![]() ![]() This makes it impossible to generate a covered plane given the notation alone. However, this notation has two main problems related to ambiguous conformation and uniqueness First, when it comes to k-uniform tilings, the notation does not explain the relationships between the vertices. With a final vertex 3 4.6, 4 more contiguous equilateral triangles and a single regular hexagon. Broken down, 3 6 3 6 (both of different transitivity class), or (3 6) 2, tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). For example: 3 6 3 6 3 4.6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling. ![]() This notation represents (i) the number of vertices, (ii) the number of polygons around each vertex (arranged clockwise) and (iii) the number of sides to each of those polygons. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi ( Latin: The Harmony of the World, 1619).Įuclidean tilings are usually named after Cundy & Rollett’s notation. Subdivision of the plane into polygons that are all regular Example periodic tilingsĪ regular tiling has one type of regular face.Ī semiregular or uniform tiling has one type of vertex, but two or more types of faces.Ī k-uniform tiling has k types of vertices, and two or more types of regular faces.Ī non-edge-to-edge tiling can have different-sized regular faces.Įuclidean plane tilings by convex regular polygons have been widely used since antiquity. ![]()
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