Euclidean tilings by convex regular polygons

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

Regular tilings[edit]

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
1-uniform n11.svg 1-uniform n1.svg 1-uniform n5.svg
Vertex type 3-3-3-3-3-3.svg
36
(t=1, e=1)
Vertex type 6-6-6.svg
63
(t=1, e=1)
Vertex type 4-4-4-4.svg
44
(t=1, e=1)

Archimedean, uniform or semiregular tilings[edit]

Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second.[1]

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 demiregular 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
1-uniform n4.svg


3.122
(t=2, e=2)
t{6,3}
1-uniform n6.svg


3.4.6.4
(t=3, e=2)
rr{3,6}
1-uniform n3.svg


4.6.12
(t=3, e=3)
tr{3,6}
1-uniform n7.svg


(3.6)2
(t=2, e=1)
r{6,3}
1-uniform n2.svg


4.82
(t=2, e=2)
t{4,4}
1-uniform n9.svg


32.4.3.4
(t=2, e=2)
s{4,4}
1-uniform n8.svg


33.42
(t=2, e=3)
{3,6}:e
1-uniform n10.svg


34.6
(t=3, e=3)
sr{3,6}

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.

k-uniform tilings[edit]

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

Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are k orbits of vertices, a tiling is known as k-uniform or k-isogonal; if there are t orbits of tiles, as t-isohedral; if there are e orbits of edges, as e-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.[2]

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 (mk), 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.

k-uniform, m-Archimedean tiling counts[3]
m-Archimedean
1 2 3 4 5 6 7 8 9 10 11 12 13 14 ≥ 15 Total
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 ? ? ? ? ? 7 0 0 0 0 0 0 0 0 ?
8 0 ? ? ? ? ? 20 0 0 0 0 0 0 0 0 ?
9 0 ? ? ? ? ? ? 8 0 0 0 0 0 0 0 ?
10 0 ? ? ? ? ? ? 27 0 0 0 0 0 0 0 ?
11 0 ? ? ? ? ? ? ? 1 0 0 0 0 0 0 ?
12 0 ? ? ? ? ? ? ? ? 0 0 0 0 0 0 ?
13 0 ? ? ? ? ? ? ? ? ? ? ? 0 0 0 ?
14 0 ? ? ? ? ? ? ? ? ? ? ? ? 0 0 ?
≥ 15 0 ? ? ? ? ? ? ? ? ? ? ? ? ? 0 ?
Total 11 0

Dissected regular polygons[edit]

Some of the k-uniform tilings can be derived by symmetrically dissecting the tiling polygons with interior edges, for example (direct dissection):

Dissected polygons with original edges
Hexagon Dodecagon
(each has 2 orientations)

Some k-uniform tilings can be derived by dissecting regular polygons with new vertices along the original edges, for example (indirect dissection):

Dissected with 1 or 2 middle vertex
Face figure 3-333.svg Dissected triangle-36.png Dissected triangle-3b.png Vertex type 4-4-4-4.svg Dissected square-3x3.png Dissected hexagon 36a.png Dissected hexagon 36b.png Dissected hexagon 3b.png
Triangle Square Hexagon

Finally, to see all types of vertex configurations, see Planigon.

2-uniform tilings[edit]

