Euclidean tilings by convex regular polygons
subdivision of the plane into polygons that are all regular
Tilings of the Euclidean plane by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonice Mundi (Latin: The Harmony of the World, 1619).
Notation of Euclidean tilings
Euclidean tilings are usually named after Cundy & Rollett's notation. 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. For example: 36; 36; 34.6, 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. Broken down, 36; 36 (both of different transitivity class), or (36)2, tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 34.6, 4 more contiguous equilateral triangles and a single regular hexagon.
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. This makes it impossible to generate a covered plane given the notation alone.
Begin with the source’s own compact description: “Euclidean tilings by convex regular polygons” is subdivision of the plane into polygons that are all regular. The dossier treats that line as a proposition to test through Euclidean, tilings and convex, not as a finished interpretation.
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