On inside-out Dissections of Polygons and Polyhedra

TL;DR

This paper investigates inside-out dissections of polygons and polyhedra, presenting improved bounds for dissecting polygons into fewer pieces and extending results to certain polyhedra, advancing geometric dissection theory.

Contribution

It introduces a new upper bound of 2n+1 pieces for dissecting polygons and shows that some polyhedra can also be dissected inside-out, which were not previously known.

Findings

Polygon can be inside-out dissected with 2n+1 pieces

Regular polygon can be dissected with at most 6 pieces

Polyhedra decomposable into regular tetrahedra and octahedra can be dissected inside-out

Abstract

In this work we study inside-out dissections of polygons and polyhedra. We first show that an arbitrary polygon can be inside-out dissected with $2n+1$ pieces, thereby improving the best previous upper bound of $4(n-2)$ pieces. Additionally, we establish that a regular polygon can be inside-out dissected with at most $6$ pieces. Lastly, we prove that any polyhedron that can be decomposed into finitely many regular tetrahedra and octahedra can be inside-out dissected.