Volume VII, Number 1, Summer, 1996.

SUMMER, 1996



Founding Editor-in-Chief:
     Sandra Lach Arlinghaus, University of Michigan;
	Institute of Mathematical Geography (independent)
Editorial Advisory Board:
     Michael F. Goodchild, University of California, Santa Barbara
     Daniel A. Griffith, Syracuse University
     Jonathan D. Mayer, University of Washington (also School of Medicine)
     John D. Nystuen, University of Michigan
William C. Arlinghaus, Lawrence Technological University
     Neal Brand, University of North Texas
     Kenneth H. Rosen, A. T. & T. Bell Laboratories
  Engineering Applications.
     William D. Drake, University of Michigan
     Frederick L. Goodman, University of Michigan
     Robert F. Austin, Austin Communications Education Services.

Technical Editor:
     Richard Wallace, University of Michigan.

Web Consultant:
     William E. Arlinghaus, UMI

Electronic address:

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     John D. Nystuen, Rhonda Ryznar, Thomas Wagner

     Sandra Lach Arlinghaus

     Daniel Jacobs

4.  INDEX TO VOLUMES I (1990) TO V (1995).

John D. Nystuen, The University of Michigan
Rhonda Ryznar, The University of Michigan
Thomas Wagner, Environmental Research Institute of Michigan

Figure 1. Landsat change image of Detroit showing changes in urban greenness from 1975 to 1992. Imagery and analysis are joint ventures between Environmental Research Institute of Michigan (ERIM) and The University of Michigan, College of Architecture and Urban Planning. Green areas show tracts with greenness increase; red areas with greenness decrease; black areas, no change. Change data derived from TM 1992 and MSS 1975 images of vegetation reflectance.

Figure 2. Ground truth, green area (reference to Figure 1). Areas of increased greenness are places where the social system is stressed. Houses are abandoned or destroyed. Much of the territory is vacant land. Sidewalks and alley surfaces are broken and overgrown with weeds. The overall effect is increased greenness over the time period. Indeed, we observed pheasants in overgrown parts of the central city. Much of the green part of the image in Figure 1, in the inner city of Detroit, are territories of this sort.

Figure 3. Ground truth, no change area--black (reference to Figure 1). In some neighborhoods in the central part of Detroit, the social structure is intact. The physical properties of these neighborhoods reflect this sustained social organization. This neighborhood has not changed much and shows as mostly black in the image (Figure 1). This is also true of cemeteries and parks where the vegetation cover has not changed over the time period.

Figure 4. Ground truth, red area (reference to Figure 1). In much of the outer edge of the city of Detroit, shown by the overlay of census tracts in Figure 1, the social structure of the neighborhoods remains intact but the physical changes indicate decline in vegetation cover. By field investigation, we noted open streets with an occasional single elm tree as shown in this figure. We attribute the decrease in greenness to the effects of the Dutch elm disease which in the decade of the 70s destroyed virtually the entire elm tree population. In 1975, at the time of the image, this process was in progress. By 1992, very few elm trees remained. In the early period, much of the street surface was shaded by these trees, whereas currently that is not the case, despite replanting of small trees.

Figure 5. Figure 5 shows a recently completed automobile plant in the City of Detroit. This location shows in Figure 1 as a red and black region with no green. The red resulted from conversion of low income neighborhoods, once green, to factory roof and parking lot surface. The black areas were pre-existing industrial areas that lacked vegetation and still have no vegetation.

Figure 6. In Figure 6, census tracts with the largest increase in greenness are shown with the darkest green and brown tones. Superimposed on this colored pattern are census tracts of high social stress (cross-hatched). High social stress census tracts are defined as those tracts with decrease in population greater than 25% between 1980 and 1990 and with more than 50% of children below the poverty level. There is substantial correspondence between the two patterns.

Landsat Data Analysis by ERIM Two Landsat data sets: MSS data from May 10, 1975 and TM data from May 16, 1992 were employed in the analysis. A restoration resampling algorithm rectified the images to the same spatial reference frame (the State Plane Coordinate system). A principal component procedure was employed to create a "greenness" vector from signals returned by several spectral bands available from the satellites' instruments. The "greenness" vector has been identified with intensity of vegetation. The resampling and rectification allows for the creation of a change image measuring the difference in "greenness" per pixel (at 25 meter resolution) between the two time points separated by 17 years. The change image was then divided into three classes: increased greenness (colored green), no change in greenness (colored black), and decrease in greenness (colored red) to create the image shown here. Detroit census tracts (1990) were superimposed on this image and counts of pixels by each class by census tracts could then be compared with census data on socioeconomic variables. A map of percent increase in vegetation was created by classifying census tracts by the proportion of green pixels to total pixels contained in each. The remote sensing analysis was carried out by the Environmental Research Laboratory of Michigan (ERIM), Ann Arbor, Michigan. Comparisons to socioeconomic data were done by the Urban and Regional Planning GIS Laboratory, University of Michigan.

