Dunce Hat 2010

by Daniel Rajchwald

Mathematics Department, University of Illinois, 2010

This is an NSF sponsored Research Experience for Undergraduates (REU).

I am a junior majoring in engineering physics at the University of Illinois at Urbana-Champaign. My interest in math stems from my exposure to problem solving in high school. I was introduced to problem solving through participation in numerous contests such as the AMC’s and the US Physics Olympiad. The elegance behind the strategies used in solving math and physics problems immediately appealed to me and led me to pursue physics in college. My first set of math classes were mostly applied in nature and it was not until I took real analysis that I learned the basics of rigorous mathematics. I continued to take more math classes and decided to pursue research to get a different perspective of math. In my free time I enjoy biking, tennis, swimming, and reading thrillers. I am the oldest of 4 boys. I was born in Mountain View, California but spent most of my growing years in the Chicago area. I look forward to meeting everyone at the REU!


The topological dunce hat is defined as a triangle with two edges oriented in the same direction with the third edge oriented in the opposite direction. It was discovered by EC Zeeman is known for being a non-collapsible, yet contractible space. John Dalbec (YSU) invented the dunce hat contraction in 1992 at MSRI, Berkeley. After forwarding the animation to George Francis (UIUC), it was translated to C to create the IlliDunce RTICA (Real Time Interactive Computer Animation). In 2006, Mimi Tsuruga (Hunter College) translated the C written animation to Mathematica in several notebooks, the most complete one being Duncehattest. Although the code was adapted from IlliDunce, there are notable differences in the structures of the programs. The purpose of this paper is to document the IlliDunce contraction and to unravel the mathematics featured in IlliDunce and Duncehattest.


Here is my final presentation on the Dunce Hat given at the Illinois Simulator Laboratory on August 3, 2010.