Tileable Surfaces

David Brander; Jens Gravesen

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arXiv · 2507.11281

Tileable Surfaces

David Brander·Jens Gravesen

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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.

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BibTeX RIS

David Brander, Jens Gravesen. 2025-07-15. Tileable Surfaces. https://arxiv.org/abs/2507.11281

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