Жазықтықты қалыпты көпбұрыштармен бөлу және мозаикалардың түрлері

Subdivision of the plane into polygons that are all regular
+ Example periodic tilingsA regular tiling has one type of regular face. A semiregular or uniform tiling has one type of vertex, but two or more types of faces. A k uniform tiling has k types of vertices, and two or more types of regular faces. A non edge to edge tiling can have different sized regular faces. 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).

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. 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 is a slightly modified version of the research and notation presented in 2012, 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.

There are 17 combinations of regular convex polygons that form 21 types of plane vertex tilings. Polygons in these meet at a point with no gap or overlap. Listing by their vertex figures, one has 6 polygons, three have 5 polygons, seven have 4 polygons, and ten have 3 polygons. As detailed in the sections above, three of them can make regular tilings (63, 44, 36), and eight more can make semiregular or archimedean tilings, (3.12.12, 4.6.12, 4.8.8, (3.6)2, 3.4.6.4, 3.3.4.3.4, 3.3.3.4.4, 3.3.3.3.6). Four of them can exist in higher k uniform tilings (3.3.4.12, 3.4.3.12, 3.3.6.6, 3.4.4.6), while six can not be used to completely tile the plane by regular polygons with no gaps or overlaps they only tessellate space entirely when irregular polygons are included (3.7.42, 3.8.24, 3.9.18, 3.10.15, 4.5.20, 5.5.10). + The plane vertex tilings 63653.3.4.3.43.3.3.4.43.3.3.3.643.3.4.123.4.3.123.3.6.6(3.6)23.4.4.63.4.6.44433.7.423.8.243.9.183.10.153.12.124.5.204.6.124.8.85.5.1063

There are twenty (20) 2 uniform tilings of the Euclidean plane. (also called 2 isogonal tilings or demiregular tilings) Vertex types are listed for each. If two tilings share the same two vertex types, they are given subscripts 1,2. + 2 uniform tilings (20)p6m, *632 p4m, *442 [36; 32.4.3.4]3 4 3/m30/r(c3) (t = 3, e = 3) [3.4.6.4; 32.4.3.4] 6 4 3,3/m30/r(h1)(t = 4, e = 4) [3.4.6.4; 33.42]6 4 3 3/m30/r(h5) (t = 4, e = 4) [3.4.6.4; 3.42.6]6 4 3,4 6/m30/r(c4) (t = 5, e = 5) [4.6.12; 3.4.6.4]12 4,6 3/m30/r(c3) (t = 4, e = 4) [36; 32.4.12]12 3,4 3/m30/r(c3) (t = 4, e = 4) [3.12.12; 3.4.3.12]12 0,3,3 0,4/m45/m(h1) (t = 3, e = 3) p6m, *632p6, 632p6, 632cmm, 2*22pmm, *2222cmm, 2*22pmm, *2222 [36; 32.62]3 6/m30/r(c2) (t = 2, e = 3) [36; 34.6]16 3,3 3/m30/r(h1) (t = 3, e = 3) [36; 34.6]26 3 3,3 3/r60/r(h8) (t = 5, e = 7) [32.62; 34.6]6 3/m90/r(h1) (t = 2, e = 4) [3.6.3.6; 32.62]6 3,6/m90/r(h3) (t = 2, e = 3) [3.42.6; 3.6.3.6]26 3,4 6 3,4 6,4/m90/r(c6) (t = 3, e = 4) [3.42.6; 3.6.3.6]16 3,4/m90/r(h4) (t = 4, e = 4)p4g, 4*2pgg, 22×cmm, 2*22cmm, 2*22pmm, *2222cmm, 2*22 [33.42; 32.4.3.4]14 3,3 4,3/r90/m(h3) (t = 4, e = 5) [33.42; 32.4.3.4]24 3,3,3 4,3/r(c2)/r(h13)/r(h45)(t = 3, e = 6) [44; 33.42]14 3/m(h4)/m(h3)/r(h2)(t = 2, e = 4) [44; 33.42]24 4 3 3/m90/r(h3)(t = 3, e = 5) [36; 33.42]14 3,4 3,3/m90/r(h3)(t = 3, e = 4) [36; 33.42]24 3 3 3/m90/r(h7)/r(h5)(t = 4, e = 5)

k uniform tilings have been enumerated up to 6. There are 673 6 uniform tilings of the Euclidean plane. Brian Galebach's search reproduced Krotenheerdt's list of 10 6 uniform tilings with 6 distinct vertex types, as well as finding 92 of them with 5 vertex types, 187 of them with 4 vertex types, 284 of them with 3 vertex types, and 100 with 2 vertex types.

Convex regular polygons can also form plane tilings that are not edge to edge. Such tilings can be considered edge to edge as nonregular polygons with adjacent colinear edges. There are seven families of isogonal each family having a real valued parameter determining the overlap between sides of adjacent tiles or the ratio between the edge lengths of different tiles. Two of the families are generated from shifted square, either progressive or zig zagging positions. Grünbaum and Shephard call these tilings uniform although it contradicts Coxeter's definition for uniformity which requires edge to edge regular polygons. Such isogonal tilings are actually topologically identical to the uniform tilings, with different geometric proportions. + Periodic isogonal tilings by non edge to edge convex regular polygons1234567Rows of squares with horizontal offsetsRows of triangles with horizontal offsetsA tiling by squaresThree hexagons surround each triangleSix triangles surround every hexagon. Three size trianglescmm (2*22)p2 (2222)cmm (2*22)p4m (*442)p6 (632)p3 (333)Hexagonal tilingSquare tilingTruncated square tilingTruncated hexagonal tilingHexagonal tilingTrihexagonal tiling