Polyhedral Characterization of Reversible Hinged Dissections

TL;DR

This paper characterizes reversible hinged dissections between polygons as noncrossing nets of a common polyhedron, providing a geometric foundation for designing hinged dissections and their monotone variants.

Contribution

It establishes a precise geometric criterion linking reversible hinged dissections to noncrossing nets of polyhedra, including convex cases, advancing the theoretical understanding of hinged dissections.

Findings

Reversible hinged dissections correspond to noncrossing nets of a common polyhedron.

Monotone reversible hinged dissections correspond to noncrossing nets of convex polyhedra.

The characterization simplifies the design of hinged dissections through envelope/parcel methods.

Abstract

We prove that two polygons $A$ and $B$ have a reversible hinged dissection (a chain hinged dissection that reverses inside and outside boundaries when folding between $A$ and $B$) if and only if $A$ and $B$ are two noncrossing nets of a common polyhedron. Furthermore, monotone reversible hinged dissections (where all hinges rotate in the same direction when changing from $A$ to $B$) correspond exactly to noncrossing nets of a common convex polyhedron. By envelope/parcel magic, it becomes easy to design many hinged dissections.