Abstract
Euclidean tilings are constantly applied to many fields of engineering (mechanical, civil, chemical, etc.). These tessellations are usually named after Cundy & Rollett’s notation. However, this notation has two main problems related to ambiguous conformation and uniqueness. This communication explains the GomJau-Hogg’s notation for generating all the regular, semi-regular (uniform) and demi-regular (k-uniform, up to at least k = 3) in a consistent, unique and unequivocal manner. Moreover, it presents Antwerp v3.0, a free online application, which is publicly shared to prove that all the basic tilings can be obtained directly from the GomJau-Hogg’s notation.
Highlights
Euclidean tiling is the covering of a plane where the repetition of regular polygons make up tiles, which through symmetry operations can be extended indefinitely without any overlapping [1]
Similar placed at the origin of the plane so that the two sides that intersect the horizontal axis to the Cundy & Rollett’s notation, each number represents the number of sides on the polygon
The notation of the tilings can be automatically generated without ambiguity There is no repetition on the names of the tilings, they are unique
Summary
Euclidean tiling is the covering of a plane where the repetition of regular polygons make up tiles, which through symmetry operations can be extended indefinitely without any overlapping [1]. Regular tilings consist of a single polygon type, with each vertex surrounded by the same kinds of polygons (vertex-transitive) (Figure 1). Semi-regular tilings (Archimedean or uniform) are polymorphic (several polygon types) and vertex-transitive (Figure 2). Tiling configurations are usually named using Cundy & Rollett’s notation [2], which adapted the Schläfli symbol 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)’. 36 ; 36 (both of different transitivity class), or (36 ) , tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles).
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