ALGEBRAIC ASPECTS OF RATIOS Sandra Lach Arlinghaus Institute of Mathematical Geography; The University of Michigan; Community Systems Foundation
To Douglas R. McManis, Editor of the Geographical Review, whose cleverness with words has helped this author tame many a distorted thought! Congratulations on your unparalleled service to the American Geographical Society (May 30, 1996). Article to appear in Solstice: An Electronic Journal of Geography and Mathematics, Vol. VI, No. 1, June, 1996 ( A particularly exciting feature of the Internet permits documents stored on the World Wide Web to incorporate graphics, as well as text, on the same page. It is generally easy to cut and paste text from a word processor into the online file in which the home page is being created (into the html file). It is almost as easy to upload an image of a scanned photo (for example, saved in gif or jpg format) from a computer in one's home to that same online file (or to one linked to it). Indeed, the whole technical process is so easy, once one has a small set of keys that open merely a few doors, that documents can be created in advance of even simple knowledge about enduring concepts. Thus, one might anticipate a host of web sites that do not take account of simple cartographic principles or of drafting lessons, let alone elements of written linguistic or mathematical style. From a more positive standpoint, however, one might see that the creation of web documents can serve as a favorable opportunity to cast new light on traditional material. Practice can motivate theory: in mathematics, geography, linguistics, or elsewhere. Consider the simple task of creating web stationery incorporating an established logo. The American Geographical Society (New York) has given permission to experiment with its relatively symmetric logo that includes a sphere (Bird, M. L., 1996). When the logo is scanned at a relatively low resolution that even older or cheaper scanners can produce, and loaded as a scanned image onto a Web page, the result appears as it does below (Figure 1). The shape of the globe appears correct to the eye, but the image is too large to be viewed on one moderately sized computer screen. Clearly, this situation is unsatisfactory for general use: a large image requires not only too much visual space, but also too much computer space, and too much time to come up on the screen. The obvious answer is to scale the image down in size.

Figure 1. Logo of the American Geographical Society, height 922 pixels, width 810 pixels.

With a rectangular image such scaling is of course a simple matter: divide the width in half (for example). To retain the same shape of image as the original, also divide the height in half (Figure 2). More generally, to retain shape when altering the scale of a two-dimensional rectangular shape, preserve the aspect ratio--the ratio of the horizontal dimension of an image to the vertical dimension of an image. Varying the aspect ratio of an image causes distortion, as in Figure 3, in which the width is divided by 2 and the height is divided by 3.

Figure 2. Logo of the American Geographical Society, height of Figure 1 divided by 2 and width divided by 2. Correctly scaled image.

Figure 3. Logo of the American Geographical Society, height of Figure 1 divided by 3 and width divided by 2. Distorted image.

