraints, and, within the set of Platonic solids? \vskip 0.2cm 2. Sandra L. Arlinghaus. {\it Down the Mail Tubes: the Pressured Postal Era, 1853-1984\/}, 1986. \vskip 0.1cm The history of the pneumatic post, in Europe and in the United States, is examined for the lessons it might offer to the technological scenes of the late twentieth century. As Sylvia L. Thrupp, Alice Freeman Palmer Professor Emeritus of History, The University of Michigan, commented in her review of this work Such brief comment does far less than justice to the intelligence and the stimulating quality of the author's writing, or to the breadth of her reading. The detail of her accounts of the interest of American private enterprise, in New York and other large cities on this continent, in pushing for construction of large tubes in systems to be leased to the government, brings out contrast between American and European views of how the new technology should be managed. This and many other sections of the monograph will set readers on new tracks of thought.'' \vskip 0.2cm 3. Sandra L. Arlinghaus. {\it Essays on Mathematical Geography\/}, 1986. \vskip 0.1cm A collection of essays intended to show the range of power in applying pure mathematics to human systems. There are two types of essay: those which employ traditional mathematical proof, and those which do not. As mathematical proof may itself be regarded as art, the former style of essay might represent traditional'' art, and the latter, surrealist'' art. Essay titles are: The well-tempered map projection,'' Antipodal graphs,'' Analogue clocks,'' Steiner transformations,'' Concavity and urban settlement patterns,'' Measuring the vertical city,'' Fad and permanence in human systems,'' Topological exploration in geography,'' A space for thought,'' and Chaos in human systems--the Heine-Borel Theorem.'' \vskip 0.2cm 4. Robert F. Austin, {\it A Historical Gazetteer of Southeast Asia\/}, 1986. \vskip 0.1cm Dr. Austin's Gazetteer draws geographic coordinates of Southeast Asian place-names together with references to these place-names as they have appeared in historical and literary documents. This book is of obvious use to historians and to historical geographers specializing in Southeast Asia. At a deeper level, it might serve as a valuable source in establishing place-name linkages which have remained previously unnoticed, in documents describing trade or other communications connections, because of variation in place-name nomenclature. \vskip 0.2cm 5. Sandra L. Arlinghaus, {\it Essays on Mathematical Geography--II\/}, 1987. \vskip 0.1cm Written in the same format as IMaGe Monograph \#3, that seeks to use pure'' mathematics in real-world settings, this volume contains the following material: Frontispiece--the Atlantic Drainage Tree,'' Getting a Handel on Water-Graphs,'' Terror in Transit: A Graph Theoretic Approach to the Passive Defense of Urban Networks,'' Terrae Antipodum,'' Urban Inversion,'' Fractals: Constructions, Speculations, and Concepts,'' Solar Woks,'' A Pneumatic Postal Plan: The Chambered Interchange and ZIPPR Code,'' Endpiece.'' \vskip 0.2cm 6. Pierre Hanjoul, Hubert Beguin, and Jean-Claude Thill, {\it Theoretical Market Areas Under Euclidean Distance\/}, 1988. (English language text; Abstracts written in French and in English.) \vskip 0.1cm Though already initiated by Rau in 1841, the economic theory of the shape of two-dimensional market areas has long remained concerned with a representation of transportation costs as linear in distance. In the general gravity model, to which the theory also applies, this corresponds to a decreasing exponential function of distance deterrence. Other transportation cost and distance deterrence functions also appear in the literature, however. They have not always been considered from the viewpoint of the shape of the market areas they generate, and their disparity asks the question whether other types of functions would not be worth being investigated. There is thus a need for a general theory of market areas: the present work aims at filling this gap, in the case of a duopoly competing inside the Euclidean plane endowed with Euclidean distance. \vskip 0.1cm (Bien qu'\'ebauch\'ee par Rau d\es 1841, la th\'eorie \'economique de la forme des aires de march\'e planaires s'est longtemps content\'ee de l'hypoth\ese de co\^uts de transport proportionnels \a la distance. Dans le mod\ele gravitaire g\'en\'eralis\'e, auquel on peut \'etendre cette th\'eorie, ceci correspond au choix d'une exponentielle d\'ecroissante comme fonction de dissuasion de la distance. D'autres fonctions de co\^ut de transport ou de dissuasion de la distance apparaissent cependant dans la litt\'erature. La forme des aires de march\'e qu'elles engendrent n'a pas toujours \'et\'e \'etudi\'ee ; par ailleurs, leur vari\'et\'e am\ene \a se demander si d'autres fonctions encore ne m\'eriteraient pas d'\^etre examin\'ees. Il para\^it donc utile de disposer d'une th\'eorie g\'en\'erale des aires de march\'e : ce \a quoi s'attache ce travail en cas de duopole, dans le cadre du plan euclidien muni d'une distance euclidienne.) \vskip 0.2cm 7. Keith J. Tinkler, Editor, {\it Nystuen---Dacey Nodal Analysis\/}, 1988. \vskip.1cm Professor Tinkler's volume displays the use of this graph theoretical tool in geography, from the original Nystuen---Dacey article, to a bibliography of uses, to original uses by Tinkler. Some reprinted material is included, but by far the larger part is of previously unpublished material. (Unless otherwise noted, all items listed below are previously unpublished.) Contents:  Foreward' " by Nystuen, 1988; Preface" by Tinkler, 1988; Statistics for Nystuen---Dacey Nodal Analysis," by Tinkler, 1979; Review of Nodal Analysis literature by Tinkler (pre--1979, reprinted with permission; post---1979, new as of 1988); FORTRAN program listing for Nodal Analysis by Tinkler; A graph theory interpretation of nodal regions'' by John D. Nystuen and Michael F. Dacey, reprinted with permission, 1961; Nystuen---Dacey data concerning telephone flows in Washington and Missouri, 1958, 1959 with comment by Nystuen, 1988; The expected distribution of nodality in random (p, q) graphs and multigraphs,'' by Tinkler, 1976. \vskip.2cm 8. James W. Fonseca, {\it The Urban Rank--size Hierarchy: A Mathematical Interpretation\/}, 1989. \vskip.1cm The urban rank--size hierarchy can be characterized as an equiangular spiral of the form$r=ae^{\theta \, \hbox{cot}\alpha}$. An equiangular spiral can also be constructed from a Fibonacci sequence. The urban rank--size hierarchy is thus shown to mirror the properties derived from Fibonacci characteristics such as rank--additive properties. A new method of structuring the urban rank--size hierarchy is explored which essentially parallels that of the traditional rank--size hierarchy below rank 11. Above rank 11 this method may help explain the frequently noted concavity of the rank--size distribution at the upper levels. The research suggests that the simple rank--size rule with the exponent equal to 1 is not merely a special case, but rather a theoretically justified norm against which deviant cases may be measured. The spiral distribution model allows conceptualization of a new view of the urban rank--size hierarchy in which the three largest cities share functions in a Fibonacci hierarchy. \vskip.2cm 9. Sandra L. Arlinghaus, {\it An Atlas of Steiner Networks \/}, 1989. \vskip.1cm A Steiner network is a tree of minimum total length joining a prescribed, finite, number of locations; often new locations are introduced into the prescribed set to determine the minimum tree. This Atlas explains the mathematical detail behind the Steiner construction for prescribed sets of n locations and displays the steps, visually, in a series of Figures. The proof of the Steiner construction is by mathematical induction, and enough steps in the early part of the induction are displayed completely that the reader who is well--trained in Euclidean geometry, and familiar with concepts from graph theory and elementary number theory, should be able to replicate the constructions for full as well as for degenerate Steiner trees. \vskip.2cm 10. Daniel A. Griffith, {\it Simulating$K=3$Christaller Central Place Structures: An Algorithm Using A Constant Elasticity of Substitution Consumption Function\/}, 1989. \vskip.1cm An algorithm is presented that uses BASICA or GWBASIC on IBM compatible machines. This algorithm simulates Christaller$K=3\$ central place
structures, for a four--level hierarchy.  It is based upon earlier published
work by the author.  A description of the spatial theory, mathematics, and
sample output runs appears in the monograph.  A digital version is available
from the author, free of charge, upon request; this request must be
accompanied by a 5.5--inch formatted diskette.  This algorithm has been
developed for use in Social Science classroom laboratory situations,
and is designed to (a) cultivate a deeper understanding of central place
theory, (b) allow parameters of a central place system to be altered and
then graphic and tabular results attributable to these changes viewed,
without experiencing the tedium of massive calculations, and (c) help
promote a better comprehension of the complex role distance
plays in the space--economy.  The algorithm also should facilitate
intensive numerical research on central place structures; it is expected
that even the sample simulation results will reveal interesting insights
into
abstract central place theory.

