Abstracts of Invited Addresses
H.S.M. Coxeter (Univ. of Toronto)
Title: Isohedra With Rhombic Or Rectangular Faces
Chandler Davis (Univ. of Toronto)
Title: Wavelet Systems Which Respect Symmetry Groups
A wavelet or multi-wavelet generates a convenient orthonormal system of functions, starting with a few seed functions. In this talk a new procedure is exhibited for finding the functions. The reward is that the expansions behave pleasantly under some familiar groups of symmetries of 2- or 3- or 4-space. These are discontinuous functions, but hope is held out that continuous wavelets with the same symmetry properties may be in the offing.
Greg N. Frederickson (Purdue University)
Title: Geometric Dissections That Swing and Twist
A geometric dissection is a cutting of a geometric figure into pieces that can be rearranged to form another figure. Some dissections can be connected with hinges so that the pieces form one figure when swung one way, and form the other figure when swung another way. This talk explores two fundamental ways to hinge dissections of 2-dimensional figures such as regular polygons and stars. The first way uses "swing hinges", which allow rotation in the plane. Techniques based on tessellations and infinite strips are presented and analyzed. The second way relies on "twist hinges," which allow one piece to be turned over relative to another, using rotation by 180 degrees through the third dimension. Techniques are introduced to convert swing-hingeable dissections to be twist-hingeable, to change the length (and thus the height) of a parallelogram, and to apply "pseudo-tessellations". The natural goal of minimizing the number of pieces, subject to the dissection being hingeable, is used throughout. The generality of such hinging schemes is discussed.
Sue Whitesides (McGill University)
Title: Geometry in Motion: a Survey of Linkage Movement Problems
A linkage is a physical realization of a graph with positive edge weights. The vertices of the graph are realized as ball joints, and the edges are realized as rods of fixed lengths. Even linkages formed by a single chain or cycle of rods exhibit a variety of interesting properties that have long intrigued investigators. This talk surveys the landscape.
Abstracts of Contributed Papers
George Baloglou (SUNY at Oswego)
Title: Classroom Symmetries
I discuss a General Education course on symmetry that we have been teaching at SUNY Oswego since 1991. The course is mainly focused on the 17 wallpaper patterns and their isometries, and the approach is Euclidean Geometric rather than Group Theoretic. A fairly detailed study of the 63 two-colored wallpaper patterns is viewed not only as a springboard for artistic creations and multicultural explorations, but also as an incentive to delve into the isometry structure of each of the 17 types. That structure is further investigated by way of composing isometries experimentally in the context of colored tilings.
Darrah Chavey (Beloit College)
Title: Symmetries in the Geometrical Diagrams of Malekula
Robert Dawson (St. Mary's University)
Title: Some New Tilings of the Sphere with Congruent Triangles
In this paper, several new tilings of the sphere with congruent spherical triangles will be presented. Unlike those enumerated by Sommerville in 1923 (and again by Davies in 1967), these tilings are not edge-to-edge. In particular, I will present a complete classification of the isosceles triangles that tile the sphere.
Leroy Dickey (University of Waterloo)
Title: A Family of Conics and The Configuration of Pappus
In the real projective plane six conics closely connected with the Configuration of Pappus are identified. These six conics meet by threes in two points, both on the "base lines of the Pappus Configuration. These four points determine a complete quadrangle, and two of its three diagonal points are on all six conics.
Douglas Dunham (University of Minnesota-Duluth)
Title: A Unified Classification Scheme for Repeating Patterns
Repeating patterns have been created in each of the three "classical geometries": Euclidean geometry, spherical geometry, and hyperbolic geometry. Many of these patterns are based on regular tessellations, which exist in each of those geometries. The regular tessellation {p,q} is the tiling by regular p-sided polygons meeting q at a vertex. The values of p and q for the underlying tessellation of a pattern specify the geometry and can be used to classify that pattern among similar patterns in the same geometry. The symmetry group of a pattern provides a finer degree of classification. We show how repeating Escher patterns and Celtic knot patterns can be classified using this scheme.
Gary Ebert (University of Delaware)
Title: Replaceable Nests
In the mid 60's Ted Ostrom described a general method called "net replacement" for creating new affine planes from old ones, the best known example of which is Ostrom's technique of "derivation." Here we survey work of R.D. Baker and the author over the last decade or so concerning certain net replacements in the Desarguesian plane which we call "nests." Several infinite families of translation planes so obtained will be described, including a discussion of their collineation groups and some related geometrical properties. A brief survey of recent attempts at characterizing these planes in terms of the collineation groups admitted will also be given. A key ingredient to the construction process is the relationship developed by R.H. Bruck between the points and circles of a miquelian inversive plane and the lines and reguli of a regular spread.
