Inflating the Cube Without Stretching

Abstract

Here by isometric we mean that the geodesic distance between pairs of points on the non-convex polyhedron is always equal to the geodesic distance between of the corresponding pairs of points on a cube. Alternatively, it means that two surfaces can be triangulated in such a way that they now consist of congruent triangles which are glued according to the same combinatorial rules. For example, if we push a vertex v of a cube C inside as shown in the Figure below, we obtain a polyhedron P whose surface ∂P is isometric to S = ∂C. Of course, vol(P ) < vol(C) in this case.