PIXEL ALGEBRA In the case of the AGS logo, the scanned image in Figure 1 (a .gif file) measured 810 pixels in width and 922 pixels in height. Pixels are the fundamental units in which these images are measured; fractions of pixels are not permitted. The algebra of pixels is an integer algebra that can be viewed to operate on a bounded, finite set of pixels (the dimensions of the cathode ray tube). In the image of the AGS logo, both dimensions are even numbers and so clearly both are divisible by 2. The aspect ratio, 810/922, can be preserved by taking 1/2 of each dimension; 810/922 = (405*2)/(461*2) = 405/461. Are both numerator and denominator divisible by 3? In times when flashy applications relied more on elegant abstractions, the distributive law was often a source of the unusual or the cute. DISTRIBUTIVE LAW For integers in a domain of two operations, + and *: Suppose that a, b, and c are integers. Then a*(b+c)=a*b + a*c. One application of the distributive law yields the trick that to determine whether or not a number is divisible by three, add the digits which form it; repeat the process until a single digit is reached. If that single digit is divisible by 3, then the entire number is divisible by 3. In the case of the AGS logo dimensions 810 becomes 8+1+0=9 which is divisible by 3 so that 810 is also divisible by 3. The number 922 becomes 9+2+2=13 which becomes 1+3=4 which is not divisible by 3 so that 922 is not divisible by 3. Thus, 1/3 would not be a scaling factor for the scanned AGS logo that would preserve the aspect ratio: 3 is not a factor of both the numerator (810) and the denominator (922) of the aspect ratio of 810/922. To see why this procedure is an application of the distributive law, work through 810 as an example, and note that numbers such as 9, 99, 999 are always divisible by 3 (conversation with W. C. Arlinghaus): 810 = 8*100 + 1*10 + 0*1 --- first use of distributive law = 8*(99+1) + 1*(9+1) + 0*1 --- partitioning of powers of ten into convenient summands = 8*99 + 8*1 + 1*9 + 1*1 + 0*1 --- second use of distributive law = 8*99 + 1*9 + 8*1 + 1*1 + 0*1 --- commutative law of addition (a+b=b+a) notice that the sum 8*99 + 1*9 is divisible by 3 because 9 and 99 are divisible by 3: 8*99 + 1*9 = (8*33 + 1*3)*3--third use of the distributive law, so now, 810 = (8*33 + 1*3)*3 + (8*1 + 1*1 + 0*1) and for the number 810 to be divisible by 3 (fourth use of the distributive law) all that is thus required is for the right summand in parentheses, 8*1 + 1*1 + 0*1 to be divisible by 3. Hence, the rule of three, for determining whether or not a given integer is divisible by 3 becomes clear. A corresponding rule of nine is not difficult to understand, as are numerous other shortcuts for determining divisibility criteria. Clearly, one need test candidate divisors only up to the square root of the number in question. However, when one is faced with an image on the screen, it would be nice not to have taken the trouble (however little) of finding that 810 is divisible by 3, only to find that 922 is not. A far better approach is to rewrite each number using some systematic procedure and then compare a pair of expressions to determine, all at once, which numbers are divisors of BOTH 810 and 922. For this purpose, the Fundamental Theorem of Arithmetic is critical. FUNDAMENTAL THEOREM OF ARITHMETIC Any positive integer can be expressed uniquely as a product of powers of prime numbers (numbers with no integral divisors other than themselves and one). Thus, in the AGS logo example, 810=2*405=2*5*81=2*3*3*3*3*5 (superscripts avoided by repetitive multiplication) and, 922=2*461. The number 810 was easy to reduce to its unique factorization into powers of primes; one might not know whether or not 461 is a prime number or whether further reduction is required to achieve the prime power factorization of 922. To this end, the Sieve of Eratosthenes (the same Librarian at Alexandria who measured the circumference of the Earth) works well. To use the sieve, simply test the prime numbers less than the square root of the number in question. The square root of 461 is about 21.47. So, the only primes that can possibly be factors are: 2, 3, 5, 7, 11, 13, 17, and 19. Clearly, 2, 3, and 5 are not factors of 461. A minute or two with a calculator shows that 7, 11, 13, 17, and 19 are also not factors of 461. Notice that these calculations need only be made once--when one has the unique factorization all divisors of both numbers are known from looking at the two factorizations, together. Thus: 810 = 2*3*3*3*3*5 922 = 2 * 461 and 2 is the only factor common to both numbers. PRESERVATION OF THE ASPECT RATIO Because the only factor the two numbers 810 and 922 have in common is 2 it follows that the only scaling factor that can be used, THAT WILL PRESERVE THE ASPECT RATIO, is 1/2. Had 1/3 or some other ratio been employed, a distorted view of the AGS logo would have been the result. Figure 4 shows the result of using a scaling factor of 1/10, so that the image is 81 pixels wide and 92 pixels high (rounded off from 92.2 pixels). The image looks good, but in fact the aspect ratio was not preserved. Level of sensitivity to image distortion will vary with the individual; it is important to have absolute techniques that guarantee correct answers. Once one knows them, then one can choose when, and when not, to violate them.

Figure 4. Logo of the American Geographical Society, height and width divided by 10, aspect ratio not preserved. Distortion present but not evident.

