Perimeter-minimizing Tilings by Convex and Non-convex Pentagons

TL;DR

This paper investigates whether the convexity assumption in a known theorem about perimeter-minimizing pentagon tilings is necessary, proving the theorem in some cases without this assumption and suggesting future research directions.

Contribution

It extends the perimeter-minimizing tiling theorem to include certain non-convex pentagons, removing the convexity requirement in specific cases.

Findings

The theorem holds for some non-convex pentagons.

Convexity may not be necessary for perimeter-minimizing tilings in certain scenarios.

Provides directions for further research on non-convex tilings.

Abstract

We study the presumably unnecessary convexity hypothesis in the theorem of Chung et al. [CFS] on perimeter-minimizing planar tilings by convex pentagons. We prove that the theorem holds without the convexity hypothesis in certain special cases, and we offer direction for future research.