\hsize = 6.5 true in %THE FIRST 14 LINES ARE TYPESETTING CODE. \input fontmac %delete to download, except on Univ. Mich. (MTS) equipment. \setpointsize{12}{9}{8}%same as previous line; set font for 12 point type. \parskip=3pt \baselineskip=14 pt \mathsurround=1pt \headline = {\ifnum\pageno=1 \hfil \else {\ifodd\pageno\righthead \else\lefthead\fi}\fi} \def\righthead{\sl\hfil SOLSTICE } \def\lefthead{\sl Summer, 1991 \hfil} \def\ref{\noindent\hang} \font\big = cmbx17 \font\tn = cmr10 \font\nn = cmr9 %The code has been kept simple to facilitate reading as e-mail \centerline{\big SOLSTICE:} \vskip.5cm \centerline{\bf AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS} \vskip5cm \centerline{\bf SUMMER, 1991} \vskip12cm \centerline{\bf Volume II, Number 1} \smallskip \centerline{\bf Institute of Mathematical Geography} \vskip.1cm \centerline{\bf Ann Arbor, Michigan} \vfill\eject \hrule \smallskip \centerline{\bf SOLSTICE} \line{Founding Editor--in--Chief: {\bf Sandra Lach Arlinghaus}. \hfil} \smallskip \centerline{\bf EDITORIAL BOARD} \smallskip \line{{\bf Geography} \hfil} \line{{\bf Michael Goodchild}, University of California, Santa Barbara. \hfil} \line{{\bf Daniel A. Griffith}, Syracuse University. \hfil} \line{{\bf Jonathan D. Mayer}, University of Washington; joint appointment in School of Medicine.\hfil} \line{{\bf John D. Nystuen}, University of Michigan (College of Architecture and Urban Planning).} \smallskip \line{{\bf Mathematics} \hfil} \line{{\bf William C. Arlinghaus}, Lawrence Technological University. \hfil} \line{{\bf Neal Brand}, University of North Texas. \hfil} \line{{\bf Kenneth H. Rosen}, A. T. \& T. Bell Laboratories. \hfil} \smallskip \line{{\bf Business} \hfil} \line{{\bf Robert F. Austin}, Director, Automated Mapping and Facilities Management, CDI. \hfil} \smallskip \hrule \smallskip The purpose of {\sl Solstice\/} is to promote interaction between geography and mathematics. Articles in which elements of one discipline are used to shed light on the other are particularly sought. Also welcome, are original contributions that are purely geographical or purely mathematical. These may be prefaced (by editor or author) with commentary suggesting directions that might lead toward the desired interaction. Individuals wishing to submit articles, either short or full-- length, as well as contributions for regular features, should send them, in triplicate, directly to the Editor--in--Chief. Contributed articles will be refereed by geographers and/or mathematicians. Invited articles will be screened by suitable members of the editorial board. IMaGe is open to having authors suggest, and furnish material for, new regular features. \vskip2in \noindent {\bf Send all correspondence to:} \vskip.1cm \centerline{\bf Institute of Mathematical Geography} \centerline{\bf 2790 Briarcliff} \centerline{\bf Ann Arbor, MI 48105-1429} \vskip.1cm \centerline{\bf (313) 761-1231} \centerline{\bf IMaGe@UMICHUM} \vfill\eject This document is produced using the typesetting program, {\TeX}, of Donald Knuth and the American Mathematical Society. Notation in the electronic file is in accordance with that of Knuth's {\sl The {\TeX}book}. The program is downloaded for hard copy for on The University of Michigan's Xerox 9700 laser-- printing Xerox machine, using IMaGe's commercial account with that University. Unless otherwise noted, all regular features are written by the Editor--in--Chief. \smallskip {\nn Upon final acceptance, authors will work with IMaGe to get manuscripts into a format well--suited to the requirements of {\sl Solstice\/}. Typically, this would mean that authors would submit a clean ASCII file of the manuscript, as well as hard copy, figures, and so forth (in camera--ready form). Depending on the nature of the document and on the changing technology used to produce {\sl Solstice\/}, there may be other requirements as well. Currently, the text is typeset using {\TeX}; in that way, mathematical formul{\ae} can be transmitted as ASCII files and downloaded faithfully and printed out. The reader inexperienced in the use of {\TeX} should note that this is not a ``what--you--see--is--what--you--get" display; however, we hope that such readers find {\TeX} easier to learn after exposure to {\sl Solstice\/}'s e-files written using {\TeX}!} {\nn Copyright will be taken out in the name of the Institute of Mathematical Geography, and authors are required to transfer copyright to IMaGe as a condition of publication. There are no page charges; authors will be given permission to make reprints from the electronic file, or to have IMaGe make a single master reprint for a nominal fee dependent on manuscript length. Hard copy of {\sl Solstice\/} 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--as a kind of academic newsstand at which one might browse, prior to making purchasing decisions. Whether or not it will be feasible to continue distributing complimentary electronic files remains to be seen. Thank you for participating in this project focussing on environmentally-sensitive publishing.} \vskip.5cm Copyright, June, 1991, Institute of Mathematical Geography. All rights reserved. \vskip1cm ISBN: 1-877751-52-9 \vfill\eject \centerline{\bf SUMMARY OF CONTENT} \smallskip {\bf 1. FULL--LENGTH ARTICLE.} \smallskip \noindent Sandra L. Arlinghaus, David Barr, John D. Nystuen. \smallskip \noindent {\bf 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 context based on the postulates of the science of space of William Kingdon Clifford (reprinted in {\sl Solstice\/}, Vol. I, No. 1). \smallskip {\bf 2. REGULAR FEATURES} \smallskip \item{i.} {\bf Construction Zone} --- The logistic curve \item{ii.} {\bf Educational feature} --- Lectures on ``Spatial Theory." \smallskip \vfill\eject \centerline{THE SPATIAL SHADOW: LIGHT AND DARK---WHOLE AND PART} \smallskip \centerline{``Life's but a walking shadow"} \centerline{Shakespeare, {\sl Macbeth\/}.} \smallskip \centerline{Sandra L. Arlinghaus, David Barr, John D. Nystuen} \smallskip \line{\bf Introduction \hfil} Sunlight and shadow, day and night, solstice and equinox, lunar and solar eclipse--all are astronomical events that transform the surface of the earth into an event focused on the contrast between light and dark. The diurnal dynamics of the sweeping edge of the darkness are a foundation critical to the well-being of life on earth. Artistic expressions are numerous, ranging from Amish quilt patterns (``sunlight and shadow") to Indonesian shadow puppets. From a spatial standpoint, the mantle of night serves as a continuum linking disparate elements of the earth's surface; it is a whole composed of unseen parts. \smallskip \centerline{WHOLE AND PART:} \centerline{\bf A Sculptural Unification of Unseen Parts.} \smallskip \line{The Four Corners Project \hfil} ``The Four Corners Project," conceived in 1976 and completed in 1985, consists of an invisible tetrahedron spanning the inside of the earth with the four separate corners, made of marble, protruding from the crust of the earth (Figure 1). [1] These individual marble corner-markers (each about four inches high) were positioned in Easter Island, South Africa, Greenland, and New Guinea, with imaginary planes extending through the earth from each corner to the other three. The length of the imaginary line planned to link each pair of terrestrial markers is approximately 6465 miles. [2] One must know what a tetrahedron looks like and expand the scale of this knowledge to the scale of the entire earth to view this sculpture. In this respect, the art follows the pattern of the natural astronomical, global patterns of light and dark that require some sort of global perspective to envision a whole created from disparate unseen parts. \topinsert \vskip6in \noindent {\bf Figure 1.} The Four Corners Project. Four marble tetrahedra, each 4 inches high, mark the corners of a suggested, invisible, tetrahedron inscribed in the earth. Side length of the suggested large tetrahedron is about 6465 miles. Marker locations are in Easter Island, South Africa, Greenland, and New Guinea. \endinsert This tetrahedron is larger than proximate space. It is an abstraction that can be appreciated, as a whole, only in the mind; images of it created visually through written, printed, and verbal records encompass a broader view of it than does any collection of images taken from arbitrary physical vantage points in the universe. It is a shared perception, transcending language, that spans the minds of those who participate. [3] It requires abstract visualization, rather than physical vision, to ``see" the entire sculpture. This sculpture creates a conceptual unit from discrete parts that coalesces the evolutionary sequence of constructivistic, structurist art as well as the philosophical concerns of Zen gardens. In the structurist vocabulary, the art work draws the physical eye from one discrete component to another, and the unity of the work is revealed through the relationships of the components rather than through singular objects. In an early effort (1934), Henry Moore (``Four-Piece Composition") used the negative space of the sculpture to draw the physical eye, in proximate space, from one discrete component to another in order to suggest a single reclining figure [4]. The Zen garden at Ryoan-ji has stones arranged deliberately so that the whole can never be totally seen from a single perspective. Thus, the viewer, as in the Four Corners Project, must always be in a less than ``divine" physical, perceptual position. Structurist reliefs emphasize the relationships among parts rather than the characterization of the parts themselves; [5] in this regard, ``Four Corners" is a structurist concept at a global scale. In all of these cases, the unity of the entire piece unfolds naturally only when a leap of the imagination gives wholeness to the sculpture--whether that leap is in proximate or global space. \smallskip \line{Geographical Background of the Four Corners \hfil} Barr fixed the general positions for the four corners on landmasses, using a globe and dividers; Nystuen pin-pointed each, using rotation matrices to align the North-South pole-based graticule with one using Easter Island and its antipodal point in the Thar Desert as poles. [6] Easter Island was chosen as the initial corner on account of its numerous cultural connections to the history of sculpture. Embedding this tetrahedron in the earth-sphere (using the Clarke ellipsoid circumference of 24,873.535 miles [7]) required theoretical assumptions but also reflected the empirical facts of land/water distribution on earth--no corner was to be submerged in a lake or ocean. The environment and local surface materials surrounding the chosen corners are apt--from the igneous rock below a volcanic island, to the granitic sand in a desert, to the crystalline forms in an ice cap, to the organic material of a mangrove swamp. Indeed, the choice of the tetrahedron within the earth-sphere intentionally reflects the structure of the carbon atom as a fundamental component of life. In 1980, Barr began to place the vertices of the tetrahedron; Table 1 shows the itinerary. The process that led to the completed product in 1985 involved the participation, from initial struggle to eventual respect and acceptance, of people from backgrounds not usually linked to the world of art: African veldt farmers, Eskimos, Irian Jayan missionaries, soldiers, police, politicians, and diplomats (for example, Table 1 shows the names of most of the airplane pilots who participated in the placement of these corners--they suggest the rich diversity of peoples associated with various aspects of this project). \topinsert \hrule \smallskip \centerline{\bf TABLE 1} \centerline{Log of travels associated with placement of the four corners} \centerline{Listing compiled by Heather and Gillian Barr.