Volume II, Number 1.  Summer, 1991.
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 \centerline{\big SOLSTICE:}
 \centerline{\bf SUMMER, 1991}
 \centerline{\bf Volume II, Number 1}
 \centerline{\bf Institute of Mathematical Geography}
 \centerline{\bf Ann Arbor, Michigan}
 \centerline{\bf SOLSTICE}
 \line{Founding Editor--in--Chief:  {\bf Sandra Lach Arlinghaus}. \hfil}
 \centerline{\bf EDITORIAL BOARD}
 \line{{\bf Geography} \hfil}
 \line{{\bf Michael Goodchild}, University of California, Santa Barbara. 
 \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).}
 \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.
 \line{{\bf Business} \hfil}
 \line{{\bf Robert F. Austin},
 Director, Automated Mapping and Facilities Management, CDI. \hfil}
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 ISBN: 1-877751-52-9
 \centerline{\bf SUMMARY OF CONTENT}
 \noindent Sandra L. Arlinghaus, David Barr, John D. Nystuen.
 \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). 
 \item{i.} {\bf Construction Zone} ---  The logistic curve
 \item{ii.} {\bf Educational feature} ---  Lectures on ``Spatial
 \centerline{``Life's but a walking shadow"}
 \centerline{Shakespeare, {\sl Macbeth\/}.}
 \centerline{Sandra L. Arlinghaus, David Barr, John D. Nystuen}
 \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.
 \centerline{WHOLE AND PART:}
 \centerline{\bf A Sculptural Unification of Unseen Parts.}
 \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 
 \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.

 \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).
 \centerline{\bf TABLE 1}
 \centerline{Log of travels associated with placement of the four corners}
 \centerline{Listing compiled by Heather and Gillian Barr.}
 \settabs\+\qquad\qquad&Frobisher to Sonderstrom Fjord\qquad&\cr
 \centerline{DECEMBER AND JANUARY, 1980-81:}
 \+&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
 \centerline{              JULY, 1981}
 \centerline{               GREENLAND}
 \+&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
 \centerline{              JANUARY, 1985}
 \centerline{               IRIAN JAYA}
 \+&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

     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 
 elevation just above sea level and is in a former leper colony.
 \centerline{\bf TABLE 2}
 \centerline{Geographic coordinates of the Four Corners}
 \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
 \+&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

     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.
 \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.
 \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. 
 \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.
 \centerline{               TABLE 3}
 \centerline{   Geographic coordinates of SunSweep}
 \settabs\+\qquad\qquad&Campobello Island, NB\qquad\quad
                    &44 50'10" N\qquad\quad
                    &123 05'00" W&\cr
 \+&Site                  &Latitude            &Longitude\cr
 \+&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

      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."
 \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. 
 \centerline{                TABLE 4}
 \centerline{Great Circle Distances between SunSweep Sites.}
 \settabs\+\qquad\qquad&Campobello Island to Lake-of-the-Woods\qquad\quad
 \+&Sites                                     &Distance in miles\cr
 \+&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
      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.
 \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)
 \centerline{``History [Art] consists of relationships rather than 
 \centerline{A. Einstein.}
 \noindent It seems therefore,  inappropriate to forge a linkage with 
 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--
 \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
 \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,
 \centerline{``Out  of pictures,  we imagine a world of solid things,"} 
 \noindent a statement reminiscent of Plato's ``Den". [27]  That is, a 
 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
 \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]  
 \item{``1.}   Postulate of Continuity.  Space is a continuous aggregate 
 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.
 \line{Shadow Postulates \hfil}
 \item{1.}   Postulate  of Continuity.   Total darkness is  a  continuous 
 aggregate  of shadow,  and not a discrete aggregate of individual 

