Английская Википедия:Euclidean tilings by convex regular polygons
| Файл:1-uniform n1.svg A regular tiling has one type of regular face. |
Файл:1-uniform n2.svg A semiregular or uniform tiling has one type of vertex, but two or more types of faces. |
| Файл:2-uniform n1.svg A k-uniform tiling has k types of vertices, and two or more types of regular faces. |
Файл:Distorted truncated square tiling.svg 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 Шаблон:Lang (Latin: The Harmony of the World, 1619).
Notation of Euclidean tilings
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.
Regular tilings
Following 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. This means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an edge-to-edge tiling by congruent regular polygons. There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations.
| p6m, *632 | p4m, *442 | |
|---|---|---|
| Файл:1-uniform n11.svg | Файл:1-uniform n1.svg | Файл:1-uniform n5.svg |
| Файл:Vertex type 3-3-3-3-3-3.svg C&R: 36 GJ-H: 3/m30/r(h2) (t = 1, e = 1) |
Файл:Vertex type 6-6-6.svg C&R: 63 GJ-H: 6/m30/r(h1) (t = 1, e = 1) |
Файл:Vertex type 4-4-4-4.svg C&R: 44 GJ-H: 4/m45/r(h1) (t = 1, e = 1) |
C&R: Cundy & Rollet's notation
GJ-H: Notation of GomJau-Hogg
Archimedean, uniform or semiregular tilings
Шаблон:Further Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second.[5]
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. Note that there are two mirror image (enantiomorphic or chiral) forms of 34.6 (snub hexagonal) tiling, only one of which is shown in the following table. All other regular and semiregular tilings are achiral.
| p6m, *632 | |||||
|---|---|---|---|---|---|
| Файл:1-uniform n4.svg Файл:Vertex type 3-12-12.svg C&R: 3.122 GJ-H: 12-3/m30/r(h3) (t = 2, e = 2) t{6,3} |
Файл:1-uniform n6.svg Файл:Vertex type 3-4-6-4.svg C&R: 3.4.6.4 GJ-H: 6-4-3/m30/r(c2) (t = 3, e = 2) rr{3,6} |
Файл:1-uniform n3.svg Файл:Vertex type 4-6-12.svg C&R: 4.6.12 GJ-H: 12-6,4/m30/r(c2) (t = 3, e = 3) tr{3,6} |
Файл:1-uniform n7.svg Файл:Vertex type 3-6-3-6.svg C&R: (3.6)2 GJ-H: 6-3-6/m30/r(v4) (t = 2, e = 1) r{6,3} | ||
| Файл:1-uniform n2.svg Файл:Vertex type 4-8-8.svg C&R: 4.82 GJ-H: 8-4/m90/r(h4) (t = 2, e = 2) t{4,4} |
Файл:1-uniform n9.svg Файл:Vertex type 3-3-4-3-4.svg C&R: 32.4.3.4 GJ-H: 4-3-3,4/r90/r(h2) (t = 2, e = 2) s{4,4} |
Файл:1-uniform n8.svg Файл:Vertex type 3-3-3-4-4.svg C&R: 33.42 GJ-H: 4-3/m90/r(h2) (t = 2, e = 3) {3,6}:e |
Файл:1-uniform n10.svg Файл:Vertex type 3-3-3-3-6.svg C&R: 34.6 GJ-H: 6-3-3/r60/r(h5) (t = 3, e = 3) sr{3,6} | ||
C&R: Cundy & Rollet's notation
GJ-H: Notation of GomJau-Hogg
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. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform.
Plane-vertex tilings
There are 17 combinations of regular convex polygons that form 21 types of plane-vertex tilings.[6][7] 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.[8]
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).[9]
k-uniform tilings
Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are Шаблон:Mvar orbits of vertices, a tiling is known as Шаблон:Mvar-uniform or Шаблон:Mvar-isogonal; if there are Шаблон:Mvar orbits of tiles, as Шаблон:Mvar-isohedral; if there are Шаблон:Mvar orbits of edges, as Шаблон:Mvar-isotoxal.
k-uniform tilings with the same vertex figures can be further identified by their wallpaper group symmetry.