} \smallskip \settabs\+\qquad\qquad&Frobisher to Sonderstrom Fjord\qquad&\cr \+&DESTINATIONS &NAME OF AIRPLANE CAPTAIN\cr \smallskip \hrule \smallskip \centerline{DECEMBER AND JANUARY, 1980-81:} \centerline{MACHU PICCHU, EASTER ISLAND, AND SOUTH AFRICA} \smallskip \+&Detroit to Miami &John Bosh\cr \+&Miami to Lima &Dick Rudman\cr \+&Lima to Cuzco &Hugo Bisso\cr \+&Cuzco to Lima &Eduardo Camino\cr \+&Lima to Santiago &Javier Mesa\cr \+&Santiago to Easter Island &Alphonso Estay\cr \+&Easter Island to Santiago &Gustavo Vila\cr \+&Santiago to Buenos Aires &Sergio Kurth\cr \+&Buenos Aires to Cape Town &Carlos Bustamante\cr \+&Cape Town to Johannesburg &Steev Kaup\cr \+&Johannesburg to New York &Tony Laas\cr \+&New York to Detroit &Hal Grenddin\cr \smallskip \centerline{ JULY, 1981} \centerline{ GREENLAND} \smallskip \+&Windsor to Montreal &Mr. Golze\cr \+&Montreal to Frobisher &Mr. Savage\cr \+&Frobisher to Sonderstrom Fjord &Patty Doyle\cr \+&Sonderstrom Fjord to Ice Cap &Patty Doyle\cr \+&Ice Cap to Sonderstrom &Patty Doyle\cr \+&Sonderstrom to Godthab (Nuuk) &Patty Doyle\cr \+&Godthab to Frobisher &Sven Syversen\cr \+&Frobisher to Montreal &Carl Gitto\cr \+&Montreal to Windsor &Louis Ghyrmothy\cr \smallskip \centerline{ JANUARY, 1985} \centerline{ IRIAN JAYA} \smallskip \+&Djajpura to Danau Bira &Poambang Kuncaro a.k.a.\cr \+&\phantom{}& &``Bang Bang Koon"\cr \+&Danau Bira to Djajpura &Bang Bang Koon\cr \+&Djajpura to Biac &Mr. Fujiono\cr \+&Biac to Ujung Pandang &Mr. Darynato\cr \+&Ujung Pandang to Bali &Angus Tiansyah\cr \+&Bali to Djakarta &Mr. Sunarto\cr \+&Djakarta to Singapore &Mr. Tan\cr \smallskip \hrule \endinsert In December of 1980, Barr and his party (which included other fine artists and a professional dancer) went to Machu Picchu, where the tetrahedral marble pinnacles were washed at the ancient ceremonial site (at the sundial called ``ini-huatana" (``hitching post of the sun")), prior to placement in the ground. From there they went to Easter Island; surveying equipment of William Mulloy [8], a member of Thor Heyerdahl's expedition to that island, was used to place the first vertex of the tetrahedron on January 4, 1981 (Table 2), one minute of longitude from the calculational center of $109^{\circ} 25'30"$. This location has elevation just above sea level and is in a former leper colony. \topinsert \hrule \smallskip \centerline{\bf TABLE 2} \centerline{Geographic coordinates of the Four Corners} \smallskip \hrule \smallskip \settabs\+\qquad\qquad&New Guinea [planned]\qquad\quad &$27^{\circ} 06'20"$ S\qquad\quad &$109^{\circ} 24'30"$ W&\cr \+&Site &Latitude &Longitude\cr \smallskip \hrule \smallskip \+&Easter Island &$27^{\circ} 06'20"$ S &$109^{\circ} 24'30"$ W\cr \+&South Africa &$27^{\circ} 30'36"$ S &$024^{\circ} 06'00"$ E\cr \+&Greenland &$72^{\circ} 38'24"$ N &$041^{\circ} 55'12"$ W\cr \+&New Guinea [actual] &$02^{\circ} 20'50"$ S &$138^{\circ} 00'00"$ E\cr \+&New Guinea [planned]&$02^{\circ} 06'36"$ S &$137^{\circ} 23'24"$ E\cr \smallskip \hrule \endinsert From Easter Island the group traveled to South Africa, where on January 11, 1981 the second vertex was placed on a farm called Karee Boom (near the town of Reivilo--see Table 2) using detailed maps of this region in South Africa between the Kimberly diamond mines and the Kalahari Desert (elevation above 1200 meters). Black and white people from the indigenous population sat together in harmony on this South African farm watching the ceremonial placement of this second corner. Part of that celebration included the second stage of an evolving dance, ``The Four Corners Dance," written to commemorate this sculpture. [9] On July 19, 1981, Barr flew from the inhabited shoreline of Greenland, to a position high on the icecap; only Barr, one art colleague, and the pilot could make the trip in the bush airplane with skis. They placed this vertex, with the aid of the plane's Loran navigation equipment (see Table 2), on the Greenland icecap at an elevation of over 3200 meters. The three fled after less than one hour, as a bitter storm formed around them; the storm, combined with the thin air of the high elevation, forced them to jettison extra gear and fuel. The final vertex was placed in Irian Jaya (New Guinea) in January of 1985, after years of struggle with a maze of political regulations. Because of this struggle, this vertex could not be placed at precisely the planned location (see Table 2) and was positioned, instead, at a site about 45 miles (great circle distance) from the planned site. The actual site in New Guinea is at an elevation of over 150 meters. In the end, the political barriers formed by the Indonesian government proved more difficult than any to overcome--even the systematic error introduced by using the sphere rather than the ellipsoid, the travel logistics, the differences in elevation, and an initial blunder causing the Easter Island vertex to be placed at one minute of longitude due east of the calculational center. The Indonesian political concerns forced the largest distortion of the abstract tetrahedron. The ``Four Corners Project" is a real sculpture, in place, close to an ideal in an imperfect world. Indeed, when a model of the global sculpture is produced in proximate space, these imperfections are imperceptible. It is only with our imaginations that we can appreciate the difference between the ideal and terrestrial forms. \smallskip \centerline{Mathematical Uniqueness of the Four Corners} \centerline{---Extensions of the idea} When spherical trigonometry was applied to a map showing all landmasses whose antipodal points are also land-based (Figure 2) [10] it was possible to prove that the choice of a tetrahedron as a shape for this sculpture is unique within the set of regular polyhedra called ``Platonic" solids. [11] Plato linked the set of five regular polyhedra (tetrahedron, cube, octahedron--polyhedron with eight triangular faces, dodecahedron--polyhedron with twelve pentagonal faces, and icosahedron--polyhedron with twenty triangular faces) with five basic components from which he believed the earth to have been formed. [12] No Platonic solid, other than the tetrahedron, can be embedded in the earth with all corners on land, one of which is on Easter Island. [13] \topinsert \vskip7.5in Figure 2. Terrae Antipodum. Dark areas represent landmasses whose antipodal points are on land. Fragmented antipodal landmasses (archipelagos) are encircled by dashed lines. Antipodal continental outlines are shown (where needed to understand the map) over the ocean as dashed lines. The base map is a Peters projection. The equator bisects the vertical neat line. This map was used to establish uniqueness of the choice of a tetrahedron within Barr's constraints. \endinsert It also follows from the mathematics that, although the tetrahedron is unique as a choice, there are an infinite number of possible positions in which it might have been oriented within the earth (Figure 3). The possibilities for the corners other than Easter Island are, however, tightly constrained within the arcs of the circle of ``latitude" (centered on C, the antipodal point of Easter Island in the Thar Desert) shown in Figure 3. (An azimuthal equidistant projection was used because distances measured from the center are true.) Once a point is chosen within one of these arcs as a corner site, the choices for the other two corners are forced (as the remaining vertices of an equilateral triangle inscribed in the circle of ``latitude"). [14] These three sites form the triangular base of a tetrahedron with Easter Island (unseen in Figure 3) at the apex of the solid, on the other side of the earth from the center of the circle in Figure 3. \topinsert \vskip8in Figure 3. Shaded intervals show all possible land-based locations for three corners of the base of a tetrahedon with Easter Island as apex of the solid inscribed in the earth. Easter Island is antipodal to the center of the circle, C. The base map is an azimuthal equidistant projection. Any distance measured from the center, C, is true. \endinsert The after-the-fact discoveries that the choice of the tetrahedron was unique within the set of Platonic solids, and that the extent of infinite ``play" in site selection could be constrained within specified bounded intervals, enhance the planned selection of Easter Island as the choice for the initial vertex of the tetrahedron. Indeed, other choices were considered as an initial vertex; however, the idea of using this tiny patch of land in the Pacific hemisphere as the anchor for this ``titanic" tetrahedron of terrestrial sites, not only proved possible, but irresistible as well. \smallskip \centerline{LIGHT AND DARK:} \centerline{\bf A problem of boundary.} Natural boundaries, such as those between water and land, are often crenulated and complex. Many words are necessary to translate a natural boundary into a cadastral survey description. At places where the abstract and natural boundaries intersect, interesting arrangements can arise. \smallskip \line{\sl SunSweep \hfil} {\sl SunSweep\/} is a sculpture in three separate locations along the U.S./Canadian border that was designed to commemorate the peaceful interaction across this border. Its three parts are located at places where natural and abstract boundaries intersect. The western terminus is on a bit of U.S. territory which can only be reached, on land, by passing through Canada. The eastern terminus is on a bit of Canadian territory which can only be reached, on land, by passing through the United States. Thus, a nice symmetry is created by the intersection of a natural and an abstract boundary; this symmetry is intentionally reflected in the choices for the locations and in the physical shapes of the elements of the SunSweep sculpture (Figure 4). The sculpture represents the arch of the sun in the sky from east to west. Coincidentally, perhaps, Barr noted a common social outlook among the people inhabiting these anomalous locations-- they appeared to share a kind of independence coming from this blurred boundary, suggesting a unity in social perspective associated with this sculpture. \topinsert \vskip7.5in Figure 4. SunSweep. The 5-foot high earth-markers set out on a lawn, prior to placement along the U.S./Canada border. \endinsert \line{Geographical Background of {\sl SunSweep} \hfil} The eastern-most piece, arching inland, is situated on Campobello Island in New Bruns\-wick; the western-most piece, also arching inland, is on Point Roberts in the State of Washington; and, the keystone of the arch, composed of two separate stone elements, is on an island in the Lake-of-the-Woods in Minnesota (Figure 5: a, b, c). Each piece is about five feet tall and is formed from selectively polished flame-finished black Canadian granite carved, in Michigan, from one mass. \topinsert \vskip7.5in Figure 5. a, b, c. Maps of the three SunSweep sites (a, New Brunswick; b, Minnesota; c, Washington) emphasizing interdigitation associated with anomalous locations along the U.S./Canada boundary. \endinsert These markers that trace the sweep of the sun across the celestial sphere were sited close to the U.S./Canadian border to commemorate the spirit of cooperation between these two countries. A hand print, suggesting ``I was here," has been lasered into the polished stone--a ``Canadian" print on one side pressing against its mirror image ``United States" print on the other side. The choice of locations for the sculpture suggests the path of the sun; they were selected with an eye to displaying the interplay of ideas between astronomical sweep and political boundary--as geographic ``boundary dwellers" in the world of art. [15] They were also selected for their characteristic of physically forcing (in terms of access) interdigitation between U.S. and Canadian