## The Geometries of Escher

Sample Abstract
A recurrent theme in the intoxicating images of M.C. Escher is the division of the plane into animal figures. Underlying each division is a tiling of the plane by simple polygons. Escher's spirit of artistic adventure led him to explore more exotic patterns, and we will follow his path into tilings of the sphere and of the hyperbolic plane.

If you know that the sum of the angles in a triangle is 180 degrees, then you can follow the math in this talk.

Description
This talk is closely related my talk on the history of non-Euclidean geometry. I developed this presentation for Ohio State University's Undergraduate Recognition Ceremony. The department invited me to give a talk pertaining to the Mathematical Association of America theme for the year, Mathematics and Art, and this was the result.

After a brief survey of Escher's mathematical and artistic themes, we discuss the polygonal tilings underlying Escher's tesselations, focusing on the triangular reflection tilings. We see that in the Euclidean plane and on the surface of the sphere, there are strong restrictions on the types of triangles that allow reflection tilings, and only finitely many triangular shapes can tile each surface. However, in the non-Euclidean plane, there are infinitely many such tilings. As we go, we see Escher prints and sculptures constructed in all three geometries.

I am indebted to Daniel Shapiro of the Ohio State University for suggesting this focus for the talk.

Level
High School and up.

The claim in the abstract that it is sufficient to know the angle sum in a Euclidean triangle is accurate. At some points in the talk I do use radians instead of degrees, but not in a way that involves any higher math.

Mechanics
For this talk, I need a computer projector. I also exhibit and pass out models, so the room needs not to be too dark.

Appearances