# 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.

## Key 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.