7 papers
On Pappus and Anosov Representations of the Modular Group
Richard Evan Schwartz
Let . Let be the space of discrete faithful representations of the modular group into ${\rm Isom\/}(X)$ which map the order generator to an isometr…
The Four Color Theorem meets Shapes of Polyhedra
Richard Evan Schwartz
We consider solutions to the -color problem for the vertices of sphere triangulations with degree sequence . We sort these solutions into combinatorial type…
Continued Fractions and the 4-Color Theorem
Richard Evan Schwartz
We study the geometry of some proper 4-colorings of the vertices of sphere triangulations with degree sequence 6,...,6,2,2,2. Such triangulations are the simplest examples which ha…
The Optimal Twisted Paper Cylinder
Noah Montgomery, Richard Evan Schwartz
An embedded twisted paper cylinder of aspect ratio is a smooth isometric embedding of a flat cylinder into such that the images of the boundary components…
Symplectic Tiling Billiards, Planar Linkages, and Hyperbolic Geometry
Richard Evan Schwartz
In this paper I will unite two games, symplectic billiards and tiling billiards. The new game is called symplectic tiling billiards. I will prove a result about periodic orbits of…
Divide and Conquer: A Distributed Approach to Five Point Energy Minimization
Richard Evan Schwartz
This work rigorously verifies the phase transition in 5-point energy minimization first observed by Melnyk-Knop-Smith in 1977. More precisely, we prove that there is a constant S =…