# Kaleidoscopic Quadrilateral Tilings

**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:** Vanishing Cycles and Kaleidoscopic Quadrilateral Tilings
**Abstract:** For the last 5 years the focus of the Rose-Hulman REU Tilings
group has been hyperbolic, kaleidoscopic tilings of Riemann surfaces by triangles.
A lot has been discovered about these objects including a complete classification
up to genus 13. Last summer we pushed beyond triangles to consider quadrilateral
tilings. On the plus side the group theory did get a bit simpler; on the minus
side we lost rigidity. A surface constructed from triangles is rigid in the
sense that there are no transformations that preserve both angles and area.
This is not true in the quadrilateral case. The euclidean analog is that all
triangles with congruent corresponding angles and the same area are congruent.
However, there is a one-parameter family of mutually non-congruent rectangles
with the same area. On hyperbolic surfaces the same holds true, but there is
an interesting twist. As we vary the quadrilaterals through an infinite family
of equiangular, equal area quadrilaterals some curves on the surface take on
arbitrarily small lengths, and shrink to a point as we go to infinity. These
are the so-called "vanishing cycles" studied in algebraic geometry.
We will show how to identify the vanishing cycles in simple geometric terms.
Much of the talk will be explaining the basic concepts in terms of small visual
examples. Students Isabel Averill, Michael Burr, John Gregoire and Kathryn Zuhr
all contributed to this project.

**Materials**

Email: allen.broughton@rose-hulman.edu
Webpage: http://www.rose-hulman.edu/~brought/
This page last updated on 21 Jan 02.