Reversible Nets of Polyhedra

TL;DR

This paper explores the concept of reversible nets of polyhedra, extending previous polygon-focused results to general connected figures and providing conditions for reversibility and tessellativity.

Contribution

It generalizes reversibility results from polygons to arbitrary convex polyhedra and establishes conditions for nets of an isotetrahedron to be both reversible and tessellative.

Findings

Nets obtained by cutting along dissection trees are reversible.

Conditions for reversibility and tessellativity of nets of an isotetrahedron.

Extension of reversibility concepts from polygons to convex polyhedra.

Abstract

An example of reversible (or hinge inside-out transformable) figures is the Dudeney's Haberdasher's puzzle in which an equilateral triangle is dissected into four pieces, then hinged like a chain, and then is transformed into a square by rotating the hinged pieces. Furthermore, the entire boundary of each figure goes into the inside of the other figure and becomes the dissection lines of the other figure. Many intriguing results on reversibilities of figures have been found in prior research, but most of them are results on polygons. This paper generalizes those results to a wider range of general connected figures. It is shown that two nets obtained by cutting the surface of an arbitrary convex polyhedron along non-intersecting dissection trees are reversible. Moreover, a condition for two nets of an isotetrahedron to be both reversible and tessellative is given.