boundaries. Thus Campobello Island, maintained as an International Park, is the site for the eastern piece; the arch is situated on Ragged Point (Table 3), a Canadian location accessible by road only through the United States. The trail leading to the sculpture is the ``SunSweep" Trail, formerly known as the ``Muskie Trail" and re-named at the suggestion of Senator Edmund Muskie of the State of Maine. The western-most piece of the arch is situated in Lighthouse Park on Point Roberts (Table 3), a United States community at the southern tip of a spit of land that is accessible (by land) only through Canada. American Point (Penasse Island), Minnesota, the northernmost U.S. island (Table 3) in the Lake-of-the-Woods (Lake situated on the U.S./Canadian border), is close to a U. S. peninsula which is accessible by land only through Canada; it is the site of the keystone for the arch in the locale referred to as ``Northwest Angle" which, other than those in Alaska, contains the only U.S. landmasses north of the 49th parallel. \midinsert \smallskip \hrule \smallskip \centerline{ TABLE 3} \centerline{ Geographic coordinates of SunSweep} \smallskip \hrule \smallskip \settabs\+\qquad\qquad&Campobello Island, NB\qquad\quad &44 50'10" N\qquad\quad &123 05'00" W&\cr \+&Site &Latitude &Longitude\cr \smallskip \hrule \smallskip \+&Campobello Island, NB &$44^{\circ}50'10"$ N&$066^{\circ}55'25"$ W\cr \+&Point Roberts, WA &$48^{\circ}58'23"$ N&$123^{\circ}05'00"$ W\cr \+&Lake-of-the-Woods, MN &$49^{\circ}21'45"$ N&$094^{\circ}57'40"$ W\cr \smallskip \hrule \endinsert Grooves lasered into the sides of one element of the keystone piece and the top edge of the sculpture offer visitors the opportunity to tie location to selected astronomical events. The top edge is angled so that a sunbeam is parallel to it on the summer solstice; a groove in one side is angled to align with the sun on both equinoxes; and, a groove on the other side is angled to align with the sun on the winter solstice. The shadows cast by a sunbeam at each astronomical event would suggest a tracing on the ground, with the succession of the seasons, in the shape of an analemma [16], calling to mind the equation of time and ultimately Kepler's Laws of planetary motion. [17] The second element of the Minnesota piece is aligned to the North Star. These markers were installed on the summer solstice of 1985. The alignments to the sun on this date and to the North Star appeared true. The pieces in New Brunswick and Washington were aligned subtly to each solstice and equinox position using the beveling planes of the granite and the orientation in the pattern of sited, smaller rocks surrounding the sculpture. The markers at each site have a bronze plaque set in the concrete base describing their metaphor. At the installation of the sculpture in Washington, the arch arrived broken and was cemented together as it was set into concrete in the ground. [18] Future generations who come across this irregular crack might wonder what it ``means," and whether or not it represents an alignment to some peculiar astronomical event. At best, it might be regarded as a remnant of a transportation system not geared to shipping heavy, brittle items with great success! The local citizenry is reconciled to the crack and in fact take delight in this sculpture as their ``Liberty Bell." \smallskip \line{Mathematical Extensions of the ideas behind {\sl SunSweep} \hfil} These three locations, selected initially for unique boundary characteristics, closely approximate ideal geometric placement along an arc of a great circle. A summary of how the actual measurements differ from the ``ideal" ones is shown in Table 4. The keystone location is, in fact, not halfway between the ends as one might hope for in a perfect arch. The great circle distance from the New Brunswick site to the Minnesota site is longer than the distance from the Minnesota site to the Washington site. \midinsert \hrule \smallskip \centerline{ TABLE 4} \centerline{Great Circle Distances between SunSweep Sites.} \smallskip \hrule \smallskip \settabs\+\qquad\qquad&Campobello Island to Lake-of-the-Woods\qquad\quad &1302.5]&\cr \+&Sites &Distance in miles\cr \smallskip \hrule \smallskip \+&Campobello Island to Lake-of-the-Woods &1347\cr \+&Lake-of-the-Woods to Point Roberts &1263\cr \+&[SUM: &2610]\cr \+&Campobello Island to Point Roberts &2605\cr \+&[Mid-point of entire great circle sweep &1302.5]\cr \smallskip \hrule \endinsert In addition, the three locations, as a set, do not lie along a single great circle; ideally, it might have been desirable to have them do so in order to keep the arch within a single plane passing through the earth's center. This sort of ideal arrangement was not possible, however, because of the requirement of interdigitation of U.S. and Canadian boundaries. Still, the actual placement of the markers is quite close to the ideal: the great circle distance from the New Brunswick location to the Washington location is 2605 miles--only 5 miles shorter than the sum of the component distances. Indeed, the midpoint of the great circle arc joining the New Brunswick location to the Washington location is at about 49 degrees 5 minutes North Latitude, 93 degrees 56 minutes West Longitude--a great circle distance of about 60 miles to a site east and slightly south of the actual location of the sculpted keystone. As was the case with ``Four Corners," the unity of the entire ``SunSweep" unfolds naturally only when a leap of the imagination gives wholeness to the sculpture; in this case that wholeness is suggested by a sequence of anomalous locations along a political boundary. Political boundaries are abstract and often simply defined, an advantage in conflict resolution. The ``Oregon Question" that agitated England and the United States for a generation was resolved during the James Polk administration (1846) by the simple agreement to extend the northwestern boundary along the 49th parallel from the Lake-of-the-Woods to the Pacific, [19] an arc of 1263 miles (great circle distance 1256 miles). Vancouver Island extends south of this line but the continental boundary ends where the 49th parallel reaches Puget Sound. The fact that the great circle distance between the western and middle sculpture sites rounds off to the same length as the length of the U.S./Canadian land border along the 49th parallel was unplanned in the sculpture. As was the case with the uniqueness of the choice of the tetrahedron for the Four Corners Project, this too was an after-the-fact discovery, linking both geography and mathematics to sculpture. \smallskip \centerline{THE SPATIAL SHADOW:} \line{\bf A theoretical framework. \hfil} The emergence of the after-the-fact discoveries surrounding these sculptures suggests the suitability of looking for theory to link the concepts underlying these particular art projects, much as poetry might be after-the-fact theory linking already- existing word-images. To do so, we draw on the interdisciplinary ties linking mathematics to geography, and linking both to art. Thus, we adopt a view in which mathematics includes the science of abstract space; in which geography ties this science of space to the real world; and, in which art offers abstract means to appreciate these ties. A set of postulates of the ``science of space" were created in the late nineteenth century by William Kingdon Clifford drawing only on common-sense notions of continuity and discreteness, flatness, magnification and contraction, and similarity, that formed part of the foundation of the non-Euclidean geometries at the base of modern physics. [20] By considering a set of fundamental relations, simply expressed, it became possible to analyze spatial relations in a fashion that did not rely solely on Euclid's postulates, and particularly not necessarily on Euclid's parallel postulate. [21] We consider a transformational approach to theory, echoing the emphasis of contemporary ``global" mathematics in seeking properties which remain invariant when carried via transformation from one space to another. It might be tempting to consider sunlight as a basic unit, because light coming through the sculpture is what links the geometry of the sculpture with the reality of the earth. With the sun at an ``infinite" distance from earth, its beams are parallel to each other (from our vantage point). Incoming solar radiation might therefore be considered an ``affine" transformation (in which sets of parallel lines are invariant) that maps elements protruding from the earth's surface as shadows onto the earth's surface (as in a structurist relief). [22] There are a number of appealing elements to this particular transformational approach. The affine transformation is the basis of much computer software for displaying graphics, suggesting a natural alignment of theory and computer mapping in order to merge the mathematics of sculptural structures with the spatial relations of the earth. [23] Because such an approach has the concept of affine transformation at its heart, however, it necessarily emphasizes the notion of parallelism. Our emphasis is, rather, on separate pieces whose relationship creates a single unit of art composed of separate parts intentionally devoid of interest in order to focus on that relationship, as (quotation attributed to Einstein) \smallskip \centerline{``History [Art] consists of relationships rather than events"} \centerline{A. Einstein.} \noindent It seems therefore, inappropriate to forge a linkage with theory based on parallelism. Far more suitable is to follow the lesson learned from Clifford and find basic elements that better match that which we seek to characterize. [24] The concept of shadow, rather than the affine transformation that creates the shadow, seems a better choice as a fundamental unit with which to work. Single spatial shadows (of physical objects) are discrete units of individual character; yet, they change in response to diurnal fluctuations, eventually to become united in a single nighttime continuum under the global spatial shadow of the earth on itself. Indeed, the concept of shadow, itself, also embodies the notion of transformation-- \smallskip \centerline{``The shadows now so long do grow,} \centerline{That brambles like tall cedars show,} \centerline{Molehills seem mountains, and the ant} \centerline{Appears a monstrous elephant."} \centerline{Charles Cotton, {\sl Evening Quatrains\/}.} ``Shadow" is dynamic mathematically, as a transformation, as well as geographically, as the sweeping boundary separating light from dark that refreshes the earth on a daily cycle. Shadow is tied directly to time through the diurnal motions of the Earth, and it is tied indirectly to time, at a personal level, as well. Each individual casts a personal time-shadow--a long trail of experiences representing accumulated wisdom over a period of years (and growing longer all the time), together with a short extension into a ``cone" of opportunity, generated by a space-time continuum, into the near future. [25] The analysis of the manner in which these temporal shadows might become unified in some global manner [26] is no doubt better left to philosopy and religion as \smallskip \centerline{``Time watches from the shadow".} \centerline{W. H. Auden, Birthday Poem.} With spatial shadows and temporal shadows, one might recast Clifford's postulates for a Science of Space as Postulates for light and dark based on the concept of shadow. The contrast between light and dark, and sunlight and shadow, gives insight into the shape of things; or, as Clifford put it, \smallskip \centerline{``Out of pictures, we imagine a world of solid things,"} \noindent a statement reminiscent of Plato's ``Den". [27] That is, a shadow is a creature that exists as a transformation of a three dimensional object onto a two-dimensional surface much as the relief format is the transitional step from two-dimensional paintings to full three dimensional art. The shape and position of the shadow are a function of \smallskip \item{1.