# Kaleidoscopic Tilings on Surfaces, This Time with the Groups

**S. Allen Broughton**
[Rose-Hulman Talk]

###
Abstract

This web page contains the notes and supporting scripts for a
talk given in the Rose-Hulman Mathematics faculty-student seminar.

### Rose-Hulman Mathematics
Faculty Student Seminar - Winter 2002-03

**Title:** Kaleidoscopic tilings on surfaces, this time with the groups.
**Abstract:** In the past I have given several lectures on kaleidoscopic
tilings by triangles and quadrilaterals on surfaces, and asserted in these talks
that the tiling group completely determined the combinatorial and topological
structure of a tiling. However, I have never really talked about the influence
of the group theory! In this series of two talks I will give two examples of
determining combinatorial and topological structure, by group computations.
Each talk will focus on a problem I intend to give to REU students this summer.
Thus, there will be no general theorems just problems statements with suggestions
of attack, the talks will focus on developing the background to get to the problem
statements. The first talk will include the necessary review of tilings and
hyperbolic geometry. You don't need to know much about group theory or hyperbolic
geometry.

First talk: Constructing a fundamental domain for kaleidoscopically tiled surfaces.
We are all familiar with the process of creating a torus by identifying opposite
sides of a euclidean rectangle. For higher genus surfaces of genus s > 1,
a surface may be constructed by identifying sides of a hyperbolic 4s-gon. For
a kaleidoscopically tiled surface can this be done so that the polygon is a
"nice" collection of tiles? The group theory computation will be focus
on relating the infinite tiling group on the hyperbolic plane to the finite
tiling group on the surface.

Second Talk: When are kaleidoscopic tilings separating? Every edge of a kaleidoscopic
tiling generates a reflection of the surface to itself fixing the edge. In the
case of a sphere the fixed point set (or mirror) of the reflection is a great
circle which separates the sphere into two pieces. This is very misleading example,
since for higher genus the mirror very rarely separates the surface. The question
is: is there a fast way to determine this splitting property from the properties
of the tiling group? The talk will present a method of attack using the group
algebra of the talk. Again, no previous knowledge of group theory is assumed.

**Materials**

Email: allen.broughton@rose-hulman.edu
Webpage: http://www.rose-hulman.edu/~brought/
This page last updated on 7 May 03.