Tileable surfaces

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.