Richard Esterle (Architect)
Title: Dynamic Polystring Transformahedra Modeling
Dynamic Polystring Transformahedra Modeling (DPTM) U.S. Patent 5,316,483 is a new modeling concept in the exploration of 3-dimensional space in general and polyhedra specifically. Polyhedra are built with tubes passing through a polyhedral core and colored loops of string through these tubes. The color loops of string define the edges of the resulting polyhedron. The color patterns mirrotate at opposite ends of the tubes as well as opposing faces of the polyhedra due to intrinsic structural requirements of the modeling process. The polyhedra deform and/or interpenetrate as a result of sliding the tubes inline (inline translation). This paper will look at a few important embodiments of DPTM and is meant to introduce the concept, but is not comprehensive.
David Flesner (Gettysburg College)
Title: The Klein Model in Dynamic Format
The first part of the presentation is a demonstration of the toolbar of macros that I have written to implement the Klein model of hyperbolic geometry within the dynamic format of the Cabri Geometry software package. We construct midpoints, parallels, perpendiculars, circles, horocycles, equidistant curves, and more =97 all within the context of Klein's model, where all the lines are straight, and almost all the angle measures as well as distances are distorted. The second part of the presentation consists of dynamic helicopter rides over hyperbolic planes filled with various regular and semiregular patterns.
Ellen Gethner (University of British Columbia)
Title: M.C. Escher: Comutational Geometer and Graph Theorist Extraordinaire
We all know that the works of M.C. Escher were derived from deep artistic and mathematical insights. Recently, Doris Schattschneider has called attention to a combinatorial technique used by Escher to create periodic tilings of the plane. Ultimately, a square tile is decorated by a finite set of overlapping polygonal regions, which intersect the boundary of the tile symmetrically and aesthetically. The plane is then tiled by standard horizontal and vertical translations, yielding a doubly periodic wallpaper pattern. That the original tile is made up of overlapping regions lends an air of mystery to the final outcome. Escher deepened the mystery by adding color to the mix, and in doing so aroused the curiosity of Rick Mabry, Stan Wagon, and Doris Schattschneider. Their question: is there arectangular prototile (concatenated copies of the original square tile), which can be colored so that colors on the sides and top and bottom match appropriately? Can this be done in such a way that overlapping components in the resulting plane pattern always receive different colors? I will talk about the history of this problem, and how Escher himself revealed a key component of the solution.
Chaim Goodman-Strauss (University of Arkansas)
Title: Compass and Straightedge Constructions in the Poincare' disk
This is a talk in "forensic geometry". We give a synthetic approach to the development of hyperbolic geometry;all our constructions are constructions with a Euclidean compass and Euclidean straightedge, and can be carried out by hand. Indeed, M.C. Escher used something like the methods we give here to produce his well known Circle Limit I--IV prints. The technique for his constructions is quite ingenious but has been described only incidentally, in a 1979 paper by Coxeter.
Peter Hamburger (Indiana-Purdue University Fort Wayne)
Title: Pretty Drawings
If you or somebody in your family is a compulsive doodler, then you will enjoy this presentation; if not, you can still appreciate it. In this talk, we will learn how to create pretty symmetrical drawings from doodles. In this journey, there will be some geometry, number theory, group theory, and some graph theory and combinatorics topics such as codes, dual graphs, symmetrical and maximal chains and others. This also will answer a conjecture of B. Grunbaum that goes back to D.W. Henderson.
Norman Johnson (Wheaton College)
Title: Coxeter Subgroups of Coxeter Groups
All discrete groups generated by reflections in the facets of a spherical or Euclidean simplex were described by Coxeter in the 1930s, and the analogous hyperbolic groups were enumerated a few years later by Coxeter & Whitrow, Lanner, and Koszul. Such _Coxeter groups_ include the symmetry groups of all regular polytopes and honeycombs, as well as many uniform figures. A number of regular or uniform compounds are associated with the occurrence of one Coxeter group as a subgroup of another of the same rank. In an early paper Coxeter himself determined all the instances involving spherical (finite) groups. Later investigations, some quite recent, have extended these results to Euclidean and hyperbolic groups.
Jay Kappraff (New Jersey Institute of Technology)
Title: The Relationship between the Pythagorean Musical Scale and Theories of Proportion as seen through the Arithmetic of Nichomachus
A number series appeared in "Introduction to Arithmetic" by Nichomachus of Gerasa in approximately 100 A.D. This series contains the elements of the Pythagorean musical scale and the system of proportion of Leon Battista Alberti. Its generalization contains the roots of the ancient Roman system of proportions documented in the writings of Theon of Smyrna, a contemporary of Nichomachus. The relation of music and systems of proportion to number will be described. A system of proportions based on the golden mean will be shown to be a limiting case. Nichomachus's arithmetic will be shown to suggest an early awareness of Farey series.