1-uniform tilings include 3 regular tilings, and 8 semiregular ones, with 2 or more types of regular polygon faces. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 332 5-uniform tilings and 673 6-uniform tilings. Each can be grouped by the number m of distinct vertex figures, which are also called m-Archimedean tilings.[10]
Finally, if the number of types of vertices is the same as the uniformity (m = k below), then the tiling is said to be Krotenheerdt. In general, the uniformity is greater than or equal to the number of types of vertices (m ≥ k), as different types of vertices necessarily have different orbits, but not vice versa. Setting m = n = k, there are 11 such tilings for n = 1; 20 such tilings for n = 2; 39 such tilings for n = 3; 33 such tilings for n = 4; 15 such tilings for n = 5; 10 such tilings for n = 6; and 7 such tilings for n = 7.
Below is an example of a 3-unifom tiling:
| Файл:3-uniform 57.svg by sides, yellow triangles, red squares (by polygons) |
Файл:3-uniform n57.svg by 4-isohedral positions, 3 shaded colors of triangles (by orbits) |
2-uniform tilings
There are twenty (20) 2-uniform tilings of the Euclidean plane. (also called 2-isogonal tilings or demiregular tilings) Шаблон:R [13][14] Vertex types are listed for each. If two tilings share the same two vertex types, they are given subscripts 1,2.
| p6m, *632 | p4m, *442 | |||||
|---|---|---|---|---|---|---|
| Файл:2-uniform n18.svg [36; 32.4.3.4] 3-4-3/m30/r(c3) (t = 3, e = 3) |
Файл:2-uniform n9.svg [3.4.6.4; 32.4.3.4] 6-4-3,3/m30/r(h1) (t = 4, e = 4) |
Файл:2-uniform n8.svg [3.4.6.4; 33.42] 6-4-3-3/m30/r(h5) (t = 4, e = 4) |
Файл:2-uniform n5.svg [3.4.6.4; 3.42.6] 6-4-3,4-6/m30/r(c4) (t = 5, e = 5) |
Файл:2-uniform n1.svg [4.6.12; 3.4.6.4] 12-4,6-3/m30/r(c3) (t = 4, e = 4) |
Файл:2-uniform n13.svg [36; 32.4.12] 12-3,4-3/m30/r(c3) (t = 4, e = 4) |
Файл:2-uniform n2.svg [3.12.12; 3.4.3.12] 12-0,3,3-0,4/m45/m(h1) (t = 3, e = 3) |
| p6m, *632 | p6, 632 | p6, 632 | cmm, 2*22 | pmm, *2222 | cmm, 2*22 | pmm, *2222 |
| Файл:2-uniform n10.svg [36; 32.62] 3-6/m30/r(c2) (t = 2, e = 3) |
Файл:2-uniform n19.svg [36; 34.6]1 6-3,3-3/m30/r(h1) (t = 3, e = 3) |
Файл:2-uniform n20.svg [36; 34.6]2 6-3-3,3-3/r60/r(h8) (t = 5, e = 7) |
Файл:2-uniform n12.svg [32.62; 34.6] 6-3/m90/r(h1) (t = 2, e = 4) |
Файл:2-uniform n11.svg [3.6.3.6; 32.62] 6-3,6/m90/r(h3) (t = 2, e = 3) |
Файл:2-uniform n6.svg [3.42.6; 3.6.3.6]2 6-3,4-6-3,4-6,4/m90/r(c6) (t = 3, e = 4) |
Файл:2-uniform n7.svg [3.42.6; 3.6.3.6]1 6-3,4/m90/r(h4) (t = 4, e = 4) |
| p4g, 4*2 | pgg, 22× | cmm, 2*22 | cmm, 2*22 | pmm, *2222 | cmm, 2*22 | |
| Файл:2-uniform n16.svg [33.42; 32.4.3.4]1 4-3,3-4,3/r90/m(h3) (t = 4, e = 5) |
Файл:2-uniform n17.png [33.42; 32.4.3.4]2 4-3,3,3-4,3/r(c2)/r(h13)/r(h45) (t = 3, e = 6) |
Файл:2-uniform n4.svg [44; 33.42]1 4-3/m(h4)/m(h3)/r(h2) (t = 2, e = 4) |
Файл:2-uniform n3.svg [44; 33.42]2 4-4-3-3/m90/r(h3) (t = 3, e = 5) |
Файл:2-uniform n14.svg [36; 33.42]1 4-3,4-3,3/m90/r(h3) (t = 3, e = 4) |
Файл:2-uniform n15.svg [36; 33.42]2 4-3-3-3/m90/r(h7)/r(h5) (t = 4, e = 5) | |
Higher k-uniform tilings
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.