} the shape of the three-dimensional object, \item{2.} the orientation of the three-dimensional object in relation to the light source, and \item{3.} the curvature of the receiving surface. \noindent The concept of shadow links these elements and therefore represents a relationship that is ``structurist" in nature. Clifford's statement of his postulates for a Science of Space follows. [28] \smallskip \item{``1.} Postulate of Continuity. Space is a continuous aggregate of points, not a discrete aggregate. \item{2.} Property of Elementary Flatness. Any curved surface which is such that the more you magnify it, the flatter it gets is said to possess elementary flatness. \item{3.} Postulate of Superposition. A body can be moved about in space without altering its size or shape. \item{4.} Postulate of Similarity. According to this postulate, any figure may be magnified or diminished in any degree without altering its shape." Both ``space" and ``darkness" are diffuse, rather than linear, as concepts; their ``lateral" character suggests that they, and other concepts possessing this characteristic, such as time, continuity, or inclusion/exclusion, have the power to unify. Thus, we rethink Clifford's postulates within his stated context, to see if they can be reasonably recast as a different set of postulates concerning light and dark. \smallskip \line{Shadow Postulates \hfil} \item{1.} Postulate of Continuity. Total darkness is a continuous aggregate of shadow, and not a discrete aggregate of individual shadows. \noindent Indeed, total darkness on the earth is continuous as it is formed from a single global shadow of the earth on itself; all other shadows are lesser. This global shadow is a limiting position that a sum of discrete aggregates of shadow might approach but never reach; the whole is greater than the sum of its parts. \item{2.} Postulate of Equinox. On every surface which has this property, all but a finite number of points are such that they are in darkness and light an equal amount of time. \noindent Clifford notes that any surface that possesses his property of elementary flatness is one on which ``the amount of turning necessary to take a direction all round into its first position is the same for all points on the surface." This is suggestive of what happens on earth at the time of the equinoxes in which all parallels of latitude are bisected by the edge of darkness so that all but the poles spend half the diurnal cycle in light and half in dark. Hence the restatement of ``Elementary Flatness" as ``Equinox." \item{3.} Postulate of Unique Position. The length and angle of individual shadows impart information, in a unique fashion, as to position on earth. \noindent One consequence of Clifford's Postulate of Superposition is that ``all parts of space are exactly alike." A body can be moved about in space without altering its size or shape, but its shadow changes at every different location on earth (at a given instant). Thus the Postulate of Unique Position is parallel to that of Superposition. \item{4.} Postulate of Solstice. On every surface which has this property, all but a finite number of points are such that they are in darkness and light an unequal amount of time. \noindent Using the idea in Clifford's Postulate of Similarity, any shadow of a single object may become magnified or diminished in any degree, through time. However, the shape of the object which casts the shadow remains unchanged. The Earth's shadow always covers exactly half of the earth-sphere (in theory). The dark/light boundary slips over the Earth's surface covering half of it in darkness, altering the extent to which shadows of unchanged objects become magnified or diminished. During this process, not all points experience the same amount of darkness. Hence, ``Similarity" is replaced with ``Solstice." The dynamics of this process are bounded between two parallels (the Tropics), so that there is also implied parallelism associated with this Postulate, just as Lobatschewsky noted implied parallelism associated with Clifford's fourth postulate and rejected it in order to consider using his geometry to understand astronomical space. [29] Now this set of postulates ``fits" with the earth and its shadow (indeed, the earth motivated it). The reader wishing to determine where the dark/light boundary appears at a given time at a given location need only perform the following construction, [30] using a globe on a sunny day. Point the north pole of the globe toward the earth's north pole (make compensating adjustments for southern hemisphere locations), where meridians of longitude converge. Rotate the globe on this north/south axis until your location appears on top of the globe--where a plane ``parallel" to the surface of the earth is tangent to the globe. The shadow cast by the sun on the globe will trace out accurately the position of the light/dark boundary on the earth at that moment. This construction works because it amounts to putting the globe in exactly the position that the earth is in relative to the sun--it is a good example of Shadow Postulate 3 concerning Unique Position because the globe position required is unique for each point on earth, even though each unique position will generate the same position for the shadow. (Postulate 1 applies, and Postulate 2 applies on two days of the year and Postulate 4 applies otherwise.) A natural next step is then to turn these postulates back around on the style of sculpture (that of discrete units that suggest unity) that motivated them. Shadow is a sort of underlying, continuous and rhythmical, [31] phrasing in a poetry of dark and light. The postulates offer a strategy to see what ``poetic images" can be formed within this poetic phrasing. SunSweep is a sculpture in three discrete parts. Thus, Shadow Postulates 1, 2, and 4, which are tied to continuity are not of particular interest, though they are significant in explaining the sun-sighting from each position. Shadow Postulate 3, dealing with Unique Position, is the natural, abstract ``line" of logic joining the sites, as the ``Sunsweep." Light coming through the keystone is what merges its geometry with the reality of the earth, as a seasonal analemma traced out on the earth by pencils of sunlight. The concept of light and dark, viewed within the concept of Unique Position, is what abstractly links the three SunSweep sites, and their sun-sighting capability, as a unit. With the Four Corners Project, we have the possibility of considering the more global postulates because of the requirement of a global view from which to visualize the entire sculpture. In this case, the interesting alignments of sculpture with theory appear to be in the Equinox and Solstice Shadow Postulates. Four Corners may be referenced using standard geographic latitude and longitude, but it is most easily referenced using a spherical coordinate system of latitude and longitude based on a polar axis through one of the four corners and its antipodal point. Rotation matrices, from linear algebra, may then be used to move from one coordinate system to the other. Thus, if one views the Four Corners Project as having a ``North" Pole at the Greenland corner, it seems natural to ask whether or not ``Equinox" and ``Solstice" relative to this coordinate system coincide with astronomical equinox and solstice positions of the earth. Indeed, the concepts apply, but the results are different. Because the only parts of this earth-scale sculpture touched by sunlight are the corners: ``equinox" occurs when exactly two of the corners are illuminated and two are in the earth's shadow; ``solstice" occurs otherwise. ``Equinox" is clearly a more frequent occurrence with the Four Corners than it is with the Earth. In this view, the natural concept drawing the Four Corners together as a unit is that of spatial relations between Earth and the Solar System as Equinox and Solstice, and at the same time, this human construct of ``Four Corners" enlightens the natural occurence of equinox and solstice. In both cases, the postulates of light and dark serve as a natural abstract line to suggest unity, much as the physical positioning of proximate discrete pieces suggests natural lines along which to sight in a wide range of artistic efforts. This is an alignment of fundamental ideas. It is reasonable to consider therefore where this might lead, both in terms of art and in terms of formal theory. Further directions appear two-fold: first, in the world of art, it may be useful to consider other existing art in this after-the-fact mode and then to employ these postulates as part of a plan in developing discrete sculpture to suggest unity; and second, in the world of formal theory, it seems appropriate to extend abstract theory from the postulates with an eye to possibly turning it back around on art. One direction that is currently being investigated by Kenneth Snelson is in the arena of mathematics applied to spheres, particularly to those applications developed in analogy with the earth's position in the solar system. Pauli's Exclusion Principle of quantum mechanics, which rests on likening the spin of an electron to the diurnal spinning of the earth on its axis, serves as a sort of spatial starting point for his alignments of modern physics and sculpture. [32] (According to Pauli's principle, no two electrons can be in the same orbit of the nucleus. [33]) In a related, but different, direction, the use of Clifford's postulates suggests that a suitable extension of ideas might arise in the world of various non-Euclidean geometries and particularly in those whose Euclidean models are often cast in terms of a sphere. \vfill\eject \line{\bf Notes \hfil} \ref 1. David Barr, 1979. ``Notes on celebration," {\sl The Structurist}, 8: pp. 52-56. David Barr, 1981, ``The four corners of the world." {\sl Coevolution\/} 5: 5. David Barr, 1982, ``The four corners project," {\sl Museum Catalogue\/}, Meadow Brook Art Gallery. \ref 2. For published documentation of Nystuen's original calculations, estimated originally by Barr, see Sandra L. Arlinghaus and John D. Nystuen, {\sl Mathematical Geography and Global Art: the Mathematics of David Barr's `Four Corners Project'\/}. (Ann Arbor: Institute of Mathematical Geography, 1986), Monograph \#1. \ref 3. Susan Ager, 1984. ``It's a titanic tetrahedron." {\sl Detroit Free Press\/}, Sunday, July 15: A-1, A-11. Marsha Miro, 1985. ``David Barr's amazing cosmic art adventure." {\sl Detroit Free Press\/}, Sunday Magazine Section. October 6: 6-12; 18-22. Smithsonian Institution Documentary Film. 1986. ``In celebration: David Barr's Four Corners Project." Archives of American Art. Released in April at The Detroit Institute of Arts. \ref 4. James J. Sweeney, {\sl Henry Moore\/}, (New York: The Museum of Modern Art, 1946), p. 31. \ref 5. Eli Bornstein, ``The search for continuity in art and connectedness with nature," {\sl The Structurist: Continuity and Connectedness\/}, No. 29/30, 1989-90, pp. 38-45. \ref 6. Ibid., all of note 2. \ref 7. Simo H. Laurila, {\sl Electronic Surveying and Navigation\/}. (New York: Wiley, 1976). \ref 8. William Mulloy (Ph.D.) late Professor of Archaeology, University of Wyoming. \ref 9. Szykula, D. and Dwaihy, E. The Four Corners Dance. 