Voronoi tessellations are all about us. In crystallography, they 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.