Fractalizing k-uniform tilings
There are many ways of generating new k-uniform tilings from old k-uniform tilings. For example, notice that the 2-uniform [3.12.12; 3.4.3.12] tiling has a square lattice, the 4(3-1)-uniform [343.12; (3.122)3] tiling has a snub square lattice, and the 5(3-1-1)-uniform [334.12; 343.12; (3.12.12)3] tiling has an elongated triangular lattice. These higher-order uniform tilings use the same lattice but possess greater complexity. The fractalizing basis for theses tilings is as follows:[15]
The side lengths are dilated by a factor of <math>2+\sqrt{3}</math>.
This can similarly be done with the truncated trihexagonal tiling as a basis, with corresponding dilation of <math>3+\sqrt{3}</math>.
Fractalizing examples
| Truncated Hexagonal Tiling | Truncated Trihexagonal Tiling | |
|---|---|---|
| Fractalizing |
Tilings that are not edge-to-edge
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.[16] Such isogonal tilings are actually topologically identical to the uniform tilings, with different geometric proportions.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|
| Файл:Square brick pattern.png Rows of squares with horizontal offsets |
Файл:Half-offset triangular tiling.png Rows of triangles with horizontal offsets |
Файл:Distorted truncated square tiling.svg A tiling by squares |
Файл:Gyrated truncated hexagonal tiling.png Three hexagons surround each triangle |
Файл:Gyrated hexagonal tiling2.png Six triangles surround every hexagon. |
Файл:Trihexagonal tiling unequal2.svg Three size triangles | |
| cmm (2*22) | p2 (2222) | cmm (2*22) | p4m (*442) | p6 (632) | p3 (333) | |
| Hexagonal tiling | Square tiling | Truncated square tiling | Truncated hexagonal tiling | Hexagonal tiling | Trihexagonal tiling | |
See also
- Grid (spatial index)
- Uniform tilings in hyperbolic plane
- List of uniform tilings
- Wythoff symbol
- Tessellation
- Wallpaper group
- Regular polyhedron (the Platonic solids)
- Semiregular polyhedron (including the Archimedean solids)
- Hyperbolic geometry
- Penrose tiling
- Tiling with rectangles
- Lattice (group)
References
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite journal
- Order in Space: A design source book, Keith Critchlow, 1970 Шаблон:ISBN
- Шаблон:Cite book Chapter X: The Regular Polytopes
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, Шаблон:ISBN, pp. 50–57
External links
Euclidean and general tiling links:
- n-uniform tilings, Brian Galebach
- Шаблон:Cite web
- Шаблон:Cite web
- Шаблон:Mathworld
- Шаблон:MathWorld
- Шаблон:MathWorld
- ↑ Шаблон:Cite book
- ↑ 2,0 2,1 Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite book
- ↑ Шаблон:Citation
- ↑ Tilings and Patterns, Figure 2.1.1, p.60
- ↑ Tilings and Patterns, p.58-69
- ↑ Шаблон:Cite web
- ↑ k-uniform tilings by regular polygons Шаблон:Webarchive Nils Lenngren, 2009
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite OEIS
- ↑ Tilings and Patterns, Grünbaum and Shephard 1986, pp. 65-67
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite journal
- ↑ Tilings by regular polygons p.236