Reduction of the AGS logo by 1/2, producing an image 1/4 of the original size, may still not be desirable. One can use the unique factorization to build what is desired. The value of 810 has a number of factors; thus, one might choose to rescan the image, holding the width at 810 pixels and shaving just a bit off the height (there appears to be room to do so in the background without touching the line drawing of the globe). Now, to create the possibility of various scaling factors, consider a tiny sliver removed to create a height of 920 = 2*2*2*5*23; there is an extra factor of 5 that 920 has in common with 810 that 922 did not, so all of 1/2, 1/5, and 1/10 (one over 2*5) are scaling factors that will preserve the aspect ratio. However, one might wish still more possible scaling factors; if a slightly larger sliver can be removed, so that the height of the scanned image is 900 pixels, with 900 = 2*2*3*3*5*5, then 810 and 900 have common prime power factors of 2, 3, 3*3, and 5, so that the set of scaling factors has been substantially expanded to include all of: 1/2, 1/3, 1/9, 1/5, combination of four things one at a time--4!/(1!*3!) 1/6, 1/18, 1/10; 1/27, 1/15; 1/45 combination of four things two at a time--4!/(2!*2!) 1/54, 1/30, 1/90, 1/135 combination of four things three at a time--4!/(3!*1!) 1/270 combination of four things four at a time--4!/(4!*0!). This set of values for the height, in pixels, should offer enough choices in various size ranges for the stationery to have a logo of reasonable size on it. IMAGE SECURITY In creating images, it is useful to have an aspect ratio with numerator and denominator with many common factors; greater flexibility in changing the image is a consequence. However, there may be situations, particularly on the Internet where downloading is just a right-click of the mouse away, in which one wishes to inhibit easy rescaling of an image. When that is the case, unique factorization via the Fundamental Theorem of Arithmetic again yields an answer: choose to scan the image so that the numerator and denominator of the aspect ratio are relatively prime--that is, so that numerator and denominator have no factors in common other than 1. In the case of the AGS logo, altering the aspect ratio from the original of 810/922 to 810/923 would serve the purpose: 923 is not divisible by any of 2, 3, or 5. Anyone downloading the AGS logo could not resize it without distortion: distortion, for once, becomes the mapmaker's friend. FINALE The abstract lessons learned in Modern Algebra are as important in the electronic world as they are elsewhere; indeed, the current technical realm therefore offers a refreshing host of new example to motivate theory, as suggested in this simple scaling example that drew on a variety of algebraic concepts including: the distributive law, the commutative law of addition, the Fundamental Theorem of Arithmetic and the needed associated concept of prime number, facts involving square roots and divisors, the Sieve of Eratosthenes, basic material on permutations and combinations together with the convention that 0!=1, and the idea of relatively prime numbers. The web is a rich source of example when one brings a rich source of abstract liberal arts training to it. CITATIONS Bird, M. L. Executive Director, American Geographical Society, May, 1996. REFERENCES For the reader wishing to learn more about modern algebra--not easy reading for most, but well worth looking at to gain some appreciation for the vastness of this field. Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra. Macmillan, Herstein, I., N. Topics in Algebra. Mac Lane, S. and Birkhoff, G. Algebra.

U.S. ROUTE 12 BUFFER Daniel Jacobs The University of Michigan
A child's turf is an ordered space in which the child plots data points at important spots. I will use maps generated on a geographical information system (Atlas GIS for Windows, v. 3.0) as one set of tools to broaden student perspective. The student population will be mainly 9-10 year old students from urban and suburban settings. I hope to initially probe three areas of student perspectives: a. Their sense of space immediately around them in their daily lives, including their school, home, block, city, county, and state; b. Their sense of the order of the physical world around them, including cardinal directions, directions of major roads and highways, positions of buildings, cities, and states along U.S. 12 (base map in Figure 1); c. Their knowledge of the people, buildings, and stories behind their known turf (sample of a map set shown in Figure 2). I am most interested in finding where different student maps end, where their Unknown begins, and what they think is beyond.

Figure 1. U.S. 12 county buffer.

Figure 2. Number of German speakers by county; U.S. 12 buffer outlined in magenta. The U.S. Census measures 24 distinct language categories under the heading of "Languages Spoken at Home." To ensure legibility, I have divided the U.S. into four sections and am generating four maps for each of the measured language groups. The ultimate goal of the Route 12 project is to organize stuednts, teachers, schools, and institutions along Route 12 into a coherent education community. NOTE: Teachers and students from Detroit have already used the maps and student information to visit classes in schools across Michigan. The Detroit students and their travels will serve as a template for the development of one to two day field trips in which students from Detroit travel to other route 12 schools and exchange historical and cultural information. Studnets in Detroit have been interviewing individuals who speak languages other than Enlish at home. Coupled with the maps I have generated, the interviews are intended to help students gain a sense of the variety of languages, nationalities, and ethnicities represented within their own neighborhoods.

4. INDEX TO VOLUMES I (1990) TO VOL. VI -------------------- Vol. VI, No. 2, December, 1995. TABLE OF CONTENTS Elements of Spatial Planning: Theory--Part I. Sandra L. Arlinghaus MapBank: An Atlas of On-line Base Maps Sandra L. Arlinghaus International Society of Spatial Sciences Volume VI, Number 1, June, 1995. Fifth Anniversary of Solstice New format for Solstice and new Technical Editor Richard Wallace. Motor Vehicle Transport and Global Climate Change: Policy Scenarios. Expository Article. Discrete Mathematics and Counting Derangements in Blind Wine Tastings. Sandra L. Arlinghaus, William C. Arlinghaus, John D. Nystuen -------------------- Volume V, No. 2, Winter, 1994. Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary: The Paris Metro: Is its Graph Planar? Planar graphs; The Paris Metro; Planarity and the Metro; Significance of lack of planarity. Sandra Lach Arlinghaus: Interruption! Classical interruption in mapping; Abstract variants on interruption and mapping; The utility of considering various mapping surfaces--GIS; Future directions. Reprint of Michael F. Dacey: Imperfections in the Uniform Plane. Forewords by John D. Nystuen. Original (1964) Nystuen Foreword; Current (1994) Nystuen Foreword; The Christaller spatial model; A model of the imperfect plane; The disturbance effect; Uniform random disturbance; Definition of the basic model; Point to point order distances; Locus to point order distances; Summary description of pattern; Comparison of map pattern; Theoretical model; Point to point order distances; Locus to point order distances; Summary description of pattern; Comparison of map pattern; Theoretical order distances; Analysis of the pattern of urban places in Iowa; Almost periodic disturbance model; Lattice parameters; Disturbance variables; Scale variables; Comparison of M(2) and Iowa; Evaluation; Tables. Sandra L. Arlinghaus: Construction Zone: The Brakenridge-MacLaurin Construction. William D. Drake: Population Environment Dynamics: Course and Monograph--descriptive material. ---------------------------- Volume V, No. 1, Summer, 1994. Virginia Ainslie and Jack Licate: Getting Infrastructure Built. Cleveland infrastructure team shares secrets of success; What difference has the partnership approach made; How process affects products--moving projects faster means getting more public investment; difference has the partnership approach made; How process affects products--moving projects faster means getting more public investment; How can local communities translate these successes to their own settings? Frank E. Barmore: Center Here; Center There; Center, Center Everywhere. Abstract; Introduction; Definition of geographic center; Geographic center of a curved surface; Geographic center of Wisconsin; Geographic center of the conterminous U.S.; Geographic center of the U.S.; Summary and recommendations; Appendix A: Calculation of Wisconsin's geographic center; Appendix B: Calculation of the geographical center of the conterminous U.S.; References. Barton R. Burkhalter: Equal-Area Venn Diagrams of Two Circles: Their Use with Real-World Data General problem; Definition of the two-circle problem; Analytic strategy; Derivation of B% and AB% as a function of r(B) and d(AB). Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary, John D. Nystuen. Los Angeles, 1994 -- A Spatial Scientific Study. Los Angeles, 1994; Policy implications; References; Tables and complicated figures. -------------------- Volume IV, No. 2, Winter, 1993. William D. Drake, S. Pak, I. Tarwotjo, Muhilal, J. Gorstein, R. Tilden. Villages in Transition: Elevated Risk of Micronutrient Deficiency. Abstract; Moving from traditional to modern village life: risks during transtion; Testing for elevated risks in transition villages; Testing for risk overlap within the health sector; Conclusions and policy implications Volume IV, No. 1, Summer, 1993. Sandra L. Arlinghaus and Richard H. Zander: Electronic Journals: Observations Based on Actual Trials, 1987-Present. Abstract; Content issues; Production issues; Archival issues; References John D. Nystuen: Wilderness As Place. Visual paradoxes; Wilderness defined; Conflict or synthesis; Wilderness as place; Suggested readings; Sources; Visual illusion authors. Frank E. Barmore: The Earth Isn't Flat. And It Isn't Round Either: Some Significant and Little Known Effects of the Earth's Ellipsoidal Shape. Abstract; Introduction; The Qibla problem; The geographic center; The center of population; Appendix; References. Sandra L. Arlinghaus: Micro-cell