# Foldability of simplicial surfaces onto a triangle

## TL;DR

This paper characterizes which simplicial surfaces can be folded onto a triangle, providing a mathematical framework involving vertex-3-colouring, involutions, and cyclic permutations, and establishes that only orientable surfaces are foldable.

## Contribution

It introduces a novel characterization of foldable simplicial surfaces using vertex-3-colouring, involutions, and cyclic permutations, and proves orientability is necessary for foldability.

## Key findings

- All foldable surfaces admit a vertex-3-colouring.
- Foldability is characterized by the existence of a cyclic permutation with specific properties.
- Only orientable simplicial surfaces can be folded 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.