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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). == Regular tilings == Following Grünbaum 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. == Archimedean, uniform or semiregular tilings == Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second.〔Critchlow, p.60-61〕 If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as ''Archimedean'', ''uniform'' or ''semiregular'' tilings. Note that there are two mirror image (enantiomorphic or chiral) forms of 34.6 (snub hexagonal) tiling, both of which are shown in the following table. All other regular and semiregular tilings are achiral. Grünbaum and Shephard distinguish the description of these tilings as ''Archimedean'' as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as ''uniform'' as referring to the global property of vertex-transitivity. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Euclidean tilings by convex regular polygons」の詳細全文を読む スポンサード リンク
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