1976- 1985. \ref 10. Waldo R. Tobler, 1961, ``World map on a Moebius strip." {\sl Surveying and Mapping\/}, 21:486. Sandra L. Arlinghaus, ``Terrae antipodum" in {\sl Essays on Math\-e\-mat\-i\-cal Geo\-graph\-y --- II\/}, (Ann Arbor: Institute of Mathematical Geography, 1987), Monograph \#5. \ref 11. Ibid., reference, only, note 2. \ref 12. Hermann Weyl, {\sl Symmetry\/}, (Princeton: Princeton University Press, 1952). Athelstan Spilhaus, 1975. ``Geo-art: tectonics and Platonic solids." {\sl Transactions of the American Geophysical Union\/}, 56. No. 2. Plato, ``Timaeus." \ref 13. Ibid., note 9. \ref 14. Ibid. \ref 15. John D. Nystuen, 1967, ``Boundary shapes and boundary problems." {\sl Papers of Peace Research Society International\/} 7:107- 128. \ref 16. Erwin Raisz, 1941, The analemma. {\sl The Journal of Geography\/}. 40:90-97. \ref 17. Johannes Kepler, {\sl Prodromus Dissertationem Mathematicarum continens Mysterium Cosmographicum\/}, (Tubingen, 1596). \ref 18. Grace Productions. 1986. ``Time Is No Object." Videotape, premiering on Michigan Public Television, January 15, 1989. \ref 19. J. D. Hicks, {\sl The Federal Union: A History of the United States to 1865\/}. (Cambridge, MA: The Riverside Press, 1948). \ref 20. William Kingdon Clifford, 1990 (reprint). The postulates of the science of space. {\sl Solstice: An Electronic Journal of Geography and Mathematics\/}, Vol. I, No. 1, pp. 6-16. Coxeter, H. S. M. 1965. {\sl Non-Euclidean Geometry\/}. (Toronto: University of Toronto Press, 1965) pp. 8-11. \ref 21. Ibid., Clifford. \ref 22. Sandra L. Arlinghaus, ``Solar Woks," in {\sl Essays on Mathematical Geography-II\/}, (Ann Arbor: Institute of Mathematical Geography, 1987), Monograph \#5, \ref 23. Sandra L. Arlinghaus, 1990, ``Fractal geometry of infinite pixel sequences: super-definition resolution?" {\sl Solstice: An Electronic Journal of Geography and Mathematics\/}, Vol. I, No. 2, 48-53. \ref 24. Saunders Mac Lane, ``Proof, Truth, and Confusion" The 1982 Ryerson Lecture, The University of Chicago, (Chicago: The University of Chicago Press, 1982). \ref 25. Albert Einstein, (reprint). {\sl Relativity: The Special and the General Theory\/}. (New York: Bonanza Books, 1961). \ref 26. Henry Moore and W. H. Auden, {\sl Auden Poems, Moore Lithographs: An Exhibition of a Book Dedicated by Henry Moore to W. H. Auden with Related Drawings\/}. (London: The British Museum, 1974). \ref 27. Plato's {\sl Republic\/}. The Complete and Unabridged Jowett Translation. (New York: Airmont Publishing, 1968), pp. 267-272, end book VI, beginning book VII. \ref 28. Ibid., note 21. \ref 29. Ibid., note 20. \ref 30. William W. Bunge. Personal communication to John D. Nystuen. \ref 31. Richard D. Cureton, {\sl Rhythmic Phrasing in English Verse\/}. (London: Longmans, 1991 (in press)). Andrea Voorhees Arlinghaus, 1991, personal communication to Sandra L. Arlinghaus. \ref 32. Encyclopaedia Brittanica. ``Quantum Mechanics," (Chicago: William Benton, 1966). Vol. 18, p. 929. Kenneth Snelson, {\sl The Nature of Structure\/}. (New York: New York Academy of Sciences, 1989). pp. 21-24. \ref 33. Ibid., Snelson. \vfill\eject Sandra L. Arlinghaus is Director, Institute of Mathematical Geography, 2790 Briarcliff, Ann Arbor, MI 48105 \smallskip David Barr is Professor of Art, Macomb Community College, South Campus, Warren, MI 48093 \smallskip John D. Nystuen is Professor of Geography and Urban Planning, The University of Michigan, Ann Arbor, MI 48109 \smallskip Written July, 1990, revised, March, 1991, and June, 1991. All funds for the art projects described herein were supplied by David Barr. \vfill\eject \centerline{\bf Construction Zone} \centerline{Simple analysis of the logistic function} A derivation supplied by S. Arlinghaus in response to questions from William D. Drake, School of Natural Resources, University of Michigan, concerning aspects of his interest in transition theory. Discussed Tuesday, May 6, 1991, Colloquium in Mathematical Geography, IMaGe. Present: Sandy Arlinghaus, Bill Drake, John Nystuen (this commentary is included in {\sl Solstice\/} at the request of the latter). \smallskip 1. The exponential function--unbounded population growth \smallskip Assumption: The rate of population growth or decay at any time $t$ is proportional to the size of the population at $t$. \smallskip Let $Y_t$ represent the size of a population at time $t$. The rate of growth of $Y_t$ is proportional to $Y_t$; $$ dY_t/dt = kY_t $$ where $k$ is a constant of proportionality. \smallskip To solve this differential equation for $Y_t$, separate the variables. $$ dY_t/Y_t = k \, dt; \int 1/Y_t\,dY_t = \int k \, dt. $$ Therefore, $$ \hbox{ln}|Y_t| = kt + c_0. $$ Consider only the positive part, so that $$ Y_t=e^{kt+c_0}=e^{c_0}e^{kt}. $$ Let ${Y_t}_0 = e^{c_0}$. Therefore, $$ Y_t={Y_t}_0\,e^{kt}; $$ exponential growth is unbounded as $t \longrightarrow \infty $. \smallskip Suppose $t=0$. Therefore, $$ Y_t={Y_t}_0\,e^0={Y_t}_0. $$ Thus, ${Y_t}_0$ is the size of the population at $t=0$, under conditions of growth where $k>0$ (Figure 1). \midinsert \vskip3in \centerline{\bf Figure 1} \endinsert \smallskip 2. The logistic function--bounded population growth. \smallskip Assumption appended to assumption for exponential growth. In reality, when the population gets large, envirnomental factors dampen growth. \smallskip The growth rate decreases-- $dY_t/dt$ decreases. So, assume the population size is limited to some maximum, $q$, where $0 0$ and $q-Y_t > 0$, $$ \hbox{ln}{{Y_t}\over{q-Y_t}}=qKt +qC. $$ Therefore, $$ {{Y_t}\over{q-Y_t}}=e^{qKt+qC}=e^{qKt}e^{qC}. $$ Replace $e^{qC}$ by $A$. Therefore, $$ {{Y_t}\over{q-Y_t}}=Ae^{qKt}; $$ $$ Y_t=(q-Y_t)Ae^{qKt}; $$ $$ Y_t=qAe^{qKt}-Y_tAe^{qKt}; $$ $$ Y_t(Ae^{qKt} +1) = qAe^{qKt}; $$ $$ Y_t={{qAe^{qKt}}\over {Ae^{qKt} +1}}; $$ now divide top and bottom by $Ae^{qKt}$, equivalent to multiplying the fraction by 1, so that $$ Y_t={q \over {1+{1 \over {Ae^{qKt}}}}}={q \over 1+ {1 \over A} e^{-qKt}}. $$ Replace $1/A$ by $a$ and $-qK$ by $b$ producing a common form for the logistic function (Figure 2), $$ Y_t = {q \over {1+ae^{bt}}} $$ with $b<0$ because $b=-qK$, and $q,\,\, K>0$. \smallskip 3. Facts about the graph of the logistic equation. \smallskip a. The line $Y_t=q$ is a horizontal asymptote for the graph. \smallskip This is so because, for $b<0$, $$ \lim_{t\to\infty }{q\over {1+ae^{bt}}} \longrightarrow {q\over{1+a(0)}} = q $$ Can the curve cross this asymptote? Or, can it be that $$ Y_t={{Y_t}\over {1+ae^{bt}}}? $$ Or, $$ 1=1+ae^{bt}? $$ Or, $$ ae^{bt}=0 $$ Or, that $a=0$? No, because $a=1/A$. Or, that $e^{bt}=0$--no. \smallskip Thus, the logistic growth curve described above cannot cross the horizontal asymptote so that it approaches it entirely from one side, in this case, from below. \smallskip b. Find the coordinates of the inflection point of the logistic curve. \smallskip {\bf Vertical component}: \smallskip The equation $dY_t/dt =KY_t(q-Y_t)=KqY_t-KY_t^2$ is a measure of population growth. Find the maximum rate of growth--derivative of previous equation: $$ d^2Y_t/dt^2=Kq-2KY_t $$ To find a maximum (min), set this last equation equal to zero. $$ Kq-2KY_t=0 $$ Therefore, $Y_t=q/2$. This is the vertical coordinate of the inflection point of the curve for $Y_t$, the logistic curve--$dY_t/dt$ is increasing to the left of $q/2$ ($d^2Y_t/dt>0$) and $dY_t/dt$ is decreasing to the right of $q/2$ ($d^2Y_t/dt<0$). So, the maximum rate of growth occurs at $Y_t=q/2$. [The rate at which the rate of growth is changing is a constant since the first differential equation is a quadratic (parabola)]. \smallskip {\bf Horizontal component}: \smallskip To find $t$, put $Y_t=q/2$ in the logistic equation and solve: $$ q/2={q\over {1+ae^{bt}}}. $$ Solving, $$ 1+ae^{bt}=2; e^{bt}=1/a; e^{-bt}=a; -bt=\hbox{ln}\,a, $$ $$ t={{\hbox{ln}\,a}\over {-b}} $$ Thus, the coordinates of the inflection point of the logistic curve are: $$ (\hbox{ln}\,a/(-b), q/2). $$ In order to track changes in transitions, such as demographic transitions, monitoring the position of the inflection point might be of use. To consider feedback in such systems, graphical analysis (Figure 2) of curves representing transitions might be of use. \midinsert \vskip3in {\bf Figure 2}. The intersection points of the line $y=x$ with the logistic curve are, using terms from chaos theory, attractors on either end, and a repelling fixed point in the middle, possibly near the inflection point of the curve. \endinsert \vfill\eject \centerline{\bf Educational Feature} \centerline{\bf Topics in Spatial Theory} \centerline{\bf Based on lectures given by S. Arlinghaus} \centerline{\bf as a guest speaker in John Nystuen's} \centerline{\bf Urban Planning, 507, University of Michigan} \centerline{\bf Feb. 21, 28, 1990; four hours} \smallskip \line{\hfil The people along the sand \hfil} \line{\hfil All turn and look one way. \hfil} \line{\hfil They turn their back on the land. \hfil} \line{\hfil They look at the sea all day. \hfil} \smallskip \centerline{$\cdots $} \smallskip \line{\hfil They cannot look out far. \hfil} \line{\hfil They cannot look in deep. \hfil} \line{\hfil But when was that ever a bar \hfil} \line{\hfil To any watch they keep? \hfil} \smallskip \line{\hfil {\bf Robert Frost} {\sl Neither Out Far Nor In Deep}} \smallskip {\bf I. Introduction} Theory guides the direction technology takes; mathematics is the theoretical foundation of technology. To become more than a mere user of various software packages and programming languages, which change rapidly (what is trendy in today's job market may be obsolete tomorrow), it is therefore critical to understand what sorts of decisions can be made at the theoretical level. Underlying theory is ``spatial" in character, rather than ``temporal," when the objects and processes it deals with are ordered in space rather than in time (most can be done in both--decide which is of greater interest). The focus with GIS is spatial; hence, the theory underlying it is ``spatial." This is not a new idea; D'Arcy Thompson, a biologist, saw (as early as 1917) a need for finding a systematic, theoretical organization of biological species that went beyond the classification of Linnaeus. What he found to be fundamental, to characterization along structural (spatial, morphological) lines (rather than along temporal, evolutionary lines) was the ``Theory of Transformations"--in Thompson's words: {\sl ``In a very large part of morphology, our essential task lies in the comparison of related forms rather than in the precise definition of each; and the deformation of a complicated figure may be a phenomenon easy of comprehension, though the figure itself have to be left unanalysed and undefined. This process of comparison, of recognising in one form a definite permutation or deformation of another, apart altogether from a precise and adequate understanding of the original `type' or standard of comparison, lies within the immediate province of mathematics, and finds its solution in the elementary use of certain method of the mathematician. This method is the Method of Co-ordinates, on which is based the Theory of Transformations.