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.

Twisted augmented bipyramid with isosceles-triangle faces

One of the other previously known families, shown below, has two integer parameters (the numbers of sides on the two half-bipyramids it combines), but I don’t know whether the same double parameterization is possible for the new family.

Combination of two half-bipyramids with isosceles-triangle faces

In 2001, Branko Grünbaum published an example of a polyhedron that could not be realized with congruent faces, even non-convexly,2 but it can be realized as a convex polyhedron with all faces isosceles (or equilateral), as shown below. I don’t know whether there are polyhedra that cannot be realized with all faces isosceles, if one allows the realization to be non-convex (but non-self-crossing and combinatorially equivalent to a convex polyhedron) and the faces to be non-congruent.

Grünbaum's example of a polyhedron that cannot be realized with congruent faces, realized convexly with isosceles and equilateral triangle faces

My new preprint proves that there exist polyhedra (iterated Kleetopes) that cannot be realized as convex polyhedra with isosceles faces. But the construction is a little non-explicit and I don’t know how complicated these polyhedra need to be. For instance, I don’t know whether there is a convex isosceles-face realization of the double Kleetope of the octahedron, shown below.

Double Kleetope of an octahedron

Grünbaum’s example can be realized convexly with only two edge lengths, and my non-isosceles-faced polyhedra require at least three edge lengths in any convex realization. I don’t know whether the number of required edge lengths can be unbounded, or whether non-convex realizations ever require three lengths (although certain stacked polyhedra require at least two).

  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.

    Non-convex polyhedron with congruent isosceles-triangle faces 

  2. Grünbaum, Branko (2001), “A convex polyhedron which is not equifacettable”, Geombinatorics 10: 165–171. I don’t know how to access old papers on this journal in general, but fortunately Grünbaum made his one available on his web site. 

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