5 papers
Four-digit Kaprekar dynamics in odd bases
Evan Chen, Ken Ono, Richard E. Schwartz +1
Start with four digits, arrange them in both descending and ascending order, subtract, and repeat. This simple process is known as the Kaprekar routine, famous in base ten for send…
Mostow Rigidity Made Easier
Richard Evan Schwartz
This article gives a self-contained proof of Mostow Rigidity, at least modulo undergrad real analysis. The proof should be accessible to grad students interested in geometry and to…
Vertex-Minimal Paper Tori
Richard Evan Schwartz
A paper torus is an embedded polyhedral torus that is isometric to a flat torus in the intrinsic sense. We prove that there does not exist a paper torus with vertices, and that…
Build Boy's Surface
Richard Evan Schwartz
This article describes Boy's surface in a nice way that does not make many demands on three-dimensional visualization. The article includes a kit that you can print out onto card s…
Collapsibility and Near Universality for Vertex Minimal Paper Tori
Peter Doyle, Richard Evan Schwartz
A paper torus is a piecewise linear isometric embedding of a flat torus into . Following up on the -vertex paper tori discovered by the second author, we prove universalit…