This is equivalent to the tiling being an edge-to-edge tiling by congruent regular polygons. This means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. Antwerp v3.0, 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.įollowing 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. In order to solve those problems, GomJau-Hogg’s notation is a slightly modified version of the research and notation presented in 2012, about the generation and nomenclature of tessellations and double-layer grids. Therefore, the second problem is that this nomenclature is not unique for each tessellation. 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. 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. "2D Euclidean tilings x3o6o - trat - O2".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. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 The Geometrical Foundation of Natural Structure: A Source Book of Design. (Chapter 2.1: Regular and uniform tilings, p. 58-65, Chapter 2.9 Archimedean and Uniform colorings pp. 102–107) Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-8 p. 296, Table II: Regular honeycombs ^ Coxeter, Regular Complex Polytopes, pp.^ Tilings and Patterns, from list of 107 isohedral tilings, p.473-481.^ Order in Space: A design source book, Keith Critchlow, p.74-75, pattern 1.(The truncated triangular tiling is topologically identical to the hexagonal tiling.) Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling).ĭrawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. Wythoff constructions from hexagonal and triangular tilings It is also topologically related as a part of sequence of Catalan solids with face configuration Vn.6.6, and also continuing into the hyperbolic plane. Symmetry given assumes all faces are the same color. With identical faces ( face-transitivity) and vertex-transitivity, there are 5 variations. The triangular tiling has Schläfli symbol of topology as the regular tiling (6 triangles around every vertex). Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Vertex-transitive, edge-transitive, face-transitive Regular tiling of the plane Triangular tiling
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