• ## Eberhard's theorem for bipartite polyhedra with one big face

Eberhard’s theorem is a topic in the combinatorial theory of convex polyhedra that once saw a lot of research, but has faded from more recent interest. It’s named after Victor Eberhard, a German mathematician from the late 19th and early 20th century who worked in geometry despite becoming blind at age 12 or 13. I find this hard to imagine, as my own research in geometry is based very heavily on visual thinking, but he was far from the only successful blind mathematician; Leonhard Euler, Lev Pontryagin, and Bernard Morin also come to mind, and there are more.

• ## Isosceles polyhedra

My latest arXiv preprint is “On polyhedral realization with isosceles triangles”, arXiv:2009.00116. As the title suggests, it studies polyhedra whose faces are all isosceles triangles. Despite several new results in it, there’s a lot I still don’t know. The paper finds a sort-of-new1 infinite family of polyhedra with congruent isosceles faces, shown below, but I don’t know if there are any more such families.

1. The family of polyhedra from the first image is only “sort-of-new” because the same combinatorial structure was previously described as a triangulation of the sphere by congruent spherical isosceles triangles: Dawson, Robert J. MacG. (2005), “Some new tilings of the sphere with congruent triangles”, Renaissance Banff. In exchange for re-purposing Dawson’s triangulation, my paper describes another infinite family of spherical triangulations by congruent spherical isosceles triangles, not given by Dawson, based on applying a similar $$2\pi/3$$ twist to an infinite family of non-convex bipyramids with congruent isosceles faces like the one below. Again, I don’t know whether there are other such families of spherical triangulations.

• ## Bricard's jumping octahedron

The Schönhardt polyhedron is a non-convex octahedron that can be formed from a convex regular octahedron by twisting two opposite faces, stretching and deforming the other faces as you twist. It’s well known for not having any interior diagonals, and for being impossible to subdivide into tetrahedra without introducing new vertices. But long before Erich Schönhardt described it in 1928 in connection with these properties, Raoul Bricard was investigating flexible octahedra, in connection with Cauchy’s theorem on the rigidity of polyhedra. The Schönhardt polyhedron forms an interesting example of flexibility, as I learned from a 1975 collection of lecture notes by Branko Grünbaum on “Lost Mathematics”). I’m not entirely sure that it was known to Bricard (it’s not clear from Bricard’s paper and Grünbaum doesn’t really say so) but it wouldn’t surprise me if it was.

• ## Report from CCCG

I spent the last few days participating in the Canadian Conference in Computational Geometry, originally planned for Saskatoon but organized virtually instead.

• ## Sona enumeration

The last of my CCCG 2020 papers is now on the arXiv: “New Results in Sona Drawing: Hardness and TSP Separation”, arXiv:2007.15784, with Chiu, Demaine, Diomidov, Hearn, Hesterberg, Korman, Parada, and Rudoy. (As you might infer from the long list of coauthors, it’s a Barbados workshop paper.) The paper studies a mathematical formalization of the lusona drawings of southwest Africa; in this formalization, a sona curve for a given set of points is a curve that can be drawn in a single motion, intersecting itself only at simple crossings, and surrounding each given point in a separate region of the plane, with no empty regions. The paper proves that it’s hard to find the shortest one, hard even to find whether one exists when restricted to grid edges, and gives tighter bounds for the widest possible ratio between sona curve length and TSP tour length; see the preprint or the video I already posted for more information.