Rich Schwartz - Virginia Mathematics Lectures - March 23-25, 2026
Event start date:
Virginia Mathematics Lectures series
Rich Schwartz (Brown University)
- Lecture 1 - The Optimal Paper Moebius Band: Monday, March 23, 2026 | 5-6pm | Wilson 402
- Lecture 2 - The Most Efficient Origami Tori: Tuesday, March 24, 2026 | 5-6pm | Wilson 402
- Lecture 3 - Universality and Collapsibility in Vertex-Minimal Origami Tori: Wednesday, March 25, 2026 | 5-6pm | Nau 101
Lecture 1: The Optimal Paper Moebius Band
Date: Monday, March 23, 2026 | 5-6pm
Location: Wilson 402
Abstract: If you take a long and thin rectangular strip, you can smoothly twist it in space, tape the ends together, and produce a Moebius band. If the strip is close to being a square, this is impossible. What is the cutoff? In this talk I will prove that you can make a Moebius band out of a rectangular strip of paper if and only if the aspect ratio of the paper is greater than √3. This result solves the 1978 conjecture about this, due to Ben Halpern and Charles Weaver. I will also explain that any paper Moebius band with aspect ratio close to √3 is very close to kind of 3-fold wrapping of an equilateral triangle. (I will make this precise in the talk.) So, in a limiting sense, this triangular Moebius band is the optimal paper Moebius band.
Lecture 2: The Most Efficient Origami Tori
Date: Tuesday, March 24, 2026 | 5-6pm
Location: Wilson 402
Abstract: An origami torus is a finite collection of triangles in space which fit together in such a way that they make an embedded topological torus, and such that the sum of the angles around each vertex is 2π. An origami torus is something you can make by folding up a flat torus in space along the fold lines of a triangulation, as long as you have a generous interpretation of the word "can". In other words, it is a piecewise affine isometric embedding of a flat torus. It is surprising that origami tori exist at all, but they have been known to exist since the work of Yuri Burago and Victor Zalgaller in 1960. One of the most basic questions you can ask, a long-standing open question, is about the efficiency of these constructions: What is the minimum number of vertices needed for an origami torus? In this talk I will prove that there does not exist a 7-vertex origami torus and that there does exist an 8-vertex origami torus. So, the answer is 8.
Lecture 3: Universality and Collapsibility in Vertex-Minimal Origami Tori
Date: Wednesday, March 25, 2026 | 5-6pm
Location: Nau 101
Abstract: This is joint work with Peter Doyle. Following up on the discovery of 8-vertex origami tori, Peter Doyle and I discovered strong universality and collapsibility results about them. One corollary of our results is that every flat torus without reflection symmetry is realized as the flat structure underlying an 8-vertex origami torus. Another corollary is that there exist 8-vertex origami tori which approximate as closely as you like a certain 6-fold wrapping of an equilateral triangle—a notion I will make precise in the talk. (This result, which resonates with the optimal paper Moebius result, is probably not that deeply connected to it, but it is fun to say.) Likewise, there exist 8-vertex origami tori which approximate as closely as you like a certain 4-fold wrapping of a square. In this talk I will sketch the proofs of our results and illustrate them with computer demos.
The Virginia Mathematics Lectures are supported by the University of Virginia Department of Mathematics and the Institute of Mathematical Sciences.
Last updated: Wednesday, March 4, 2026