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