Recognition of affine-equivalent polyhedra by their natural developments

Abstract

The classical Cauchy rigidity theorem for convex polytopes reads that if two convex polytopes have isometric developments then they are congruent. In other words, we can decide whether two polyhedra are isometric or not by using their developments only. In this article, we study a similar problem about whether it is possible, using only the developments of two convex polyhedra of Euclidean 3-space, to understand that these polyhedra are (or are not) affine-equivalent.

How to decide whether two convex octahedra are affinely equivalent using their natural developments only

“…An analog of Problem 1 for general polyhedra in Euclidean 3-space is considered in [2], where necessary conditions for the affine equivalence of polyhedra are found. In this article, we present a similar approach as applied to octahedra.…”

“…We choose octahedra as the object of our study for two reasons. First, because they are the simplest polyhedra in terms of combinatorial structure with no trivalent vertices (the latter simplify the problem of recognition of affinely equivalent polyhedra by their natural developments, see [2]). Second, because historically octahedra played an intriguing role in the proof of the Cauchy rigidity theorem (see [9, p. 446]) while our study is motivated by that theorem.…”