Hex-nets? Introduction; Lattices: Microcell hex-nets; References Sandra L. Arlinghaus, William C. Arlinghaus, Frank Harary: Sum Graphs and Geographic Information. Abstract; Sum graphs; Sum graph unification: construction; Cartographic application of sum graph unification; Sum graph unification: theory; Logarithmic sum graphs; Reversed sum graphs; Augmented reversed logarithmic sum graphs; Cartographic application of ARL sum graphs; Summary. -------------------- Volume III, No. 2, Winter, 1992. Frank Harary: What Are Mathematical Models and What Should They Be? What are they? Two worlds: abstract and empirical; Two worlds: two levels; Two levels: derivation and selection; Research schema; Sketches of discovery; What should they be? Frank E. Barmore: Where Are We? Comments on the Concept of Center of Population. Introduction; Preliminary remarks; Census Bureau center of population formulae; Census Bureau center of population description; Agreement between description and formulae; Proposed definition of the center of population; Summary; Appendix A; Appendix B; References. Sandra L. Arlinghaus and John D. Nystuen: The Pelt of the Earth: An Essay on Reactive Diffusion. Pattern formation: global views; Pattern formation: local views; References cited; Literature of apparent related interest. Volume III, No. 1, Summer, 1992. Harry L. Stern: Computing Areas of Regions with Discretely Defined Boundaries. Introduction; General formulation; The plane; The sphere; Numerical examples and remarks; Appendix--Fortran program. Sandra L. Arlinghaus, John D. Nystuen, Michael J. Woldenberg: The Quadratic World of Kinematic Waves. -------------------- Volume II, No. 2, Winter, 1991. Reprint of Saunders Mac Lane: Proof, Truth, and Confusion, The Nora and Edward Ryerson Lecture at The University of Chicago in 1982. The fit of ideas; Truth and proof; Ideas and theorems; Sets and functions; Confusion via surveys; Cost-benefit and regression; Projection, extrapolation, and risk; Fuzzy sets and fuzzy thoughts; Compromise is confusing. Robert F. Austin: Digital Maps and Data Bases: Aesthetics versus accuracy. Introduction; Basic issues; Map production; Digital maps; Computerized data bases; User community. Volume II, No. 1, Summer, 1991. Sandra L. Arlinghaus, David Barr, John D. Nystuen: 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 Solstice, Vol. I, No. 1.). Sandra L. Arlinghaus: Construction Zone--The Logistic Curve. Educational feature--Lectures on Spatial Theory. -------------------- Volume I, No. 2, Winter, 1990. John D. Nystuen: 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, Geographical Review, April, 1974) in the Detroit metropolitan regions of 1974. As with Clifford's Postulates, reprinted in the last issue of Solstice, note the timely quality of many of the observations. Sandra Lach Arlinghaus: 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. Sandra Lach Arlinghaus: 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. Sandra L. Arlinghaus, William C. Arlinghaus, and John D. Nystuen: 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, 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 entries when the underlying adjacency pattern of the road network was altered by the 1989 earthquake that closed the San Francisco--Oakland Bay Bridge. Sandra Lach Arlinghaus: Fractal Geometry of Infinite Pixel Sequences: "Super-definition" Resolution? Comparison of space-filling qualities of square and hexagonal pixels. Sandra Lach Arlinghaus: Construction Zone--Feigenbaum's number; a triangular coordinatiztion of the Euclidean plane; A three-axis coordinatization of the plane. Volume I, No. 1, Summer, 1990. Reprint of William Kingdon Clifford: Postulates of the Science of Space. This reprint of a portion of Clifford's lectures to the Royal Institution in the 1870s suggests many geographic topics of concern in the last half of the twentieth century. Look for connections to boundary issues, to scale problems, to self-similarity and fractals, and to non-Euclidean geometries (from those based on denial of Euclid's parallel postulate to those based on a sort of mechanical `polishing'). What else did, or might, this classic essay foreshadow? Sandra Lach Arlinghaus: Beyond the Fractal. 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. William C. Arlinghaus: Groups, Graphs, and God. Sandra L. Arlinghaus: Theorem Museum--Desargues's Two Triangle Theorem from projective geometry. Construction Zone--centrally symmetric hexagons. ------------------------------------------------------------------------