Tileable Surfaces

TL;DR

This paper investigates $C^1$-regular surfaces in three-dimensional space that can be tiled with a finite set of congruent tiles, focusing on monotilings, and explores the possibility of smoothing polyhedral surfaces into tileable surfaces.

Contribution

It introduces a systematic framework for studying tileable surfaces, classifies monotilings with up to three edges, and examines the smoothing of polyhedra into tileable surfaces.

Findings

Classified all monotilings with three or fewer edges.

Provided examples of surfaces that cannot be smoothed into tileable surfaces.

Outlined open problems in the study of tileable surfaces.

Abstract

We study $C^1$-regular surfaces in $R^3$ that admit tilings by a finite number of rigid motion congruence classes of tiles. We construct examples with various topologies and present a framework for a systematic study, mainly concentrating on monotilings. A finite edge prototile is a tile that has only a finite number of possible interfaces with adjacent copies of itself. We describe all monotilings by such tiles with three or less edges. We consider the question of whether a monohedral polyhedron can be smoothed to become a finite edge type tileable surface with the same graph structure, and we give an example where this is not possible. Finally we list some open problems.