**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**

- Ohio State University Undergraduate Recognition Ceremony, May 2003
- Amherst College MAA Student Chapter, October 2003
- University of Akron, Lecture for Undergraduates, November 2003
- Holy Cross Math Colloquium, March 2004
- Faculty Seminar, St. Mary’s College of Maryland, February 2005
- Closing Address, Mathematical Association of America MD/DC/VA Sectional Meeting, James Madison University, April 2008
- Natural Sciences and Mathematics Colloquium, St. Mary’s College of Maryland, February 2010