


Symmetry in Geometry
Org: Ted Bisztriczky (Calgary), Ferenc Fodor (Szeged, Hungary; Calgary), Richard K. Guy (Calgary) and Asia Weiss (York) [PDF]
 CATHARINE BAKER, Mount Allison University, Sackville, NB
Affine planes, symmetry and 3e.c. graphs
[PDF] 
A graph is 3e.c. if, for each triple S of vertices and for each
T Í S, there exists a vertex not in S which is adjacent to
all vertices of T and to no vertices of S \T. The
structure and symmetry of an affine plane provide us with tools for
constructing families of 3e.c. graphs on the point set of the plane
and for determining when the resulting graphs are nonisomorphic.
This is joint work with Anthony Bonato (Wilfrid Laurier), Julia Brown
(York) and Tamás Szönyi (Eötvös University).
 LEAH BERMAN, Ursinus College, Department of Mathematics and Computer
Science, P.O. Box 1000, Collegeville, PA 19426, USA
Symmetric Configurations
[PDF] 
A geometric configuration (p_{q}, n_{k}) is a collection of p points
and q straight lines, usually in the Euclidean plane, so that each
point lies on q lines and each line passes through k points. This
talk will discuss configurations with high degrees of symmetry; in
other words, configurations in which the number of symmetry classes of
points and of lines formed by isometries of the plane mapping the
configuration to itself is small. Of particular interest will be
astral configurations, where a (p_{q}, n_{k}) configuration is
astral if it has precisely ë[(k+1)/2] û symmetry
classes of points and ë[(q+1)/2] û symmetry classes
of linesthe fewest possible.
 ANDRAS BEZDEK, Auburn University, Auburn, AL 36849
On the number of mutually touching cylinders
[PDF] 
The following problem was posed by Littlewood in 1968. What is the
maximum number of congruent infinite circular cylinders that can be
arranged in R^{3} so that any two of them are touching? Is it 7? It
was proved by the author in 2005 that this maximum number is at most
24. The talk will also survey models of mutually touching cylinders
and explain the connection of this problem to Gardner's (1959)
mathematical puzzle concerning cigarettes.
 KAROLY BEZDEK, University of Calgary, 2500 University Drive NW, Calgary,
Alberta T2N 1N4
On the Xray numbers of 3dimensional convex bodies
[PDF] 
P. Soltan (1972) introduced the concept of Xray numbers of convex
bodies. According to a conjecture of K. Bezdek and T. Zamfirescu
(1991), the Xray number of any 3dimensional convex body is at most
6. I give a proof of this conjecture for any convex body with affine
symmetry.
 ROBERT DAWSON, Saint Mary's University, Halifax, NS
Cebysev sets in hyperspaces over R^{n}
[PDF] 
A set in a metric space is called a Cebysev set if it contains
a unique "nearest neighbour" to each point of the space. In this
paper we generalize this notion, defining a set to be Cebysev
relative to another set if every point in the second set has a unique
"nearest neighbour" in the first. We are interested in
Cebysev sets in some hyperspaces over R^{n}, endowed with the
Hausdorff metric, mainly the hyperspaces of compact sets, compact
convex sets, and strictly convex compact sets.
We present some new classes of Cebysev and relatively
Cebysev sets in various hyperspaces. In particular, we show
that certain nested families of sets are Cebysev; as these
families are characterized purely in terms of containment, without
reference to the semilinear structure of the underlying metric space,
their properties differ markedly from those of known Cebysev
sets. (A conjectured link with symmetry did not materialize; thus
this paper has become, in this session, something of a lucus a
non lucendo.)
 ANTOINE DEZA, McMaster University, Hamilton, Ontario, L8S 4K1
Colourful Simplicial Depth
[PDF] 
Inspired by Barany's colourful Carathéodory theorem, we introduce a
colourful generalization of Liu's simplicial depth of a point p in
R^{d} relative to a fixed set S of sample points, i.e., the
number of simplices generate by points in S that contain p. We
prove a parity property and conjecture that the minimum colourful
simplicial depth of any core point in any ddimensional
configuration is d^{2}+1 and that the maximum is d^{d+1}+1. We
exhibit configurations attaining each of these depths, and apply our
results to the problem of bounding monochrome (noncolourful)
simplicial depth. Independently found recent quadratic lower bounds
by Barany and Matousek and by Stephen and Thomas are also presented.
Joint work with Sui Huang (McMaster), Tamon Stephen (Magdeburg) and
Tamas Terlaky (McMaster).
 CHRIS FISHER, University of Regina, Regina, SK S4S 0A2
A Computer Drawing of the Complete Pascal Configuration
[PDF] 
A hexagon inscribed in a conic determines 60 Pascal lines, and these
generate a remarkable configuration of 146 points and 110 lines.
