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).

Notation of Euclidean tilings[edit]

Euclidean tilings are usually named after Cundy & Rollett’s notation.^[1] 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 ^[2] 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. And second, some tessellations have the same nomenclature, they are very similar but it can be noticed that the relative positions of the hexagons are different. Therefore, the second problem is that this nomenclature is not unique for each tessellation.

In order to solve those problems, GomJau-Hogg’s notation ^[3] is a slightly modified version of the research and notation presented in 2012,^[2] about the generation and nomenclature of tessellations and double-layer grids. Antwerp v3.0,^[4] a free online application, allows for the infinite generation of regular polygon tilings through a set of shape placement stages and iterative rotation and reflection operations, obtained directly from the GomJau-Hogg’s notation.

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n-uniform tilings

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Euclidean and general tiling links: