SUMS 2014: Math and Visualization

Visualization Revolutions in Geometry and Topology

Thomas Banchoff, Brown University

What images and models have fueled mathematical exploration in geometry and topology over the past couple of centuries, and why is visualization now taking off in so many new directions? The story starts with paper and plaster models, leading to questions only successfully answered with the advent of modern computer technology, algebraic as well as geometric. Moebius bands and Klein bottles, projective planes and Riemann surfaces all play starring roles in this ongoing story. What will come next?

Playing with Surfaces: Spheres, Monkey Pants, and Zippergons

Kelly Delp, Ithaca College

I will describe a process, inspired by clothing design, of smoothing an octahedron into a round sphere. This process was adapted to build many surfaces out of paper and craft foam. The pattern pieces for the surfaces were designed using a dynamic Mathematica notebook, and cut using a digital cutter. This project was joint with Bill Thurston.

Mating Habits of Polynomials

Sarah Koch, University of Michigan

Given two suitable complex polynomial maps, one can construct a new dynamical system by mating the polynomials; that is, by "gluing" the polynomials together in a dynamically meaningful way. In this talk, we focus on quadratic polynomials -- we begin with a brief discussion of parameter space for quadratic polynomials (where the Mandelbrot set lives), we then define the mating of two quadratic polynomials, and finally we explore examples where the mating does exist, and examples where it does not. There will be many pictures and movies in this talk.

Visualizing Outer Billiards

Richard E. Schwartz, Brown University

Outer billiards is a kind of game in which a point bounces around the outside of a shape according to a simple rule. I will describe how outer billiards works and then demonstrate a graphical user interface I made, which explores the intricate structure of outer billiards when it is defined relative to the Penrose kite.