Euclidean tilings of regular polygons

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A periodic (2-uniform) tiling of regular triangles, squares, hexagons and dodecagons

Euclidean plane tilings by 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
Vertex type 6-6-6.svg
63
Vertex type 4-4-4-4.svg
44

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 semiregular tilings. Note that there are two mirror image (enantiomorphic or chiral) forms of 34.6 (snub hexagonal) tiling, both of which are shown in the following table. All other regular and semiregular tilings are achiral.

Uniform tilings (8)
p6m, *632
1-uniform n4.svg
Vertex type 3-12-12.svg
3.122
1-uniform n6.svg
Vertex type 3-4-6-4.svg
3.4.6.4
1-uniform n3.svg
Vertex type 4-6-12.svg
4.6.12
1-uniform n7.svg
Vertex type 3-6-3-6.svg
(3.6)2
p4m, *442 p4, 442 cmm, 2*22 p6, 632
1-uniform n2.svg
Vertex type 4-8-8.svg
4.82
1-uniform n9.svg
Vertex type 3-3-4-3-4.svg
32.4.3.4
1-uniform n8.svg
Vertex type 3-3-3-4-4.svg
33.42
1-uniform n10.svg
Vertex type 3-3-3-3-6.svg
34.6

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]

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

Other types of vertices in Euclidean plane tilings[edit]

For edge-to-edge Euclidean tilings, the internal angles of the polygons meeting at a vertex must add to 360 degrees. A regular n\,\!-gon has internal angle \left(1-\frac{2}{n}\right)180 degrees. There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one types of vertex.

Only eleven of these can occur in a uniform tiling of regular polygons, given in previous sections.

In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same. If they are not, they would have to alternate around the first polygon, which is impossible if its number of sides is odd. By that restriction these six cannot appear in any tiling of regular polygons:

3 polygons at a vertex (unuseable)
Regular polygons meeting at vertex 3 3 7 42.svg
3.7.42
Regular polygons meeting at vertex 3 3 8 24.svg
3.8.24
Regular polygons meeting at vertex 3 3 9 18.svg
3.9.18
Regular polygons meeting at vertex 3 3 10 15.svg
3.10.15
Regular polygons meeting at vertex 3 4 5 20.svg
4.5.20
Regular polygons meeting at vertex 3 5 5 10.svg
5.5.10

These four can be used in k-uniform tiling:

4 polygons at a vertex (mixable with other vertex types)
Valid
vertex
types
Vertex type 3-3-4-12.svg
32.4.12
Vertex type 3-4-3-12.svg
3.4.3.12
Vertex type 3-3-6-6.svg
32.62
Vertex type 3-4-4-6.svg
3.42.6
Example
2-uniform
tilings
2-uniform 13.png
with 36
2-uniform 2.png
with 3.12.12
2-uniform 11.png
with (3.6)2
2-uniform 6.png
with (3.6)2

Dissected regular polygons[edit]

Some of the k-uniform tilings can be derived by symmetrically dissecting tiling polygons, for example:

Dissected polygons with original edges
Triangular tiling vertfig.png Hexagonal cupola flat.png Dissected dodecagon.png
Hexagon Dodecagon
(each has 2 orientations)

Some k-uniform tiligns can be derived by dissected regular polygons with new edge vertices, for example:

Dissected with 1 or 2 middle vertex
Dissected triangle-3a.png Dissected triangle-36.png Dissected triangle-3b.png Dissected square.png Dissected square-3x3.png Dissected hexagon 36a.png Dissected hexagon 36b.png Dissected hexagon 3b.png
Triangle Square Hexagon


2-uniform tilings[edit]

Shown below are the twenty 2-uniform tilings, also called 2-isogonal tilings or demiregular tilings.[2][3][4] Vertex types are listed for each. If two tilings share the same two vertex types, they are given subscripts 1,2. They are 2-isogonal, t-isohedral, e-isotoxal for the listed values of t and e.

2-uniform tilings (20)
p6m, *632 p4m, *442
2-uniform 18.png
[36; 32.4.3.4]
(t=3, e=3)
2-uniform 9.png
[3.4.6.4; 32.4.3.4]
(t=4, e=4)
2-uniform 8.png
[3.4.6.4; 33.42]
(t=4, e=4)
2-uniform 5.png
[3.4.6.4; 3.42.6]
(t=5, e=5)
2-uniform 1.png
[4.6.12; 3.4.6.4]
(t=4, e=4)
2-uniform 13.png
[36; 32.4.12]
(t=4, e=4)
2-uniform 2.png
[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 10.png
[36; 32.62]
(t=2, e=3)
2-uniform 19.png
[36; 34.6]1
(t=3, e=3)
2-uniform 20.png
[36; 34.6]2
(t=5, e=7)
2-uniform 12.png
[32.62; 34.6]
(t=2, e=4)
2-uniform 11.png
[3.6.3.6; 32.62]
(t=2, e=3)
2-uniform 6.png
[3.42.6; 3.6.3.6]2
(t=3, e=4)
2-uniform 7.png
[3.42.6; 3.6.3.6]1
(t=4, e=4)
p4g, 4*2 pgg, 2× cmm, 2*22 cmm, 2*22 pmm, *2222 cmm, 2*22
2-uniform 16.png
[33.42; 32.4.3.4]1
(t=4, e=5)
2-uniform 17.png
[33.42; 32.4.3.4]2
(t=6, e=6)
2-uniform 4.png
[44; 33.42]1
(t=2, e=4)
2-uniform 3.png
[44; 33.42]2
(t=3, e=5)
2-uniform 14.png
[36; 33.42]1
(t=3, e=4)
2-uniform 15.png
[36; 33.42]2
(t=4, e=5)

3-uniform tilings[edit]

Chavey (1989) lists 61 3-uniform tilings, 39 are 3-Archimedean with 3 distinct vertex types, while 22 have 2 identical vertex types in different symmetry orbits.