There are twenty (20) 2-uniform tilings of the Euclidean plane. (also called 2-isogonal tilings or demiregular tilings)[4][5][6] 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
2-uniform n18.svg
[36; 32.4.3.4
(t=3, e=3)
2-uniform n9.svg
[3.4.6.4; 32.4.3.4
(t=4, e=4)
2-uniform n8.svg
[3.4.6.4; 33.42]
(t=4, e=4)
2-uniform n5.svg
[3.4.6.4; 3.42.6]
(t=5, e=5)
2-uniform n1.svg
[4.6.12; 3.4.6.4]
(t=4, e=4)
2-uniform n13.svg
[36; 32.4.12]
(t=4, e=4)
2-uniform n2.svg
[3.12.12; 3.4.3.12]
(t=3, e=3)
p6m, *632 p6, 632 p6, 632 cmm, 2*22 pmm, *2222 cmm, 2*22 pmm, *2222
2-uniform n10.svg
[36; 32.62]
(t=2, e=3)
2-uniform n19.svg
[36; 34.6]1
(t=3, e=3)
2-uniform n20.svg
[36; 34.6]2
(t=5, e=7)
2-uniform n12.svg
[32.62; 34.6]
(t=2, e=4)
2-uniform n11.svg
[3.6.3.6; 32.62]
(t=2, e=3)
2-uniform n6.svg
[3.42.6; 3.6.3.6]2
(t=3, e=4)
2-uniform n7.svg
[3.42.6; 3.6.3.6]1
(t=4, e=4)
p4g, 4*2 pgg, 22× cmm, 2*22 cmm, 2*22 pmm, *2222 cmm, 2*22
2-uniform n16.svg
[33.42; 32.4.3.4]1
(t=4, e=5)
2-uniform n17.png
[33.42; 32.4.3.4]2
(t=3, e=6)
2-uniform n4.svg
[44; 33.42]1
(t=2, e=4)
2-uniform n3.svg
[44; 33.42]2
(t=3, e=5)
2-uniform n14.svg
[36; 33.42]1
(t=3, e=4)
2-uniform n15.svg
[36; 33.42]2
(t=4, e=5)

Higher k-uniform tilings[edit]

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[edit]

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:[7]

Triangle Square Hexagon Dissected
Dodecagon
Shape
A Hexagon Tile.png
A Dissected Dodecagon.png
Fractalizing
Truncated Hexagonal Fractal Triangle.png
Truncated Hexagonal Fractal Square.png
Truncated Hexagonal Fractal Hexagon.png
Truncated Hexagonal Fractal Dissected Dodecagon.png

The side lengths are dilated by a factor of .

This can similarly be done with the truncated trihexagonal tiling as a basis, with corresponding dilation of .

Triangle Square Hexagon Dissected
Dodecagon
Shape
A Hexagon Tile.png
A Dissected Dodecagon.png
Fractalizing
Truncated Trihexagonal Fractal Triangle.png
Truncated Trihexagonal Fractal Square.png
Truncated Trihexagonal Fractal Hexagon.png
Truncated Trihexagonal Fractal Dissected Dodecagon.png

Fractalizing Examples[edit]

Truncated Hexagonal Tiling Truncated Trihexagonal Tiling
Fractalizing
Planar Tiling Fractalizing the Truncated Trihexagonal Tiling.png

Tilings that are not edge-to-edge[edit]

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.[8] 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
Square brick pattern.png
Rows of squares with horizontal offsets
Half-offset triangular tiling.png
Rows of triangles with horizontal offsets
Distorted truncated square tiling.svg
A tiling by squares
Gyrated truncated hexagonal tiling.png
Three hexagons surround each triangle
Gyrated hexagonal tiling2.png
Six triangles surround every hexagon.
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[edit]

References[edit]

  1. ^ Critchlow, p.60-61
  2. ^ k-uniform tilings by regular polygons Archived 2015-06-30 at the Wayback Machine Nils Lenngren, 2009
  3. ^ "n-Uniform Tilings". probabilitysports.com. Retrieved 2019-06-21.
  4. ^ Critchlow, p.62-67
  5. ^ Tilings and Patterns, Grünbaum and Shephard 1986, pp. 65-67
  6. ^ "In Search of Demiregular Tilings" (PDF). Archived from the original (PDF) on 2016-05-07. Retrieved 2015-06-04.
  7. ^ Chavey, Darrah (2014). "TILINGS BY REGULAR POLYGONS III: DODECAGON-DENSE TILINGS". Symmetry-Culture and Science. 25 (3): 193–210. S2CID 33928615.
  8. ^ Tilings by regular polygons p.236

External links[edit]

Euclidean and general tiling links: