High Quality Content by WIKIPEDIA articles! Plane ? tilings by ? regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of ? Kepler in ? Harmonices Mundi. Following ? Grunbaum and Shephard (section 1.3), a tiling is said to be regular if the ? symmetry group of the tiling ? acts transitively on the flags of the tiling, where a flag is a triple consisting of a mutually incident vertex, edge and tile of the tiling. This means that for every pair of flags there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an edge-to-edge tiling by congruent regular polygons. There must be six ? equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations.

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