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On inside-out Dissections of Polygons and Polyhedra

Reymond Akpanya, Adi Rivkin, Frederick Stock

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.

On inside-out Dissections of Polygons and Polyhedra

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 pieces, thereby improving the best previous upper bound of pieces. Additionally, we establish that a regular polygon can be inside-out dissected with at most pieces. Lastly, we prove that any polyhedron that can be decomposed into finitely many regular tetrahedra and octahedra can be inside-out dissected.

Paper Structure

This paper contains 4 sections, 5 theorems, 1 equation, 7 figures.

Table of Contents

  1. Introduction
  2. Dissections of Polygons
  3. Dissections of Polyhedra
  4. Conclusion & Outlook

Key Result

Lemma 2.1

Any $n$-gon $P$ can be inside-out dissected with $2n + 1$ pieces. In particular, this means $\mathcal{I}(n)\leq 2n+1.$

Figures (7)

  • Figure 1: An inside-out-dissection of an obtuse triangle using four pieces by Aaron Meyerowitz.AM14
  • Figure 2: A triangle can be inside-out dissected with 7 pieces.
  • Figure 3: A generic polygon can be inside-out dissected in $2n + 1$ pieces.
  • Figure 4: An inside-out dissection of a regular hexagon.
  • Figure 5: Two regular polygons with their pieces (left) 7-gon, (right) 10-gon.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Definition 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Remark 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4