Folding Polyiamonds into Octahedra

TL;DR

This paper investigates the conditions under which polyiamonds can fold into an octahedron, providing characterizations, size bounds, and algorithms for testing foldability, along with face coverage assignments.

Contribution

It offers the first comprehensive analysis of polyiamond foldability into octahedra, including characterizations, size bounds, and polynomial-time testing methods.

Findings

Almost all polyiamonds with holes are foldable into the octahedron.

Convex polyiamonds fold into the octahedron if they contain one of five specific shapes.

Polyiamonds of size 15 or more always fold into the octahedron.

Abstract

We study polyiamonds (polygons arising from the triangular grid) that fold into the smallest yet unstudied platonic solid -- the octahedron. We show a number of results. Firstly, we characterize foldable polyiamonds containing a hole of positive area, namely each but one polyiamond is foldable. Secondly, we show that a convex polyiamond folds into the octahedron if and only if it contains one of five polyiamonds. We thirdly present a sharp size bound: While there exist unfoldable polyiamonds of size 14, every polyiamond of size at least 15 folds into the octahedron. This clearly implies that one can test in polynomial time whether a given polyiamond folds into the octahedron. Lastly, we show that for any assignment of positive integers to the faces, there exist a polyiamond that folds into the octahedron such that the number of triangles covering a face is equal to the assigned number.