Janusz Kapusta (Brooklyn, New York)
Title: The Circle, the square, and the Golden Proportion
The reason behind taking another look at the number Phi is its overwhelming appearance in art, nature, and mathematics.
Yanxi Liu (Carnegie Mellon University)
Title: Frieze and Wallpaper group Classification and Motif Generation
from Real Images
Humans have an innate ability to perceive symmetry yet it is not obvious how to automate this powerful insight. Our work aims at making computers understand repeated patterns by constructing a computer algorithm that can recognize the symmetry group (one of the seven Frieze groups or one of the 17 wallpaper groups) of a given repeated pattern image. We constructed and implemented a symmetry group classification algorithm (for computers) that takes as input a wallpaper pattern under Euclidean, affine or perspective distortions. The algorithm starts with a novel peak detection algorithm based on "regions of dominance" that can automatically detect the underlying translational lattice of a repeated pattern. This lattice construction algorithm is reliable in that it can handle local pattern irregularities due to defects and missing features, as well as global geometric distortions due to oblique viewing angles. Another aspect of our current work is making the connection between the problem of extracting visually meaningful building blocks (motifs) from a repeated pattern, and the symmetry group of the pattern itself. Study of the interplay among rotation, reflection, glide-reflection and translation symmetry in a pattern leads to a small finite set of candidate motifs that exhibit local symmetry consistent with the global symmetry group of the entire pattern, and can be automatically generated.
Our work is the first to introduce robust computations of Frieze and Wallpaper groups into the computer vision community, and the first to seek a principled method for determining a representative motif of a repeated pattern automatically. Experiments have carried out on synthetic and real images. Precise classification of 2D repeated patterns in terms of their symmetry groups provides a computational means for image indexing, image matching, object recognition, pose and motion recovery.
Joseph Malkevitch (York College (CUNY))
Title: Convex 3-polytopes with Isosceles Triangle Faces
Many interesting conditions have been found which lead to either finite collections or easy to classify infinite families of convex 3-dimensional polytopes. These conditions lead to such classes as the Platonic Solids, the Archimedean Solids, the Johnson Solids, and the convex deltahedra. Generalizing the convex deltahedra, we consider the question of classifying convex 3-polytopes whose faces are congruent isosceles triangles. This approach leads to a variety of interesting geometrical questions as well as graph theory problems.
Vince Matsko (Quincy University)
Title: Vertex-Finite Tilings in Three Dimensions
With the usual defintion of a tiling in three dimensions, tilings are only possible when the individual polyhedral tiles have tetrahedral or octahedral symmetry (such as the tiling of space with cubes or with tetrahedra and octahedra). By relaxing some restrictions to allow space to be multiply (but finitely) covered at each vertex of the tiling, numerous tilings of space emerge using tiles with icosahedral symmetry.
Ben Nicholson (Illinois Institute of Technology)
Victor Piotrowski (University of Wisconsin-Superior)
Title: Centroids in Convex Geometries
The weight of a point P in a finite convex space (X,C) is defined as the size of the largest convex set not containing P, and the centroid is the set of points with the minimum weight. The centroids are proven to be convex, and they consist of extreme points if the space is a convex geometry. A set K of vertices of a connected graph is convex if for every pair of vertices in K, all vertices of all minimal paths joining them also lie in K. Jordan proved that the centroid of a tree is either one vertex or two adjacent vertices. This result is generalized and the centroid of a chordal graph is proven to be a complete graph. Also, necessary and sufficient conditions for a graph to be a centroid of another graph as well as of itself are given. Centroids of several types of graphs are described.
Doris Schattschneider (Moravian College)
Title: A Pentagonal Tiling Challenge
A pentagonal tile that could have easily been constructed by Euclid is the star attraction in a tiling in the lobby of the MAA headquarters in Washington, DC. Although this tile can fill the plane in many simple ways, it also can produce some complex and beautiful tilings. Marjorie Rice, amateur mathematician of pentagonal tiling fame, discovered the 3-isohedral tiling that now graces the MAA lobby, as well as other tilings with this tile. I will challenge the participants to find as many edge-to-edge tilings by this tile as possible-- how many there are is not known.
Philip Straffin and Darrah Chavey (Beloit College)
Title: Sona Geometry
As part of our Ethnomathematics course at Beloit, we teach a unit on Tshokwe sona, drawings made with one continuous line through an array of guiding dots and walls. (See Ascher, Ethnomathematics or Gerdes, Geometry from Africa.) We will discuss theorems about the construction of sona and associated discovery activities for students.