The background spatial theory concerns demand and competition in the
space--economy; both linear and non--linear spatial demand functions are
discussed.  The mathematics is concerned with (a) integration of non--linear
spatial demand cones on a continuous demand surface, using a constant
elasticity of substitution consumption function, (b) solving for roots of
polynomials, (c) numerical approximations to integration and root
extraction, and (d) multinomial discriminant function classification of
commodities into central place hierarchy levels.  Sample
output is presented for contrived data sets, constructed from
artificial and empirical information, with the wide range of all possible
central place structures being generated.  These examples should facilitate
implementation testing.  Students are able to vary single or multiple
parameters of the problem, permitting a study of how certain changes
manifest
themselves within the context of a theoretical central place structure.
Hierarchical classification criteria may be changed, demand elasticities may
or may not vary and can take on a wide range of non--negative values, the
uniform transport cost may be set at any positive level, assorted fixed
costs and variable costs may be introduced, again within a rich range of
non--negative possibilities, and the number of commodities can be
altered.  Directions for algorithm execution are summarized.  An ASCII
version of the algorithm, written directly from GWBASIC, is included in
an appendix; hence, it is free of typing errors.
\vskip.2cm
11.  Sandra L. Arlinghaus and John D. Nystuen,
{\it Environmental Effects on Bus Durability\/}, 1990.
\vskip.1cm

This monograph draws on the authors'
previous publications on Climatic" and Terrain" effects on bus
durability.
Material on these two topics is selected, and reprinted, from
three published papers that appeared in the {\sl Transportation
Research Record\/} and in the {\sl Geographical Review\/}.  New
material concerning congestion" effects is examined at the
national level, to determine dense," intermediate," and
sparse" classes of congestion, and at the local level of
congestion in Ann Arbor (as suggestive of how one  might use local data).
This material is drawn together in a single volume, along with a summary
of the consequences of all three effects simultaneously, in order to suggest
direction for more highly automated studies that should follow naturally
with the release of the 1990 U. S. Census data.
\vskip.2cm
12.  Daniel A. Griffith, Editor.
{\sl Spatial Statistics:  Past, Present, and Future\/},  1990.
\vskip.1cm

Proceedings of a Symposium of the same name held at Syracuse
University in Summer, 1989.  Content includes a Preface by Griffith and
the
following papers:
\vskip.1cm

\line{{\sl Brian Ripley}, Gibbsian interaction models"; \hfil}
\line{{\sl J. Keith Ord}, Statistical methods for point pattern data";
\hfil}
\line{{\sl Luc Anselin}, What is special about spatial data"; \hfil}
\line{{\sl Robert P. Haining}, Models in human geography: \hfil}
estimating, and validating models for spatial data"; \hfil}
\line{{\sl R. J. Martin},
The role of spatial statistics in geographic modelling"; \hfil}
\line{{\sl Daniel Wartenberg}, \hfil }
\line{Exploratory spatial analyses:  outliers,
leverage points, and influence functions"; \hfil}
\line{{\sl J. H. P. Paelinck},
Some new estimators in spatial econometrics"; \hfil}
\line{{\sl Daniel A. Griffith}, \hfil }
\line{A numerical simplification for estimating parameters of
spatial autoregressive models"; \hfil}
\line{{\sl Kanti V. Mardia}
Maximum likelihood estimation for spatial models"; \hfil}
\line{{\sl Ashish Sen}, Distribution of spatial correlation statistics";
\hfil}
\line{{\sl Sylvia Richardson},
Some remarks on the testing of association between spatial
processes";\hfil}
\line{{\sl Graham J. G. Upton}, Information from regional data";\hfil}
\line{{\sl Patrick Doreian},
Network autocorrelation models:  problems and prospects." \hfil}

Each chapter is preceded by an Editor's Preface" and followed by a
Discussion and, in some cases, by an author's Rejoinder to the Discussion.
