Foldability of simplicial surfaces onto a triangle

Abstract

We characterise which simplicial surfaces can be folded onto a triangle. We define a notion of folding that incorporates the non-intersection-properties of real materials. All of the surfaces foldable onto a triangle admit a vertex-3-colouring. Based on this colouring, we can describe the surface by three involutions that act on the faces of the surface. A simplicial surface is foldable onto a triangle if and only if there exists a cyclic permutation on all faces, whose products with the involutions have a specified number of cycles. In addition, we show that all simplicial surfaces that can be folded onto a triangle have to be orientable.