Euclidean tilings by convex regular polygons

Template:Short description

Example periodic tilings
File:1-uniform n1.svg A regular tiling has one type of regular face.	File:1-uniform n2.svg A semiregular or uniform tiling has one type of vertex, but two or more types of faces.
File:2-uniform n1.svg A k-uniform tiling has k types of vertices, and two or more types of regular faces.	File: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 {{#invoke:Lang|lang}} (Latin: The Harmony of the World, 1619).

Notation of Euclidean tilingsEdit

Euclidean tilings are usually named after Cundy & Rollett’s notation.<ref>Template:Cite book</ref> 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 <ref name="Gomez-Jauregui 2012">Template:Cite journal</ref> 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 <ref>Template:Cite journal</ref> is a slightly modified version of the research and notation presented in 2012,<ref name="Gomez-Jauregui 2012" /> about the generation and nomenclature of tessellations and double-layer grids. Antwerp v3.0,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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 tilingsEdit

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.

Regular tilings (3)
p6m, *632		p4m, *442
File:1-uniform n11.svg	File:1-uniform n1.svg	File:1-uniform n5.svg
File:Vertex type 3-3-3-3-3-3.svg C&R: 3⁶ GJ-H: 3/m30/r(h2) (t = 1, e = 1)	File:Vertex type 6-6-6.svg C&R: 6³ GJ-H: 6/m30/r(h1) (t = 1, e = 1)	File: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 tilingsEdit

Template:Further Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second.<ref name="Critchlow 1969">Template:Cite book</ref>

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.

Uniform tilings (8)
p6m, *632
File:1-uniform n4.svg File:Vertex type 3-12-12.svg C&R: 3.12² GJ-H: 12-3/m30/r(h3) (t = 2, e = 2) t{6,3}	File:1-uniform n6.svg File: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}	File:1-uniform n3.svg File: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}	File:1-uniform n7.svg File: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}
File:1-uniform n2.svg File:Vertex type 4-8-8.svg C&R: 4.8² GJ-H: 8-4/m90/r(h4) (t = 2, e = 2) t{4,4}	File:1-uniform n9.svg File: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}	File:1-uniform n8.svg File: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	File:1-uniform n10.svg File: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 tilingsEdit

There are 17 combinations of regular convex polygons that form 21 types of plane-vertex tilings.<ref>Template:Citation</ref><ref>Tilings and patterns, Figure 2.1.1, p.60</ref> 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.<ref>Tilings and patterns, p.58-69</ref>

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).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Plane-vertex tilings
6	File:Regular polygons meeting at vertex 6 3 3 3 3 3 3.svg 3⁶
5	File:Regular polygons meeting at vertex 5 3 3 4 3 4.svg 3.3.4.3.4	File:Regular polygons meeting at vertex 5 3 3 3 4 4.svg 3.3.3.4.4	File:Regular polygons meeting at vertex 5 3 3 3 3 6.svg 3.3.3.3.6
4	File:Regular polygons meeting at vertex 4 3 3 4 12.svg 3.3.4.12	File:Regular polygons meeting at vertex 4 3 4 3 12.svg 3.4.3.12	File:Regular polygons meeting at vertex 4 3 3 6 6.svg 3.3.6.6	File:Regular polygons meeting at vertex 4 3 6 3 6.svg (3.6)²	File:Regular polygons meeting at vertex 4 3 4 4 6.svg 3.4.4.6	File:Regular polygons meeting at vertex 4 3 4 6 4.svg 3.4.6.4	File:Regular polygons meeting at vertex 4 4 4 4 4.svg 4⁴
3	File:Regular polygons meeting at vertex 3 3 7 42.svg 3.7.42	File:Regular polygons meeting at vertex 3 3 8 24.svg 3.8.24	File:Regular polygons meeting at vertex 3 3 9 18.svg 3.9.18	File:Regular polygons meeting at vertex 3 3 10 15.svg 3.10.15	File:Regular polygons meeting at vertex 3 3 12 12.svg 3.12.12	File:Regular polygons meeting at vertex 3 4 5 20.svg 4.5.20	File:Regular polygons meeting at vertex 3 4 6 12.svg 4.6.12	File:Regular polygons meeting at vertex 3 4 8 8.svg 4.8.8	File:Regular polygons meeting at vertex 3 5 5 10.svg 5.5.10	File:Regular polygons meeting at vertex 3 6 6 6.svg 6³

k-uniform tilingsEdit

Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are Template:Mvar orbits of vertices, a tiling is known as Template:Mvar-uniform or Template:Mvar-isogonal; if there are Template:Mvar orbits of tiles, as Template:Mvar-isohedral; if there are Template:Mvar orbits of edges, as Template: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.<ref>k-uniform tilings by regular polygons Template:Webarchive Nils Lenngren, 2009</ref>

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:

Colored 3-uniform tiling #57 of 61
File:3-uniform 57.svg by sides, yellow triangles, red squares (by polygons)	File:3-uniform n57.svg by 4-isohedral positions, 3 shaded colors of triangles (by orbits)

citation
1	2			3	4	5	6	7	8	9	10	11	12	13	14	≥ 15	Total
CitationClass=web }}</ref><ref>Template:Cite OEIS</ref><ref>{{#invoke:citation/CS1\|citation	CitationClass=web }}</ref>			m-Archimedean
k-uniform	1	11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	11
	2	0	20	0	0	0	0	0	0	0	0	0	0	0	0	0	20
	3	0	22	39	0	0	0	0	0	0	0	0	0	0	0	0	61
	4	0	33	85	33	0	0	0	0	0	0	0	0	0	0	0	151
	5	0	74	149	94	15	0	0	0	0	0	0	0	0	0	0	332
	6	0	100	284	187	92	10	0	0	0	0	0	0	0	0	0	673
	7	0	175	572	426	218	74	7	0	0	0	0	0	0	0	0	1472
	8	0	298	1037	795	537	203	20	0	0	0	0	0	0	0	0	2850
	9	0	424	1992	1608	1278	570	80	8	0	0	0	0	0	0	0	5960
	10	0	663	3772	2979	2745	1468	212	27	0	0	0	0	0	0	0	11866
	11	0	1086	7171	5798	5993	3711	647	52	1	0	0	0	0	0	0	24459
	12	0	1607	13762	11006	12309	9230	1736	129	15	0	0	0	0	0	0	49794
	13	0	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	0	0	0	0	0	103082
	14	0	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	0	0	0	0	0	Template:Dunno
	≥ 15	0	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	Template:Dunno	0	Template:Dunno
	Total	11	∞	∞	∞	∞	∞	∞	∞	∞	∞	∞	∞	∞	∞	0	∞

2-uniform tilingsEdit

There are twenty (20) 2-uniform tilings of the Euclidean plane. (also called 2-isogonal tilings or demiregular tilings) Template:R <ref>Tilings and patterns, Grünbaum and Shephard 1986, pp. 65-67</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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
File:2-uniform n18.svg [3⁶; 3².4.3.4] 3-4-3/m30/r(c3) (t = 3, e = 3)	File:2-uniform n9.svg [3.4.6.4; 3².4.3.4] 6-4-3,3/m30/r(h1) (t = 4, e = 4)	File:2-uniform n8.svg [3.4.6.4; 3³.4²] 6-4-3-3/m30/r(h5) (t = 4, e = 4)	File:2-uniform n5.svg [3.4.6.4; 3.4².6] 6-4-3,4-6/m30/r(c4) (t = 5, e = 5)	File:2-uniform n1.svg [4.6.12; 3.4.6.4] 12-4,6-3/m30/r(c3) (t = 4, e = 4)	File:2-uniform n13.svg [3⁶; 3².4.12] 12-3,4-3/m30/r(c3) (t = 4, e = 4)	File: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
File:2-uniform n10.svg [3⁶; 3².6²] 3-6/m30/r(c2) (t = 2, e = 3)	File:2-uniform n19.svg [3⁶; 3⁴.6]₁ 6-3,3-3/m30/r(h1) (t = 3, e = 3)	File:2-uniform n20.svg [3⁶; 3⁴.6]₂ 6-3-3,3-3/r60/r(h8) (t = 5, e = 7)	File:2-uniform n12.svg [3².6²; 3⁴.6] 6-3/m90/r(h1) (t = 2, e = 4)	File:2-uniform n11.svg [3.6.3.6; 3².6²] 6-3,6/m90/r(h3) (t = 2, e = 3)	File: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)	File: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
File:2-uniform n16.svg [3³.4²; 3².4.3.4]₁ 4-3,3-4,3/r90/m(h3) (t = 4, e = 5)	File:2-uniform n17-1.svg [3³.4²; 3².4.3.4]₂ 4-3,3,3-4,3/r(c2)/r(h13)/r(h45) (t = 3, e = 6)	File:2-uniform n4.svg [4⁴; 3³.4²]₁ 4-3/m(h4)/m(h3)/r(h2) (t = 2, e = 4)	File:2-uniform n3.svg [4⁴; 3³.4²]₂ 4-4-3-3/m90/r(h3) (t = 3, e = 5)	File:2-uniform n14.svg [3⁶; 3³.4²]₁ 4-3,4-3,3/m90/r(h3) (t = 3, e = 4)	File:2-uniform n15.svg [3⁶; 3³.4²]₂ 4-3-3-3/m90/r(h7)/r(h5) (t = 4, e = 5)

Higher k-uniform tilingsEdit

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 tilingsEdit

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:<ref>Template:Cite journal</ref>

	Triangle	Square	Hexagon	Dissected Dodecagon
Shape	File:The Triangle.svg	File:A Square Tile.svg	File:A Hexagon Tile.svg	File:A Dissected Dodecagon.svg
Fractalizing	File:Truncated Hexagonal Fractal Triangle.svg	File:Truncated Hexagonal Fractal Square.svg	File:Truncated Hexagonal Fractal Hexagon.svg	File:Truncated Hexagonal Fractal Dissected Dodecagon.svg

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>.

	Triangle	Square	Hexagon	Dissected Dodecagon
Shape	File:The Triangle.svg	File:A Square Tile.svg	File:A Hexagon Tile.svg	File:A Dissected Dodecagon.svg
Fractalizing	File:Truncated Trihexagonal Fractal Triangle.svg	File:Truncated Trihexagonal Fractal Square.svg	File:Truncated Trihexagonal Fractal Hexagon.svg	File:Truncated Trihexagonal Fractal Dissected Dodecagon.svg

Fractalizing examplesEdit

	Truncated Hexagonal Tiling	Truncated Trihexagonal Tiling
Fractalizing	File:Planar Fractalizing Truncated Hexagonal Tiling II.png	File:Planar Tiling Fractalizing the Truncated Trihexagonal Tiling.png

Tilings that are not edge-to-edgeEdit

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 figures, 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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 polygons
1	2	3	4	5	6	7
File:Square brick pattern.png Rows of squares with horizontal offsets		File:Half-offset triangular tiling.png Rows of triangles with horizontal offsets	File:Distorted truncated square tiling.svg A tiling by squares	File:Gyrated truncated hexagonal tiling.png Three hexagons surround each triangle	File:Gyrated hexagonal tiling2.svg Six triangles surround every hexagon.	File: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