Listing by their vertex figures, one has 6 polygons, three have 5 polygons, seven have 4 polygons, and ten have 3 polygons. Polygons in these meet at a point with no gap or overlap. There are 17 combinations of regular convex polygons that form 21 types of plane-vertex tilings. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform. Grünbaum and Shephard distinguish the description of these tilings as Archimedean as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as uniform as referring to the global property of vertex-transitivity. All other regular and semiregular tilings are achiral. Note that there are two mirror image (enantiomorphic or chiral) forms of 3 4.6 (snub hexagonal) tiling, only one of which is shown in the following table. If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as Archimedean, uniform or semiregular tilings. ![]() Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second. GJ-H: Notation of GomJau-Hogg Archimedean, uniform or semiregular tilings įurther information: List of convex uniform tilings There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations. 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. 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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