* [*The mathematical Theory of Transformations is part of the Theory of Groups, of great importance in modern mathematics. A distinction is drawn between Substitution-groups and Transformation-groups, the former being discontinuous, the latter continuous--in such a way that within one and the same group each transformation is infinitely little different from another. The distinction among biologists between a mutation and a variation is curiously analogous.] I imagine that when Descartes conceived the method of co-ordinates, as a generalisation from the proportional diagrams of the artist and the architect, and long before the immense possibilities of this analysis could be foreseen, he had in mind a very simple purpose; it was perhaps no more than to find a way of translating the form of a curve (as well as the position of a point) into numbers and into words. This is precisely what we do, by the method of coordinates, every time we study a statistical curve; and conversely translate numbers into form whenever we `plot a curve', to illustrate a table or mortality, a rate of growth, or the daily variation of temperature or barometric pressure. In precisely the same way it is possible to inscribe in a net of rectangular co-ordinates the outline, for instance, of a fish, and so to translate it into a table of numbers, from which again we may at pleasure reconstruct the curve. But it is the next step in the employment of co-ordinates which is of special interest and use to the morphologist; and this step consists in the alteration, or deformation, of our system of co-ordinates, and in the study of the corresponding transformation of the curve or figure inscribed in the co-ordinate network. Let us inscribe in a system of Cartesian co-ordinates the outline of an organism, however complicated, or a part thereof: such as a fish, a crab, or a mammalian skull. We may now treat this complicated figure, in general terms, as a function of $x$, $y$. If we submit our rectangular system to deformation on simple and recognised lines, altering, for instance, the direction of the axes, the ratio of $x/y$, or substituting for $x$ and $y$ some more complicated expressions, the we obtain a new system of co-ordinates, whose deformation from the original type the inscribed figure will precisely follow. In other words, we obtain a new figure which represents to old figure under a more or less homogeneous strain, and is a function of the new co-ordinates in precisely the same way as the old figure was of the original co-ordinates $x$ and $y$. The problem is closely akin to that of the cartographer who transfers identical data to one projection or another [reference below]; and whose object is to secure (if it be possible) a complete correspondence, in each small unit of area, between the one representation and the other. The morphologist will not seek to draw his orgainc forms in a new and artificial projection; but, in the converse aspect of the problem, he will enquire whether two different but more or less obviously related forms can be so analysed and interpreted that each may be shown to be a transformed representation of the other. This once demonstrated, it will be a comparatively easy task (in all probability) to postulate the direction and magnitude of the force capable of effecting the required transformation. Again, if such a simple alteration of the system of forces can be proved adequate to meet the case, we may find ourselves able to dispense with many widely current and more complicated hypotheses of biological causation. For it is a maxim in physics that an effect ought not to be ascribed to the joint operation of many causes if few are adequate to the production of it. Reference: Tissot, M\'emoire sur la representation des surfaces, et les projections des cartes g\'eographiques (Paris, 1881)."} Sir D'Arcy Wentworth Thompson, pp. 271-272, in {\sl On Growth and Form\/}. \midinsert \vskip3in {\bf Figure I} Sample of Thompson's Transformations. Fig. I.1: Argyropelecus olfersi. Fig. I.2: Sternoptyx diaphana. \endinsert \smallskip Look at Thompson's comments concerning biological structure to see what parallels there are, already, with GIS structure and to see what they might suggest--compare to Tobler's map transformations. \smallskip 1. GIS (the digitizer) uses coordinates to translate forms (maps) into numbers. \smallskip 2. All GIS software translates numbers into maps, which may then be printed out, parallel to inscribing a fish in a set of coordinates, translating it into a set of numbers, from which the fish may be reproduced at any time (Figure I.1) \smallskip 3. Thompson's deformations correspond to the ideas of scale shifts on maps. Transformations describe shifts in scale. Figure I.2. \smallskip 4. Thompson's comments on the distinction between discontinuous and continuous reflects partitioning of mathematics into discrete and continuous. Discrete need not be finite--look at two different types of garbage bag ties--twist ties and slip-through ties, and imagine them to be of infinite extent. \smallskip 5. We see simple transformations in GIS--maps might be stretched or compressed in the vertical direction. Imagine using a small digitizing table to encode a large map by deliberately recording ``wrong" positions---then use a transformation within the computer to correct the ``wrong" positions so that the map prints out correctly on the plotter. Large digitizing tables become unnecessary. \smallskip 6. We look, for future direction, to the Theory of Groups. For today, we confine ourselves to a few simple transformations. \smallskip {\bf II. Transformations} \smallskip Transformations can allow you to relate one form to another in a systematic manner allowing retieval of all forms. To do this, you need to know how to define a transformation so that this is possible. Beyond this, one might consider a stripped-down transformation, for even more efficient compression of electronic effort [Mac Lane]. \smallskip A. Well-defined (single-valued). \smallskip Let ``tau" be a transformation carrying a set $X$ to a set $Y$: in notation, $\tau : X \longrightarrow Y$. Tau is said to be well-defined if each element of $X$ corresponds to exactly one element of $Y$. Visually, this might be thought of in terms of lists of street addresses: the set $X$ consists of house addresses used as ``return" addresses on letters. The set $Y$ consists of other house addresses. The transformation is the postal transmission of a letter from locations in $X$ to locations in $Y$. A single value of $X$ maps to single value of $Y$. \midinsert \vskip2in {\bf Figure II.1} This is a transformation--two distinct letters ($x$ and $x'$) can be posted to the same address ($y$). (Many-one map). \endinsert \midinsert \vskip2in {\bf Figure II.2} This is NOT a transformation--one letter ($x$) cannot, itself, go to two different addresses ($y$ and $y'$) (new technology of e-mail permits this--suggests for possible need for change in fundamental definitions). (One-many map). \endinsert \smallskip B. Reversible \smallskip \item{i.} One-to-one correspondence. \smallskip A one-to-one correspondence is a transformation in which each $x$ in $X$ goes to a distinct $y$ in $Y$; the situation depicted in Figure II.1 cannot hold. From the standpoint of reversibility, this is important; if the situation in II.1 could hold how would you decide, in reversing, whether to ``return" $y$ to $x$ or to $x'$?? \smallskip \item{ii.} Transformations of $X$ onto $Y$ \smallskip A transformation of $X$ onto $Y$ is such that every element in $Y$ comes from some element of $X$; there are no addresses outside the postal system (Figure II.3). \midinsert \vskip2in {\bf Figure II.3} This is a transformation--it is neither one-to-one, nor onto ($y'$ is outside the system). \endinsert \smallskip \item{iii.} A transformation $\tau $ from $X$ to $Y$ is reversible-- it has an inverse $\tau^{-1}$ from $Y$ to $X$ if $\tau $ is one-to-one and onto; it has an inverse from a subset of $Y$ to $X$ if $\tau $ is one-to-one (Figure II.4). \midinsert \vskip5in {\bf Figure II.4} In the top part, $\tau (X)=Y$. In the bottom part $\tau (X)$ is properly contained in $Y$; this is like data compression--like ZIP followed by UNZIP. \endinsert \smallskip C. Rubbersheeting \smallskip The use of transformations that have inverses is critical in rubbersheeting; associations between data sets must be made in a manner so that correct information can be gained from the process. \smallskip {\bf III. Types of Transformations} \smallskip One might consider moving objects within a fixed coordinate system, or holding the objects fixed and moving the coordinate system. Thompson did the latter; rubbersheeting does the latter; NCGIA materials (Lecture 28) comment that the latter approach is particularly well-suited to GIS purposes. \smallskip Two major types of transformations: \smallskip a. Affine transformations: these are transformations under which parallel lines are preserved as parallel lines. That is, both the concept of ``straight line" and ``parallel" remain; angles may change, however. There are four types of affine transformations as noted on suitable NCGIA handout (Figure III.1). Products of affine transformations are themselves affine transformations. \midinsert \vskip6in \centerline{Figure III.1} \endinsert Current technology employs types 1 and 2, quite clearly. CRT allows for translation of maps, and for scale change in $y$-direction only. Copier also allows for the same, and in addition, permits different shifts in scale along the two axes, allowing maps with different scales along different axes to be brought to the same scale and pieced together. (See output from Canon Color Copier.) On that output, the $x$-axis if fixed by the transformation and the $y$-axis is stretched to 200\% of the original. Thus, a circle transforms to an ellipse, a rectangle with base parallel to the $x$-axis transforms to a larger rectangle, and a rectangle with base not parallel to the $x$-axis transforms to a parallelogram with no right angles (Figure III.2). \midinsert \vskip6in \centerline{\bf Figure III.2} \endinsert \smallskip B. Curvilinear transformations; neither straightness nor parallelism is necessarily preserved (Thompson fish, Figure III.3). \midinsert \vskip6in \centerline{\bf Figure III.3} \endinsert \smallskip {\bf IV. Exercise, page 5, lecture 28, NCGIA.} \smallskip {\bf V. Steiner networks} \smallskip If centers of gravity are used as a centering scheme in a triangulated irregular network, then it is desired to have no centroid lie outside a triangular cell. Thus, no cell should have angle greater than 120 degrees, so that the Steiner network (where all angles are exactly 120 degrees) will serve as an outer edge (a limiting position) for the set of acceptable triangulations. Thus, it is important to know how to locate Steiner networks. \smallskip {\bf VI. Digital Topology} \smallskip The notion of a ``triangulation" is a fundamental concept in topology (sometimes called ``rubber sheet" geometry). ``Digital" topology is a specialization of ``combinatorial" topology in which the fundamental units are pixels. The same ``important" theorems underlie each. The Jordan Curve Theorem (which characterizes the difference between the ``inside" and the ``outside" of a curve, is an example of such a theorem). Using concepts from digital topology, ``picture" processing (as a parallel to ``data" processing) is possible. There are numerous references in this field; some include works by geographer Waldo Tobler and by mathematician Azriel Rosenfeld. Other key-words to topics of interest in this area include, Jordan Curve Theorem in higher dimensions; quadtrees; scale-free transformations; close-packings of pixels. \smallskip {\bf VII. The algebra of symmetry--some group theory} \smallskip D'Arcy Thompson commented that the theory of transformations was tied to the theory of groups. A ``group" is a mathematical system whose structure is simpler than that of the number system we customarily use in the ``real-world." In our usual number system, we have two distinct operations of ``+" and ``x"; thus, we have rules on how to use each of these operations, and rules telling us how to link these two operations (distributive law; conventions regarding order of operations). A group is composed of a finite set of elements, $S=\{a,b, c, \ldots, n\}$ that are related to each other using a single operation of ``$\star $." Under this operation, the set obeys the following rules (and is, by that fact, a group). \item{1.} The product, under $\star $, of any two elements of $S$ is once again an element of $S$---this system is ``closed" under the operation of $\star $---no new element (information) is generated. \item{2.