## Flat Surfaces

### Billiards and Flat Surfaces - Sabbatical Report Part I

S. Allen Broughton, Rose - Hulman Institute of Technology (presenter)

October 1, 2008
Rose Math Seminar

Abstract: What do flat surfaces like a cube or icosahedron have to do with billiards? The billiard question is simply: If you hit a billiard on a polygonally shaped billiard table and it continues indefinitely, will it eventually get near to every point on the table? The answer is fairly easy for rectangular shaped tables but more complicated for other shapes. In this talk we will discuss how flat surfaces arise from the discussion of billiards and look at some of the properties of flat surfaces, including a suitable interpretation of Euler's formula. This talk is the first of two sabbatical report talks from Professor Broughton's sabbatical at Indiana University last spring. The first talk is a motivational introduction to flat surfaces and is intended for a general audience of Rose faculty and students. The second talk, to be given later in the year, will discuss additional concepts and problems about flat surfaces suitable for undergraduate research topics.

### Voronoi Tesselations, Delaunay Tesselations and Flat Surfaces - Sabbatical Report Part II

S. Allen Broughton, Rose - Hulman Institute of Technology (presenter)

Decmeber 10, 2008
Rose Math Seminar

Abstract: Voronoi tessellations are all about us. In crystallography, the can be used to define a unit cell. In coding theory the can be used measure effectiveness of detection and correction of errors in transmission. The sizes of the cells can give us information about uniform placement of points on a sphere such as satellites in the sky. Delaunay tessellations are dual to Voronoi tessellations and have their own uses. In the first part of this talk we will give some examples of the tessellations and discuss algorithms for determining them. In the second part of the talk we will look at how these tessellations can be used to understand the geometry of flat surfaces, such as a cube or icosahedron.

This talk is the second of two sabbatical report talks from Professor Broughton's sabbatical at Indiana University last spring. The first talk "Billiards and Flat Surfaces" was a motivational introduction to flat surfaces intended for a general audience. This second talk, will discusses additional geometrical concepts and problems about flat surfaces suitable for undergraduate research.

#### Lecture and other materials

Email:  allen.broughton@rose-hulman.edu
Webpage:  http://www.rose-hulman.edu/~brought/