\vskip.2cm
13.  Sandra L. Arlinghaus, Editor.
{\sl Solstice---I\/},  1990.
\vskip.2cm
14.  Sandra L. Arlinghaus, {\sl Essays on Mathematical Geography--III\/},
1991.
\smallskip
A continuation of the series.  Essays in this volume are:  Table for
central place fractals; Tiling according to the Administrative"
Principle; Moir\'e maps; Triangle partitioning; An enumeration of
candidate Steiner networks; A topological generation gap;
Synthetic centers of gravity:  A conjecture.
\vskip.2cm
15.  Sandra L. Arlinghaus, Editor, {\sl Solstice---II\/}, 1991.
\vfill\eject
\centerline{\it DISCUSSION PAPERS--ORIGINAL}
\centerline{\it Editor, Daniel A. Griffith}
\centerline{\it Professor of Geography}
\centerline{\it Syracuse University}
\centerline{Founder as an IMaGe series:  Sandra L. Arlinghaus}
\smallskip
\noindent 1.  {\sl Spatial Regression Analysis on the PC:
Spatial Statistics Using Minitab}.  1989.
\vskip.5cm
\centerline{\it DISCUSSION PAPERS--REPRINTS}
\centerline{\it Editor of MICMG Series, John D. Nystuen}
\centerline{\it Professor of Geography and Urban Planning}
\centerline{\it The University of Michigan}
\smallskip
\noindent 1.  {\sl Reprint of the Papers of the Michigan InterUniversity
Community of Mathematical Geographers.}  Editor, John D. Nystuen.
\smallskip
Contents--original editor:  John D. Nystuen.
\smallskip
\noindent 1.  Arthur Getis, Temporal land use pattern analysis with the
use of nearest neighbor and quadrat methods."  July, 1963
\smallskip
\noindent 2.  Marc Anderson, A working bibliography of mathematical
geography."  September, 1963.
\smallskip
\noindent 3.  William Bunge, Patterns of location."  February, 1964.
\smallskip
\noindent 4.  Michael F. Dacey, Imperfections in the uniform plane."
June, 1964.
\smallskip
\noindent 5.  Robert S. Yuill, A simulation study of barrier effects
in spatial diffusion problems."  April, 1965.
\smallskip
\noindent 6.  William Warntz, A note on surfaces and paths and
applications to geographical problems."  May, 1965.
\smallskip
\noindent 7.  Stig Nordbeck, The law of allometric growth."
June, 1965.
\smallskip
\noindent 8.  Waldo R. Tobler, Numerical map generalization;"
and Waldo R. Tobler, Notes on the analysis of geographical
distributions."  January, 1966.
\smallskip
\noindent 9.  Peter R. Gould, On mental maps."  September, 1966.
\smallskip
\noindent 10.  John D. Nystuen, Effects of boundary shape and the
concept of local convexity;"  Julian Perkal, On the length of
empirical curves;" and Julian Perkal, An attempt at
objective generalization."  December, 1966.
\smallskip
\noindent 11. E. Casetti and R. K. Semple, A method for the
stepwise separation of spatial trends."  April, 1968.
\smallskip
\noindent 12.  W. Bunge, R. Guyot, A. Karlin, R. Martin, W. Pattison,
W. Tobler, S. Toulmin, and W. Warntz, The philosophy of maps."
June, 1968.
\vfill\eject
\centerline{\bf Reprints of out-of-print textbooks.}
\centerline{\bf Printer and obtainer of copyright permission:  Digicopy}
\centerline{Inquire for cost of reproduction---include class size}
\smallskip
1.  Allen K. Philbrick.  {\sl This Human World}.
\smallskip
\vskip.5cm
Publications of the Institute of Mathematical Geography have
been reviewed in

\item{1.} {\sl The Professional Geographer\/} published
by the Association of American Geographers;
\item{2.}  {\sl The Urban Specialty Group Newsletter\/}
of the Association of American Geographers;