} Given $a$, $b$, and $c$ in $S$: $(a \star b) \star c =a \star (b \star c)$. The manner in which parentheses are introduced is not of significance in determining the answer (information content) resulting from a string of operations under $\star $. The operation of $\star $ is said to be associative. \item{3.} There is an identity element, 1, in $S$ such that for any element of $S$, say $a$, it follows that $$ a \star 1 = 1 \star a = a. $$ \item{4.} Each element of $S$ has an inverse in $S$; that is, for a typical element $a$ of $S$, there exists another element, $b$ of $S$, such that $$ a \star b = b \star a = 1. $$ Denote the inverse of $a$ as $a^{-1}$. Thus, $a \star a^{-1} = a^{-1} \star a = 1$. \smallskip The order in which elements are related to each other, using $\star $, may matter; it need not be true that $a \star b = b \star a$. (Elements of the group do not necessarily ``commute" with each other.) The algebraic idea of ``closure" is comparable to the GIS notion of snapping a polygon shut, so that chaining of line segments does not continue forever---the system is ``closed." \smallskip A. The affine group; affine geometry. \smallskip The definition of group given above was to a set of elements and an operation linking them. These elements might be regarded as transformations. In particular, consider the set of all affine transformations of the plane that are one-to-one (translations, scalings, rotations, and reflections). These form a group, when the operation $\star $ is considered as the composition of functions: \item{i.} The product of two affine transformations is itself an affine transformation; \item{ii.} In a sequence of three affine transformations, it does not matter which two are grouped first, as long as the pattern of the three is unchanged---associativity. \item{iii.} The affine transformation which maps the plane to itself serves as an identity element. \item{iv.} Because the affine transformations dealt with here are one-to-one, they have inverses (all translations have inverses; only those linear transformations with inverses are considered here). Affine geometry is the study of properties of figures that remain invariant under the group of one-to-one affine transformatons. Here are some theorems from affine geometry. \item{i.} Any one-to-one affine transformation maps lines to lines. \item{ii.} Any affine transformation maps parallel sets of lines to parallel sets of lines. \item{iii.} Any two triangles are equivalent with respect to the affine group. \smallskip To demonstrate the theorem in iii., consider a fixed triangle with position $(OB_0C_0)$, relative to an $x$/$y$ coordinate system. Choose an arbitrary triangle, $(ABC)$. Use elements of the affine group to move $(ABC)$ to coincide with $(OB_0C_0)$: a translation slides $A$ to $O$ (Figure VII.1). Two separate scaling operations and rotations slide $B$ to $B_0$ and $C$ to $C_0$. This is possible because $O$, $B$, and $C$ are not collinear (as vectors, $OB$ and $OC$ are linearly independent). \midinsert \vskip6in \centerline{\bf Figure VII.1} \endinsert This is the theoretical origin of the GIS notion that control points must be non-collinear and that there must be at least three of them. From a mathematical standpoint, it does not, therefore, matter whether the control points are chosen close together or far apart; however, from a visual standpoint it does matter. When control points are chosen close together the scaling operation required to transform the control triangle into other triangles is generally enlargement. When the control triangle is chosen with widely spaced vertices, the scaling operations required to transform it into other triangles is generally reduction. Errors are more visible with enlargement. Therefore, it is better, for the sake of visual comfort, to rely on reduction (reducing error size, as well) whenever possible, and therefore, to choose widely-spaced control points. This is like the exercise above; there are two scalings and another affine transformation (here a translation, in the exercise, a reflection). In either case, the outcome of applying a sequence of affine transformations is still an affine transformation. In this case, it does not matter in what order the scaling operations are executed and in what order, relative to the scaling, the translation is applied. In the case of the exercise, however, this is not the case. It does not matter in what order the scalings are applied. It is the case that $\tau_1 \circ \tau_2 = \tau_2 \circ \tau_1 $. It is also the case that $\tau_1 \circ \tau_3 = \tau_3 \circ \tau_1$. However, it is not the case that $$ \tau_2 \circ \tau_3 = \tau_3 \circ \tau_2: $$ $$ (50,5) {\tau_2 \atop \longrightarrow} (50,48) {\tau_3 \atop \longrightarrow} (50, 432) $$ $$ (50,5) {\tau_3 \atop \longrightarrow} (50, 475) {\tau_2 \atop \longrightarrow} (50, 4660) $$ Observe, however, that it is possible to solve the problem applying the reflection earlier. Take $\tau_1$ to be the required reflection so that $y$ is sent to $50-y$ (reflection before the scale change on the $y$-axis). Figure VII.2 shows the solution here. In the non-commutative case here, there is a sharp difference in the ``correct" $y$-value and the other possible one. In this case, as in the previous one, it does not matter how the application of transformations are separated by parentheses, and it is guaranteed that the product will itself be affine. \midinsert \vskip6in \centerline{\bf Figure VII.2} \endinsert Thus, the order of application of affine transformations, within the group (locally), is important. This might cause difficulties (sending you off the screen), or it might be turned to an advantage in zooming-in on something. What caused the problem here was the reflection. Products of rotations of the plane are roatations of the plane; products of translations are translations, and products of scalings are scalings. Here, and as we shall see later, reflections cause non-commutativity (similar problems might have arisen in Figure VII.1, had a reflection been involved). \item{iv.} Any triangle is affine-equivalent to an equilateral triangle (choose whatever control triangle desired---can choose an underlying lattice of regularly spaced triangular points and rubber sheet them to an irregularly spaced one). \item{v.} Any ellipse is affine equivalent to a circle (demonstrated via copier technology). \item\item{a.} Parallelism and GIS: crossing lines and polygon area. Groups suggest how theoretical structure may be built from assembling simple pieces. GIS algorithms for complex processes are also often built from assembling simple pieces. \smallskip Straight lines \smallskip How can we tell if two lines intersect in a node? \smallskip Example from NCGIA Lecture 32: does the line $L_1$ from (4,2) to (2,0) cross the line $L_2$ from (0,4) to (4,0)? From a mathematical standpoint, two lines in the Euclidean plane cross if they have different slopes, $m_1$ and $m_2$, where the slope $m$ between points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as $(y_2-y_1)/(x_2-x_1)$. In this case, the slope of $L_1$ is $(0-2)/(2-4)=1$ and the slope of $L_2$ is $(0-4)/(4-0)=-1$. The slopes are different, so the lines cross in the plane. However, in the GIS context: \item{i.} Do the lines cross on the computer screen, or is the intersection point outside the bounded Euclidean region of the screen? \item{ii.} Even if the lines cross on the screen, do they intersect at a node of the data base (was that point digitized)? To answer these questions, it is necessary to determine the intersection point of the two lines. Equation of $L_1$: one form for the equation of a line between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $$ y-y_1=m(x-x_1) $$ where $m$ is the slope and $b$ is the second coordinate of the $y$-intercept. Thus, $L_1$ has equation $y-2=1(x-4)$ or $y=x-2$; $L_2$ has equation $y-4=-1(x-0)$ or $y=-x+4$. Solve these equations simultaneously to yield $x=3$ and $y=1$. Thus, if the point $(3,1)$ lies within the boundaries of the screen, the lines intersect on the screen; if the point $(3,1)$ was digitized, then another line might be hooked onto the intersection point. If it was not digitized, then the lines ``cross" but do not intersect, much as water pipes might cross but do not necessarily intersect (as in snapping a segment onto the middle of a line on the CRT). This is a graph-theoretic characteristic. Note that vertical lines are a special case; their slope is undefined because $x_2-x_1$, the denominator in the slope, is zero. Recognizing vertical lines should not be difficult, but it should be remembered that attempting to calculate slope across an entire set of lines, which might include vertical lines, can produce errors. \smallskip Chains of straight line segments. \smallskip How can we tell if chains of segments cross? \smallskip Because chains are of finite length and are bounded, it is possible to enclose them in a rectangle (no larger than the CRT screen) (Figure VII.3). This is a minimum enclosing rectangle. \midinsert \vskip6in \centerline{\bf Figure VII.3} \endinsert Thus, given two chains, $C_1$ and $C_2$, if their respective minimum enclosing rectangles do not intersect (as do straight lines) then they do not intersect, and further testing is warranted. \smallskip Polygon area: \smallskip Calculate polygon area using notion of parallelism (Figure VII.4) \midinsert \vskip6in \centerline{\bf Figure VII.4} \endinsert Simple rule, based on vertical lines, to determine if a point is inside or outside a polygon (Figure VII.5) \midinsert \vskip6in \centerline{\bf Figure VII.5} \endinsert Centroids of polygons, with attached weights are often used as single values with which to characterize the entire polygon. Centroids are preserved, as centroids, under affine transformations. These are technical procedures for determining various useful measures and are documented in NCGIA material; all are based in the theory of affine transformations applied to sets of pixels. Move now to consider the mechanics of how sets of affine transformations might affect a single pixel. \smallskip B. Group of symmetries of a square (pixel); the hexagonal pixel. \smallskip A square may have a set of rotations and of reflections applied to it as noted in Figure VII.6. Each may be represented as a permutation of the vertices, labelled clockwise. Permutations are multiplied as indicated in the example, below: multiply the permutation $(1234)$ by the permutation $(13)(24)$: \smallskip \qquad 1 goes to 2 (in the left one) \smallskip \quad and 2 goes to 4 (in the right one) \smallskip \quad so 1 goes to 4 (in the product) \smallskip \qquad 4 goes to 1 (in the left one) \smallskip \quad and 1 goes to 3 (in the right one) \smallskip \quad so 4 goes to 3 (in the product) \smallskip \qquad 3 goes to 4 (in the left one) \smallskip \quad and 4 goes to 2 (in the right one) \smallskip \quad so 3 goes to 2 (in the product) \smallskip \qquad 2 goes to 3 (in the left one) \smallskip \quad and 3 goes to 1 (in the right one) \smallskip \quad so 2 goes to 1 (in the product) \smallskip This last stage is akin to snapping a polygon closed in a GIS environment---here it is a cycle of numbers rather than of vertices. Figure VII.6 shows all the calculations; note, that no new permutations ever arise; hence, the system is closed under $\star $; the rotation $I$ serves as the identity transformation; each element has an inverse: $$ I \star I = I; I^{-1} = I $$ $$ R_1 \star R_3 = I; R_1^{-1} = R_3 $$ $$ R_2 \star R_2 = I; R_2^{-1} = R_2 $$ $$ R_3 \star R_1 = I; R_3^{-1} = R_1 $$ $$ H \star H = I; V \star V = I; D_1 \star D_1 = I; D_2 \star D_2 = I. $$ So, this system is a ``group." It is not, however, a commutative group---for example, $R_1 \star H = D_2$ and $H \star R_1 = D_1$. Once again, a reminder to be careful when combining reflections with affine transformations. Note that the set of rotations (including the identity rotation) is itself a group within this group. This is a ``subgroup"---it is commutative---the order in which rotations are applied to the square is irrelevant. \topinsert \vskip3in {\bf Figure VII.6} Group of symmetries of a square \settabs\+\indent&$D_1$: diagonal, 1 to 3\qquad &Permutation representation:&\cr \smallskip \+& Rotations: &Permutation representation \cr \+& $I$: identity &(1)(2)(3)(4) \cr \+& $R_1$: through 90 deg &(1234) \cr \+& $R_2$: through 180 deg &(13)(24) \cr \+& $R_3$: through 270 deg &(1432) \cr \smallskip \+& Reflections: &Permutation representation \cr \+& $H$: horizontal &(14)(23) \cr \+& $V$: vertical &(12)(34) \cr \+& $D_1$: diagonal, 1 to 3 &(1)(3)(24) \cr \+& $D_2$: diagonal, 2 to 4 &(2)(4)(13) \cr \smallskip Table--operation, $\star $, is multiplication of permutations. \settabs\+\indent&$R_1$\quad &$\star$\quad &$R_1$\quad &$R_1$\quad &$R_2$\quad &$R_3$\quad &$R_1$\quad &$R_1$\quad &$D_1$\quad &$D_2$&\cr \smallskip \+&{}&$\star $&$I$ &$R_1$&$R_2$&$R_3$&$H$&$V$&$D_1$&$D_2$\cr \smallskip \+&$I$ &\phantom{0}&$I$ &$R_1$&$R_2$&$R_3$&$H$ &$V$ &$D_1$&$D_2$\cr \+&$R_1$&\phantom{0}&$R_1$&$R_2$&$R_3$&$I$ &$D_2$&$D_1$&$H$ &$V$ \cr \+&$R_2$&\phantom{0}&$R_2$&$R_3$&$I$ &$R_1$&$V$ &$H$ &$D_2$&$D_1$\cr \+&$R_3$&\phantom{0}&$R_3$&$I$ &$R_1$&$R_2$ &$D_1$&$D_2$&$V$ &$H$ \cr \+&$H$ &\phantom{0}&$H$ &$D_1$&$V$ &$D_2$&$I$ &$R_2$&$R_1$&$R_3$\cr \+&$V$ &\phantom{0}&$V$ &$D_2$&$H$ &$D_1$&$R_2$&$I$ &$R_3$&$R_1$\cr \+&$D_1$&\phantom{0}&$D_1$&$V$ &$D_2$&$H$ &$R_3$&$R_1$&$I$ &$R_2$\cr \+&$D_2$&\phantom{0}&$D_2$&$H$ &$D_1$&$V$ &$R_1$&$R_3$&$R_2$&$I$ \cr \smallskip \endinsert \vfill\eject {\bf Figure VII.6, continued} \smallskip \hrule \smallskip \settabs\+\indent&(1)(2)(3)(4)\quad &$\star$\quad &(1)(2)(3)(4)\quad &$=$\quad &(1)(2)(3)(4)\quad &$R_3$\quad &\cr \+&Permutation&$\star $&Permutation&$=$&Permutation&\phantom{0}\cr \smallskip \+&(1)(2)(3)(4)&{}&(1)(2)(3)(4)&{}&(1)(2)(3)(4)&$I$ \cr \+&(1234) &{}&(1)(2)(3)(4)&{}&(1234) &$R_1$\cr \+&(13)(24) &{}&(1)(2)(3)(4)&{}&(13)(24) &$R_2$\cr \+&(1432) &{}&(1)(2)(3)(4)&{}&(1432) &$R_3$\cr \+&(14)(23) &{}&(1)(2)(3)(4)&{}&(14)(23) &$H$ \cr \+&(12)(34) &{}&(1)(2)(3)(4)&{}&(12)(34) &$V$ \cr \+&(1)(3)(24) &{}&(1)(2)(3)(4)&{}&(1)(3)(24) &$D_1$\cr \+&(2)(4)(13) &{}&(1)(2)(3)(4)&{}&(2)(4)(13) &$D_2$\cr \smallskip \+&(1)(2)(3)(4)&{}&(1234)&{}&(1234) &$R_1$\cr \+&(1234) &{}&(1234)&{}&(13)(24) &$R_2$\cr \+&(13)(24) &{}&(1234)&{}&(1432) &$R_3$\cr \+&(1432) &{}&(1234)&{}&(1)(2)(3)(4) &$I$ \cr \+&(14)(23) &{}&(1234)&{}&(1)(3)(24) &$D_1$\cr \+&(12)(34) &{}&(1234)&{}&(2)(4)(13) &$D_2$\cr \+&(1)(3)(24) &{}&(1234)&{}&(12)(34) &$V$ \cr \+&(2)(4)(13) &{}&(1234)&{}&(14)(23) &$H$ \cr \smallskip \+&(1)(2)(3)(4)&{}&(13)(24)&{}&(13)(24) &$R_2$\cr \+&(1234) &{}&(13)(24)&{}&(1432) &$R_3$\cr \+&(13)(24) &{}&(13)(24)&{}&(1)(2)(3)(4) &$I$ \cr \+&(1432) &{}&(13)(24)&{}&(1234) &$R_1$\cr \+&(14)(23) &{}&(13)(24)&{}&(12)(34) &$V$ \cr \+&(12)(34) &{}&(13)(24)&{}&(14)(23) &$H$ \cr \+&(1)(3)(24) &{}&(13)(24)&{}&(2)(4)(13) &$D_2$\cr \+&(2)(4)(13) &{}&(13)(24)&{}&(1)(3)(24) &$D_1$\cr \smallskip \+&(1)(2)(3)(4)&{}&(1432)&{}&(1432) &$R_3$\cr \+&(1234) &{}&(1432)&{}&(1)(2)(3)(4) &$I$ \cr \+&(13)(24) &{}&(1432)&{}&(1234) &$R_1$\cr \+&(1432) &{}&(1432)&{}&(13)(24) &$R_2$\cr \+&(14)(23) &{}&(1432)&{}&(12)(34) &$D_2$\cr \+&(12)(34) &{}&(1432)&{}&(14)(23) &$D_1$\cr \+&(1)(3)(24) &{}&(1432)&{}&(2)(4)(13) &$H$ \cr \+&(2)(4)(13) &{}&(1432)&{}&(1)(3)(24) &$V$ \cr \smallskip \+&(1)(2)(3)(4)&{}&(14)(23)&{}&(14)(23) &$H$ \cr \+&(1234) &{}&(14)(23)&{}&(2)(4)(13) &$D_2$\cr \+&(13)(24) &{}&(14)(23)&{}&(12)(34) &$V$ \cr \+&(1432) &{}&(14)(23)&{}&(1)(3)(24) &$D_1$\cr \+&(14)(23) &{}&(14)(23)&{}&(1)(2)(3)(4) &$I$ \cr \+&(12)(34) &{}&(14)(23)&{}&(13)(24) &$R_2$\cr \+&(1)(3)(24) &{}&(14)(23)&{}&(1432) &$R_3$\cr \+&(2)(4)(13) &{}&(14)(23)&{}&(1234) &$R_1$\cr \smallskip \+&(1)(2)(3)(4)&{}&(12)(34)&{}&(12)(34) &$V$ \cr \+&(1234) &{}&(12)(34)&{}&(1)(3)(24) &$D_1$\cr \+&(13)(24) &{}&(12)(34)&{}&(14)(23) &$H$ \cr \+&(1432) &{}&(12)(34)&{}&(2)(4)(13) &$D_2$\cr \+&(14)(23) &{}&(12)(34)&{}&(13)(24) &$R_2$\cr \+&(12)(34) &{}&(12)(34)&{}&(1)(2)(3)(4) &$I$ \cr \+&(1)(3)(24) &{}&(12)(34)&{}&(1234) &$R_1$\cr \+&(2)(4)(13) &{}&(12)(34)&{}&(1432) &$R_3$\cr \smallskip \+&(1)(2)(3)(4)&{}&(1)(3)(24)&{}&(1)(3)(24) &$D_1$\cr \+&(1234) &{}&(1)(3)(24)&{}&(14)(23) &$H$ \cr \+&(13)(24) &{}&(1)(3)(24)&{}&(2)(4)(13) &$D_2$\cr \+&(1432) &{}&(1)(3)(24)&{}&(12)(34) &$V$ \cr \+&(14)(23) &{}&(1)(3)(24)&{}&(1234) &$R_1$\cr \+&(12)(34) &{}&(1)(3)(24)&{}&(1432) &$R_3$\cr \+&(1)(3)(24) &{}&(1)(3)(24)&{}&(1)(2)(3)(4)&$I$ \cr \+&(2)(4)(13) &{}&(1)(3)(24)&{}&(13)(24) &$R_2$\cr \smallskip \+&(1)(2)(3)(4)&{}&(2)(4)(13)&{}&(2)(4)(13) &$D_2$\cr \+&(1234) &{}&(2)(4)(13)&{}&(12)(34) &$V$ \cr \+&(13)(24) &{}&(2)(4)(13)&{}&(1)(3)(24) &$D_1$\cr \+&(1432) &{}&(2)(4)(13)&{}&(14)(23) &$H$ \cr \+&(14)(23) &{}&(2)(4)(13)&{}&(1432) &$R_3$\cr \+&(12)(34) &{}&(2)(4)(13)&{}&(1234) &$R_1$\cr \+&(1)(3)(24) &{}&(2)(4)(13)&{}&(13)(24) &$R_2$\cr \+&(2)(4)(13) &{}&(2)(4)(13)&{}&(1)(2)(3)(4)&$I$ \cr \smallskip \hrule \smallskip Are there any other subgroups? Yes, $I$, $R_2$, $H$, $V$ also form a commutative subgroup. Note that the product of two reflections is a rotation. A similar style of analysis might be executed for the pixel viewed as a hexagon. Other theoretical issues arise concerning the possibility of using a crt display with hexagonal pixels. \quad{i.} Issues involving centroids \qquad{a.} Transformation to generate a centrally- symmetric hexagon from an arbitrary (convex) hexagon (rubbersheeting; TIN). \noindent One such issue involves concern for taking a set of irregularly--spaced data points and converting them into some sort of more regular distribution (as with rubbersheeting and a TIN). This procedure illustrates how to transform an arbitrary convex hexagon ($V_1$, $V_2$, $V_3$, $V_4$, $V_5$, $V_6$) into a centrally symmetric hexagon ($S_1$, $S_2$, $S_3$, $S_4$, $S_5$, $S_6$) centered on a point that is easy to find. (See construction in {\sl Solstice I\/}---Summer, 1990, Vol. I, No. 1., pp. 41-42.) Thus, rubbersheeting would appear possible with an hexagonal pixel. \qquad{b.} Area algorithm generalizes to hexagons: regular hexagon is two isosceles trapezoids (one on either side of a single diameter of the hexagon). \noindent What else might generalize from the square pixel format to the hexagonal pixel format? A hexagon can be decomposed into two trapezoids; thus one might imagine using an algorithm similar to that for the square pixel to find polygon areas relative to an hexagonal pixel display. \qquad{c.} Steiner networks as boundaries of sets of hexagonal pixels; given a set of points, find a minimal hexagonal network linking them. If centers of gravity (centroids) are used as a centering scheme in a triangulated irregular network (or other network of polygons), then it would be nice to have no centroid lie outside a triangular cell (or other polygon). A centroid is the intersection point of medians; it is the balance point on which the figure would rest. Sometimes the centroid lies outside the polygon; Coxeter suggests viewing the centroid as a balance point among electrical charges, thereby allowing for this possibility. Another point that is useful for using as a ``central" weight is a Steiner point; in a triangle, it is that point which minimizes total network length joining the three vertices. It is always within the triangle when no angle of the triangle is greater than or equal to 120 degrees. (See {\sl Solstice--I\/}, Vol. I., no. 2, ``Super-definition resolution.") Assigning point weights to represent polygon values is one way to compare them; another way is to assign centrally-located networks traversing underlying grid lines (Manhattan lines with square pix\-els, Stein\-er networks with hexagonal pixels); another way is to overlay the areas---again, a point-line-area classification as mentioned in detail in one of Nystuen's earlier lectures. \quad{ii.} Issues involving polygon overlays. \qquad{a.} Close-packings of hexagons; central place geometry. \qquad{b.} Fractal approach; space-filling; data compression. Polygon overlay is familiar from OSUMAP. Look at some abstract geographic/geometric issues that might suggest directions to consider in looking at ideas behind the process of overlays. Geometry of central place theory--including fractal generation of these layers. Look for a number of issues of this sort, that are theoretical, in using GIS-type equipment. Below is an outline of material in these lectures and of suggestions for future directions in which to look. \noindent I. Introduction: the role of theory. Mathematics is fundamental, and in dealing with spatial phenomena, geometry in particular, is fundamental. Historical precedent from Biology in works of D'Arcy Thompson; Tobler's map transformations. \quad A. Statement of Thompson regarding the role of theory. \quad B. Visual evidence: one species of fish is transformed into another actual species by choosing a suitable coordinate transformation. \noindent II. Transformations. \quad{A.} Well-defined (single-valued). \quad{B.} Reversible \qquad{i.} One-to-one correspondence \qquad{ii.} Transformations of $X$ onto $Y$. \quad{C.} ``Rubbersheeting"---example from Nystuen lecture, with fire stations. What is involved is creating a transformaton from an irregular scatter of locations to a regular one, locating new points (fire stations) and snapping the surface back to the irregular scatter. This requires transformations that are reversible. \noindent III. Types of transformations and examples. \quad{A.} Affine \qquad{i.} Translation \qquad{ii.} Scaling \qquad{iii.} Rotation \qquad{iv.} Reflection \quad{B.} Curvilinear \noindent IV. Exercise---scaling to make digitized map mesh with CRT scale. \noindent V. GIS tie to Steiner networks. \noindent VI. Digital topology. Quadtrees--Rosenfeld, Tobler. Jordan Curve Theorem; American Mathematical Society special sessions on digital topology (run by Rosenfeld). Hexagonal pixels---scanner technology. \noindent VII. Local scale of mathematical extension of the concept of ``affine transformation." The algebra of symmetry: definition of a group. \quad{A.} The affine group; affine geometry. \qquad{i.} Parallelism and GIS: crossing lines and polygon area. \qquad{ii.} Projective geometry; any two lines intersect in a point; no parallels. Here for completeness--not really discussed. \quad{B.} Group of symmetries of a square (pixel); the hexagonal pixel. \qquad{i.} Issues involving centroids. \qquad\quad{a.} Transformation to generate a centrally-symmetric hexagon from an arbitrary (convex) hexagon (rubbersheeting; TIN). \qquad\quad{b.} Area algorithm generalizes to hexagons; hexagon is two trapezoids. \qquad\quad{c.} Steiner networks as boundaries of sets of hexagonal pixels; given a set of points, find a minimal hexagonal network linking them--dealt with in a third lecture, not presented here. \qquad{ii.} Issues involving polygon overlays. \qquad\quad{a.} Close-packings of hexagons; central place geometry. \qquad\quad{b.} Fractal approach; space-filling; data compression. \noindent VIII. Global scale of mathematical extension of the concept of ``affine transformation." Topology. \quad{A.} Combinatorial topology. \qquad{i.} Jordan curve theorem. GIS connection, inside and outside of polygons. \qquad{ii.} Cell complexes; 0, 1, and 2 cells of GIS. \qquad{iii.} Hexagons derived from barycentric subdivision of a complex. \quad{B.} Point-set topology. \qquad{i.} Definitions. \qquad{ii.} Consequences of Definitions interpreted in GIS context. \quad{C.} Digital topology. \qquad\quad{i.} Jordan curve theorem--3-dimensions. \qquad\quad{ii.} Quadtrees. \noindent III. Further extension at different scales. Commutative diagrams---entry to different level of mathematical thought and spatial theory. \bye