


Discrete and Computational Geometry
Org: Leroy J. Dickey (Waterloo) and Asia Ivic Weiss (York) [PDF]
 LEAH BERMAN, Ursinus College, Department of Mathematics & Computer
Science, PO Box 1000, Collegeville, PA 19426, USA
Odd Astral Configurations
[PDF] 
An astral configuration (p_{q}, n_{k}) is a collection of p points and
n straight lines, usually in the Euclidean plane, where every point
is incident with q straight lines, every line is incident with k
points, and there are precisely ë[(q+1)/2] û
symmetry classes (transitivity classes) of lines and ë[(k+1)/2] û symmetry classes of points. An odd
astral configuration is an astral configuration where at least one of
q and k is odd. This talk will discuss current known results
about the existence of odd astral configurations where q and k are
at least 4.
 TED BISZTRICKY, University of Calgary
On edgeantipodal polytopes
[PDF] 
A convex polytope is edgeantipodal if any two vertices,that determine
an edge of the polytope, lie on distinct parallel supporting
hyperplanes of the polytope. We survey recent results about such
polytopes of dimension three or four.
 JÜRGEN BOKOWSKI, Darmstadt University of Technology, Germany
Haskell and the Theory of Oriented Matroids
[PDF] 
The use of functional programming within the theory of oriented
matroids helps very much to deepen the understanding of various data
structures within this fundamental part of Discrete Geometry. The
author uses the functional programming language "Haskell" and main
aspects of his forthcoming book, Computational Oriented
Matroids (Cambridge University Press, 2005), to underline the above
assertion. His intention is to invite the novice, perhaps even a
novice in both disciplines, in Functional Programming and in the
Theory of Oriented Matroids, to understand the benefit of putting
these disciplines to work together.
With a 3page list of basic Haskell functions, we cannot only write
the Sieve of Erastostenes for finding prime numbers as an executable
line ps = sieve[2...] where sieve(p:ns) = p:sieve[n  n < ns, n¢mod¢p > 0], we can also reduce the burden of dealing with sign
vectors that seem to have no geometric meaning for the novice.
 ROBERT DAWSON, Saint Mary's University, Halifax, Nova Scotia
NonEdgetoEdge Triangulations of the Sphere
[PDF] 
The tilings of the sphere with congruent spherical triangles meeting
edgetoedge were enumerated under certain assumptions by Sommerville
in 1923, and completely by Davies in 1965. If we do not require the
triangles to meet edgetoedge, further tilings are possible. In this
talk I will report (with plenty of pictures) on a probablycomplete
enumeration of spherical triangles that tile in such a fashion.
 LEE DICKEY, University of Waterloo
Steiner Conics
[PDF] 
Given a Pappus configuration, six distinct points 3 by 3, on two
distinct lines, we find six different Pappus lines by permuting the
points of each 3 set. These Pappus lines meet by threes in two
points, which we call Steiner points. We study the conic that is the
locus of such a Steiner point as a function of any one of the original
six points.
 BOB ERDAHL, Mathematic and Statistics, Queen's University
Voronoi's Conjecture on Parallelotopes
[PDF] 
A parallelotope is a polytope with the following, very restrictive,
combinatorial property: a selection of translates of the polytope fit
together facettofacet to tile space. It is easy to show that the
centers of the parallelotopes of such a tiling form a lattice. On the
other hand, starting with a geometric lattice, the Voronoi polytope is
determined by the Euclidean metric: the Voronoi polytope for a
particular lattice point is the set of points as close to that lattice
point as to any other. The Voronoi polytopes for all lattice points
fit together facettofacet to tile space, so a Voronoi polytope is a
special type of parallelotope. Voronoi conjectured in 1909 that these
two tilings, definded is such different ways, are in fact equivalent;
Voronoi conjectured each parallelotope tiling is affinely equivalent to
a Voronoi tiling.
I will report on the recent work of Andrei Ordine who established that
the Voronoi Conjecture holds for a new special casefor the case
where the parallelotope is 3irreducible, a condition that will be
defined during the course of my lecture. I will also review the
status of the Voronoi Conjecture, describing the various cases for
which the Conjecture has been proven to hold, and discuss prospects
for further developments.
 WENDY FINBOWSINGH, Acadia University
Low Dimensional Neighbourly Polytopes
[PDF] 
Among the dpolytopes with v vertices, the neighbourly polytopes
have the greatest number of facets. This maximum property of
neighbourly polytopes has prompted researchers to compile lists of
them. In this talk, we will discuss the simplicial neighbourly
5polytopes with nine vertices. We show that there are at least one
hundred, twentysix of them. We discuss the connection between the
neighbourly 4polytopes with eight vertices, the neighbourly
5polytopes with nine vertices, and the neighbourly 6polytopes with
ten vertices.