Although the complete Pascal figure has been studied since the middle
of the 19th century, it is only with today's computer graphics that we
are easily able to see it. For the computer to do its job a
systematic notation is needed; this was provided by J. J. Sylvester
(1844), with improvements from Stanley Payne (1973). Norma Fuller and
I will soon have a web page ready to display the figure. In my talk I
will demonstrate how the viewer will be able to explore the numerous
subconfigurations and see how they fit together to form the whole
figure.
 BRANKO GRÜNBAUM, University of Washington, Seattle, WA 98195, USA
Are all symmetry groups present in the Alhambra, and related
questions
[PDF] 
The question which of the seventeen wallpaper groups are represented
in the fabled ornamentation of the Alhambra has been raised and
discussed quite often, with widely diverging answers. Some of the
arguments from these discussions will be presented in detail. This
leads to the more general problem about the validity and meaning of
the answers to such questions. The second part of the presentation
will deals with the approaches to symmetry in ornaments of various
cultures that should replace the mechanical counting of the wallpaper
groups that occur. A more reasonable investigation would deal with
symmetries as they may be considered and understood by the people of
the societies in question. Such an approach to the remarkable orderly
decorations of ancient Peruvian fabrics will be presented.
 HEIKO HARBORTH, Techn. Univ. Braunschweig, Germany
Integral Distances in Geometrical Figures
[PDF] 
Can one find n points in dimension m with pairwise distances being
integral (rational)? What is the minimum largest distance (diameter)
for n points, for n points in semigeneral position (no m+1
points in a hyperplane), and for n points in general position (no
m+1 points in a hyperplane, no m+2 points on a hypersphere)? What
about a rational box? What about graph drawings in the plane with
straight line edges of integral length?
 BARRY MONSON, University of New Brunswick, Fredericton, NB
Down with Symmetry!
[PDF] 
... a little. In a natural way, the faces of ranks 1 and 2 in a
4polytope P provide the vertices of a bipartite graph
G. Recently, Asia Weiss and I have examined this
construction when P is a finite, abstract regular (or
chiral) polytope of Schläfli type {3,q,3}. If in this case
P is also selfdual, then G must be
3transitive (or 2transitive). Here I discuss further work with
the additional help of Egon Schulte and Tomaz Pisanski. We show
that when P is not selfdual, then G
is no more symmetric then it has right to be. Indeed, G
is a trivalent semisymmetric graph, so that Aut
(G) is transitive on edges but not on vertices.
 DEBORAH OLIVEROS, Universidad Nacional Autónoma de México, Circuito
Exterior CU., México DF, México
Polygons enclosing point sets
[PDF] 
We will discuss some problems of the following type: Let R a set of
red points and B be a set of black points in the plane all in
general position. Find a simple polygon P with vertex set R such
that the interior of P contains as many points of B as possible.
 TOMAZ PISANSKI, University of Ljubljana and University of Primorska, Slovenia
Grünbaum incidence calculus
[PDF] 
One of the key problems in the theory of configurations is the
question whether a given combinatorial configuration has a geometric
realization. In the last decade Branko Grünbaum systematically
investigated a more tractable variant of this problem, namely for
which types (v_{r},b_{k}) do there exist geometric configurations. In a
series of papers he developed a large number of interesting
constructions that produce large configurations from small ones,
thereby giving positive answer in majority of cases. In this talk we
present some of his methods as a theory under the name of
Grünbaum incidence calculus.
 EGON SCHULTE, Northeastern University, Department of Mathematics, Boston,
MA 02115
The Classification of Chiral Polyhedra
[PDF] 
This talk presents the complete enumeration of chiral polyhedra in
Euclidean 3space. Chiral, or irreflexibly regular, polyhedra are
nearly regular polyhedra; their geometric symmetry groups have two
orbits on the flags (regular polyhedra have just one orbit), such that
adjacent flags are in distinct orbits. There are several (very)
infinite families of chiral polyhedra, each either with finite skew
polygonal faces and vertexfigures or with infinite helical faces and
planar vertexfigures. Their geometry and combinatorics are rather
complicated.
 ARTHUR SHERK, University of Toronto
A Class of Finite ThreeDimensional Metric Spaces
[PDF] 
Let K be the cubic extension of the finite field F=GF(q), where
Q is any prime power. Being a vector space, K is also an affine
3space. The mapping from K to F known as the Norm imposes a
metric on K. The unit surface C in K is the set of all elements
of K with norm 1. Using the symmetries of the unit surface, we
cover C with a map of type 3,6 (triangular faces, six at each
vertex). It then follows that C is a torus. The covering map has
considerable symmetry, but is not regular.

