Which convex polyhedra can be made by gluing regular hexagons? (Q2308504)

The starting point of this paper is Alexandrov's theorem, saying roughly that every metric which has the topology and local geometry that are necessary for a convex 3D polyhedron is in fact the inner metric of such a polyhedron. The authors ask the special question which convex 3D polyhedra can be obtained by cutting and gluing edge-to-edge finitely many regular hexagons. Such a polyhedron can have at most six vertices, and the authors find that there cannot be more than 15 graphs of such polyhedra. They characterize the five doubly covered 2D polygons obtainable in this way, and they give examples of all (four) simplicial and two non-simplicial polyhedra that are constructed by creasing and gluing regular hexagons edge-to-edge. The realizability of the remaining types remains open. The article closes with the results of computer experiments, on the enumeration of non-isomorphic gluings of \(n\) regular hexagons, and with some open questions.