  \noindent Indeed, total darkness on the earth is continuous as it is 
 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  
 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 
 \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  
 ``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 
 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 
     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.
 \line{\bf Notes \hfil}
 \ref 1.   David Barr, 1979.  ``Notes on celebration,"  {\sl The 
 8:   pp.  52-56.   David  Barr,  1981,  ``The four corners of  the 
 world."  {\sl Coevolution\/} 5:  5.   David Barr, 1982, ``The four 
 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\/},  
 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 
 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-
 \ref 10.  Waldo  R. Tobler,  1961,  ``World  map on a  Moebius  strip."  
 {\sl Surveying and Mapping\/},  21:486.    Sandra L.  Arlinghaus, 
 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  
 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-
 \ref  16.  Erwin Raisz, 1941,  The analemma. {\sl The Journal of 
 \ref  17.  Johannes  Kepler, {\sl Prodromus  Dissertationem  
 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 
 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 
 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, 
 \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 
 the General Theory\/}.  (New York:  Bonanza Books, 1961).
 \ref  26.  Henry Moore and W. H. Auden, {\sl Auden Poems, Moore 
 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,"  
 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.
 Sandra L. Arlinghaus is
 Director, Institute of Mathematical Geography,
 2790 Briarcliff,
 Ann Arbor, MI 48105
 David Barr is
 Professor of Art,
 Macomb Community College, South Campus,
 Warren, MI 48093
 John D. Nystuen is
 Professor of Geography and Urban Planning,
 The University of Michigan,
 Ann Arbor, MI 48109
 Written July, 1990, revised, March, 1991, and June, 1991.
 All funds for the art projects described herein were supplied by 
 David Barr.
 \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).
 1.  The exponential function--unbounded population growth
 Assumption:  The rate of population growth or decay at any
 time $t$ is proportional to the size of the population at $t$.
 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.
 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.
 \hbox{ln}|Y_t| = kt + c_0.
 Consider only the positive part, so that
 Let ${Y_t}_0 = e^{c_0}$.  Therefore,
 exponential growth is unbounded as $t \longrightarrow \infty $.
 Suppose $t=0$.  Therefore,
 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}
 2.  The logistic function--bounded population growth.
 Assumption appended to assumption for exponential growth.
 In reality, when the population gets large, envirnomental
 factors dampen growth.  
 The growth rate decreases--
 $dY_t/dt$ decreases.  So, assume the population size is
 limited to some maximum, $q$, where $00$ and $q-Y_t > 0$,
 \hbox{ln}{{Y_t}\over{q-Y_t}}=qKt +qC.
 Replace $e^{qC}$ by $A$.  Therefore,
 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 $q,\,\, K>0$. 
 3.  Facts about the graph of the logistic equation.
 a.  The line $Y_t=q$ is a horizontal asymptote for the graph.
 This is so because, for $b<0$,
 \lim_{t\to\infty }{q\over {1+ae^{bt}}}
 {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, that $a=0$?  No, because $a=1/A$.
 Or, that $e^{bt}=0$--no.
 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.
 b.  Find the coordinates of the inflection point of the
     logistic curve.
 {\bf Vertical component}:
     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:
 To find a maximum (min), set this last equation equal to zero.
 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)].
 {\bf Horizontal component}:
      To find $t$, put $Y_t=q/2$ in the logistic equation and
 q/2={q\over {1+ae^{bt}}}.
 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.
 \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}
 \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}
 \centerline{$\cdots $}
 \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}
 \line{\hfil {\bf Robert Frost} {\sl Neither Out Far Nor In Deep}}

 {\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.
 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.
 1.  GIS (the digitizer) uses coordinates to translate forms (maps)
 into numbers.

 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)
 3.  Thompson's deformations correspond to the ideas of scale
 shifts on maps.  Transformations describe shifts in scale.  Figure

 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.

 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. 

 6.  We look, for future direction, to the Theory of Groups.
 For today, we confine ourselves to a few simple transformations.
 {\bf II. Transformations}
 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].
 A.  Well-defined (single-valued).
 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).
 B. Reversible
 \item{i.} One-to-one correspondence.
 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 $x'$??
 \item{ii.}  Transformations of $X$ onto $Y$
 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).
 \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
 C.  Rubbersheeting
 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.
 {\bf III.  Types of Transformations}

 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.
 Two major types of transformations:
 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}

 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}
 B.  Curvilinear transformations; neither straightness nor
 parallelism is necessarily preserved (Thompson fish, Figure III.3).
 \midinsert \vskip6in 
 \centerline{\bf Figure III.3}
 {\bf IV.  Exercise, page 5, lecture 28, NCGIA.}
 {\bf V. Steiner networks}
 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.
 {\bf VI.  Digital Topology}
 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.
 {\bf VII.  The algebra of symmetry--some group theory}
 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
 \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, $a \star a^{-1}
 = a^{-1} \star a = 1$.
 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."
 A.  The affine group; affine geometry.
     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

     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
 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}

     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}

     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.
 Straight lines
 How can we tell if two lines intersect in a node?
 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
 \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
 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
 Chains of straight line segments.
 How can we tell if chains of segments cross?
 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}

 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.
 Polygon area:
 Calculate polygon area using notion of parallelism (Figure VII.4)
 \midinsert \vskip6in
 \centerline{\bf Figure VII.4}
 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}
 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

 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.
 B.  Group of symmetries of a square (pixel); the hexagonal pixel.

 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 $(13)(24)$:
 \qquad 1 goes to 2 (in the left one)
 \quad and 2 goes to 4 (in the right one)
 \quad so 1 goes to 4 (in the product)  
 \qquad 4 goes to 1 (in the left one)
 \quad and 1 goes to 3 (in the right one)
 \quad so 4 goes to 3 (in the product)  
 \qquad 3 goes to 4 (in the left one)
 \quad and 4 goes to 2 (in the right one)
 \quad so 3 goes to 2 (in the product)  
 \qquad 2 goes to 3 (in the left one)
 \quad and 3 goes to 1 (in the right one)
 \quad so 2 goes to 1 (in the product)  

     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
 \+& 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
 \+& 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
 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$&\cr
 \+&{}&$\star $&$I$  &$R_1$&$R_2$&$R_3$&$H$&$V$&$D_1$&$D_2$\cr
 \+&$I$  &\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
 {\bf Figure VII.6, continued}
 \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 $&Permutation&$=$&Permutation&\phantom{0}\cr 
 \+&(1)(2)(3)(4)&{}&(1)(2)(3)(4)&{}&(1)(2)(3)(4)&$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
 \+&(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
 \+&(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
 \+&(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
 \+&(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
 \+&(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
 \+&(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
 \+&(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

     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

 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

 \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

 \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

 \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 

 \quad{B.}  Point-set topology.

 \qquad{i.}  Definitions.

 \qquad{ii.}  Consequences of Definitions interpreted in GIS

 \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.