 CHRIS FISHER, University of Regina, Regina, SK S4S 0A2
Fourier Series and Ovals in Finite Geometry
[PDF] 
A century ago, Hurwitz showed how Fourier series can be used to study
geometric properties of curves in the Euclidean plane that bound
convex sets. Ten years ago, Helmut Groemer wrote a book on the
subjectGeometric Applications of Fourier Series and
Spherical Harmonics. It turns out to be convenient to enlarge the
scope to include all oriented smooth closed curves (in the plane) that
have, for 0 £ f £ 2p, exactly one point where its directed
tangent makes an angle of f with the positive xaxis. Such a
curve has a parametrization z(f) : R ®C, where z(f) = z(f+2p) and the unit tangent
vector is e^{if} for all f Î R. These curves
form a vector space over R that properly contains the
boundaries of convex sets. Since many of the properties of such
curves that one can investigate using Fourier series involve only
algebra, much remains valid when R and C are
replaced by some other field and its quadratic extension. In
particular, I am interested in the geometry of the planes represented
by the finite field GF(q^{2})for x and y elements of the finite
field GF(q) of order q, the point (x,y) corresponds to z=x+iy,
where i is an element of GF(q^{2}) that is not in GF(q). In fact,
there is a point to this generalization: the use of finite Fourier
series sheds light on a 50yearold problem of Beniamino Segre, who
sought a way to classify the ovals of finite projective planes. An
oval is defined to be a set of q+1 points, no three on a line; it
can be represented by a Fourier series that resembles that of its
continuous kin.
 ISABEL HUBARD, York University
Twisting selfdual chrial 4polytopes
[PDF] 
A chrial polytope is a non reflexible polytope with maximal symmetry
by rotation. Selfdual chiral polytopes can be properly or
improperly. Properly selfdual chrial 4polytopes can be twisted to
obtain regular maps. A similar operation can be done for improperly
selfdual chiral 4polytopes which will give us chiral quotents of the
PetrieCoxeter polyhedra.
 LILY MOSHE, York University, 4700 Keele St., Toronto, ON M3J 1P3
Duality in Inductive Constructions of Circuits in 2Rigidity
via Tree Partitions
[PDF] 
Given a graph G=(V,E), a realization in the plane G(p),
as a framework, may be firstorder rigid or firstorder flexible. A
basis is a minimal firstorder rigid framework, while a circuit is a
minimally redundant framework. For all generic realizations, the
behaviour in the plane is determined by the combinatorics of the
graph G. In this talk, we will investigate inductive constructions
for extending a circuit to a larger circuit, for graphs on V
vertices which are rigid for generic realizations in the plane.
Using Crapo's characterization of circuits in terms of two spanning
trees, we explore some global properties of the circuits in terms of
local changes in the graphs around a few vertices. For circuits on
planar graphs, these inductions have a striking duality. The plane
dual of a 2circuit is a 2circuit. This duality pairs techniques
in a direct fashion and offers additional insights and corollaries.
This talk is based on joint work with Walter Whiteley and Laura
Chávez Lomelí.
 TOMAZ PISANSKI, IMFM, University of Ljubljana and University of Primorska
Dimension of unsplittable incidence structures
[PDF] 
A combinatorial incidence structure C = (P,L,I) consists of 'points'
P, 'lines' L, and an incidence relation I between them. The
corresponding bipartite incidence graph is sometimes called the Levi
graph of the structure.
A collection of points and lines in the Euclidean space defines a
geometric incidence structure. Each geometric incidence structure
determines a unique combinatorial incidence structure. The converse
is not true. An incidencepreserving mapping of 'points' and 'lines'
of a combinatorial incidence structure C to points and lines of an
Euclidean space is called a representation of C.
dim(C) is the maximum dimension of the affine span of any geometric
incidence structure representing combinatorial incidence
structure C.
Ggraph is the graph square of the Levi graph of an incidence
structure C. An incidence structure C is unsplittable if by
removing any set of vertices independent in Ggraph from the Levi
graph of C, the Levi graph remains connected. This talk will
indicate some problems concerning the dimension of unsplittable
incidence structures in particular for some combinatorial or geometric
configurations.
This is joint work in progress with Branko Grünbaum.
 DORETTE PRONK, Department of Mathematics and Statistics, Dalhousie
University, Halifax, NS B3H 3J5
Touching Woodthe Shape of Fractal Trees
[PDF] 
In this talk we study nonoverlapping symmetric binary fractal trees
as representations of the free monoid M(L,R) on two generators L
and R. Such a tree is determined by an angle q and a scaling
ratio r, such that the interiors of its branches do not overlap.
Motivated by techniques from shape theory and computational topology
we consider for each fractal tree the homology of its inverse system
of closed eneighbourhoods. We show that holes in these
eneighbourhoods are determined by certain pairs of
"contact vertices" on the tree and use this to identify different
types of holes.
This leads us to consider a coloured version of the "topological
barcode" (as introduced by Carlsson et al. in [1]) of a
fractal tree. The topological barcode encodes for each hole its
persistence interval, that is, the interval of the epsilon values for
which this hole exists. Moreover, there is a natural action of the
monoid M(L,R) on the coloured barcode. This will lead to several
new classifications of symmetric binary fractal trees. Tara Taylor
will discuss this in more detail in her presentation in this session,
including several interesting "golden" examples.
This is joint work with Tara Taylor (St. Francis Xavier University).
References
 [1]

G. Carlsson, A. Zomorodian, A. Collins and L. Guibas,
Persistence barcodes for shapes.
In: Symposium on Geometry Processing, Nice, France, July 810, 2004
(can also be found on the website
math.stanford.edu/comptop/preprints/).
 FRANCO SALIOLA, Cornell University, Ithaca, NY, USA
Geometry and Algebra associated to Hyperplane Arrangements
[PDF] 
The geometry and combinatorics of hyperplane arrangements define an
associative product on the faces of the arrangement and is used in
determining various properties of the resulting semigroup algebra. In
turn, the properties of the algebra afford geometric and combinatorial
results about the arrangement. An example of this interplay is the
result that the face semigroup algebra depends only on the
intersection lattice of the hyperplane arrangement. It is obtained by
constructing the quiver of the face semigroup algebra using the
geometry of the hyperplane arrangement.
 EGON SCHULTE, Northeastern University, Department of Mathematics, Boston,
MA 02115, USA
Reflection groups and polytopes over finite fields
[PDF] 
Any finite or infinite Coxeter group G with string diagram is the
automorphism group of an abstract regular polytope. When G is
crystallographic, its standard real representation is easily reduced
modulo an odd prime p, thus giving a finite representation in some
finite orthogonal space V over the field with p elements. The
finite group need not be polytopal; and whether or not it is depends
in an intricate way on the geometry of V. The talk presents recent
work with Barry Monson, in which we describe this construction in
considerable generality and study in depth the interplay between the
geometric properties of the polytope (if it exists) and the algebraic
structure of the overlying finite orthogonal group. The rank 4 case
has been worked out in considerable detail.
 TARA TAYLOR, St. Francis Xavier University, Antigonish, Nova Scotia
Finding Gold in the Forest: Fractal Trees and The Golden
Ratio
[PDF] 
This talk will expand on the talk presented by Dorette Pronk of
Dalhousie University (in the same session). We continue with a
discussion of the classifications of symmetric binary fractal trees
via an analysis of the closed eneighbourhoods using methods
of computational topology. We illustrate the theory we have developed
to study fractal trees by discussing an overview of the topology of
four selfcontacting trees that are related to the golden ratio.
These four trees each possess remarkable symmetry, and their
topological barcodes demonstrate the rich structure that a topological
barcode can add to a fractal tree.
 ASIA IVIC WEISS, York University
Gray graph as medial graph of a 4polytope
[PDF] 
Each combinatorial polytope of rank 4 which can be assigned
Schläfli symbol {k,q,k} yields a bipartite kvalent graph,
called medial graph of the polytope, whose vertices are the faces of
the polytope of ranks 1 and 2. Two vertices of such graph are
adjacent whenever the corresponding faces are incident. I shall
present recent work with Barry Monson, in which we prove that the
medial graph for the polytope with regular toroidal facets
{3,6}_{(3,0)} and vertexfigures {6,3}_{(1,1)} is an
edgetransitive graph with 54 vertices, known as the Gray graph. In
fact, our construction yields an infinite family of edgetransitive
graphs which are not vertextransitive. Gray graph is the smallest
known graph with this property.
 WALTER WHITELEY, York University, Toronto, Ontario
Some applications of rigidity to control of formations
[PDF] 
Recent work on control of formations of robots has used a number of
results from rigidity theory, while adding some new questions about
directed graphs (which of the end vertex agents is responsible for
maintaining the length of the edge?). In addition to graphs which can
be rigidly maintained, there are problems of `control'will noise
disrupt the convergence of the formation to the desired path?
We will present some key results drawn from rigidity theory and as
well as currrent unsolved problems for formations of agents in the
plane, with distance constraints.
This is part of joint work with groups of Steven Morse (Electrical
Engineering, Yale), Richard Yang (Computer Science, Yale) and Peter
Belhumeur and Tolga Eren (Computer Science, Columbia).
 GORDON WILLIAMS, Ursinus College, PO Box 1000, Collegeville, PA 19426, USA
Geometric Realizations of NonConvex Simplicial Spheres
[PDF] 
The study of combinatorial types of convex polytopes has been an
extremely fruitful one for twentiethcentury mathematics. In 1967,
Grünbaum and Shreedharan discovered some errors in Brückner's 1909
enumeration of the simple convex polytopes with 8 facets in
4dimensional Euclidean space. In particular, they discovered that
one of the combinatorial types listed by Brückner did not admit a
realization as the boundary of a convex polytope. Since then, much
work has been done on determining which simplicial spheres admit
realizations as the boundaries of convex polytopes; we will call such
spheres convexly realizable. This talk will review some of
that work and explore some of the interesting questions that arise
when one begins to investigate the geometric structure of those
simplicial spheres that are not convexly realizable.

