Euclidean tilings by convex regular polygons

Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi (Latin: The Harmony of the World, 1619).

Example periodic tilings
A regular tiling has one type of regular face.	A semiregular or uniform tiling has one type of vertex, but two or more types of faces.
A k-uniform tiling has k types of vertices, and two or more types of regular faces.	A non-edge-to-edge tiling can have different-sized regular faces.

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