3-uniform tilings with 2 vertex types (22)
3-uniform 26.svg
[(3.4.6.4)2; 3.426]
3-uniform 58.svg
[(36)2; 346]
3-uniform 59.svg
[(36)2; 346]
3-uniform 60.svg
[(36)2; 346]
3-uniform 61.svg
[36; (346)2]
3-uniform 57.svg
[36; (324.3.4)2]
3-uniform 28.svg
[(3.426)2; 3.6.3.6]
3-uniform 30.svg
[3.426; (3.6.3.6)2]
3-uniform 32.svg
[3.426; (3.6.3.6)2]
3-uniform 39.svg
[3262; (3.6.3.6)2]
3-uniform 45.svg
[(346)2; 3.6.3.6]
3-uniform 46.svg
[(346)2; 3.6.3.6]
3-uniform 10.svg
[3342; (44)2]
3-uniform 13.svg
[(3342)2; 44]
3-uniform 16.svg
[3342; (44)2]
3-uniform 19.svg
[(3342)2; 44]
3-uniform 53.svg
[(3342)2; 324.3.4]
3-uniform 55.svg
[3342; (324.3.4)2]
3-uniform 52.svg
[36; (3342)2]
3-uniform 51.svg
[36; (3342)2]
3-uniform 50.svg
[(36)2; 3342]
3-uniform 49.svg
[(36)2; 3342]
3-uniform tilings with 3 vertex types (39)
3-uniform 5.svg
[3.426; 3.6.3.6; 4.6.12]
3-uniform 6.svg
[36; 324.12; 4.6.12]
3-uniform 7.svg
[324.12; 3.4.6.4; 3.122]
3-uniform 8.svg
[3.4.3.12; 3.4.6.4; 3.122]
3-uniform 35.svg
[3342; 324.12; 3.4.6.4]
3-uniform 47.svg
[36; 3342; 324.12]
3-uniform 48.svg
[36; 324.3.4; 324.12]
3-uniform 56.svg
[346; 3342; 324.3.4]
3-uniform 24.svg
[36; 324.3.4; 3.426]
3-uniform 34.svg
[36; 324.3.4; 3.4.6.4]
3-uniform 36.svg
[36; 3342; 3.4.6.4]
3-uniform 37.svg
[36; 324.3.4; 3.4.6.4]
3-uniform 54.svg
[36; 3342; 324.3.4]
3-uniform 9.svg
[324.12; 3.4.3.12; 3.122]
3-uniform 22.svg
[3.4.6.4; 3.426; 44]
3-uniform 25.svg
[324.3.4; 3.4.6.4; 3.426]
3-uniform 23.svg
[3342; 324.3.4; 44]
3-uniform 11.svg
[3.426; 3.6.3.6; 44]
3-uniform 12.svg
[3.426; 3.6.3.6; 44]
3-uniform 17.svg
[3.426; 3.6.3.6; 44]
3-uniform 18.svg
[3.426; 3.6.3.6; 44]
3-uniform 27.svg
[3342; 3262; 3.426]
3-uniform 29.svg
[3262; 3.426; 3.6.3.6]
3-uniform 31.svg
[3262; 3.426; 3.6.3.6]
3-uniform 33.svg
[346; 3342; 3.426]
3-uniform 1.svg
[3262; 3.6.3.6; 63]
3-uniform 2.svg
[3262; 3.6.3.6; 63]
3-uniform 3.svg
[346; 3262; 63]
3-uniform 4.svg
[36; 3262; 63]
3-uniform 38.svg
[36; 346; 3262]
3-uniform 40.svg
[36; 346; 3262]
3-uniform 41.svg
[36; 346; 3262]
3-uniform 44.svg
[36; 346; 3.6.3.6]
3-uniform 42.svg
[36; 346; 3.6.3.6]
3-uniform 43.svg
[36; 346; 3.6.3.6]
3-uniform 14.svg
[36; 3342; 44]
3-uniform 15.svg
[36; 3342; 44]
3-uniform 20.svg
[36; 3342; 44]
3-uniform 21.svg
[36; 3342; 44]

4-uniform tilings[edit]

Brian Galebach's search reproduced Krotenheerdt's list of 33 4-uniform tilings with 4 distinct vertex types, as well as finding 85 of them with 3 vertex types, and 33 with 2 vertex types.[5]

4-uniform tilings with 2 pairs of 2 types of vertices (12)
4-uniform 29.svg
[(3464)2; (46.12)2]
4-uniform 106.svg
[(33434)2; (3464)2]
4-uniform 107.svg
[(33434)2; (3464)2]
4-uniform 125.svg
[(33336)2; (3636)2]
4-uniform 150.svg
[(333333)2; (33336)2]
4-uniform 143.svg
[(33344)2; (33434)2]
4-uniform 41.svg
[(33344)2; (4444)2]
4-uniform 52.svg
[(33344)2; (4444)2]
4-uniform 61.svg
[(33344)2; (4444)2]
4-uniform 139.svg
[(333333)2; (33344)2]
4-uniform 140.svg
[(333333)2; (33344)2]
4-uniform 141.svg
[(333333)2; (33344)2]
4-uniform tilings with 2 types (3 and 1) vertices (21)
4-uniform 33.svg
[343.12; (3.12.12)3]
4-uniform 129.svg
[(33336)3; 3636]
4-uniform 151.svg
[333333; (33336)3]
4-uniform 148.svg
[(333333)3; 33336]
4-uniform 149.svg
[(333333)3; 33336]
4-uniform 142.svg
[(33344)3; 33434]
4-uniform 144.svg
[33344; (33434)3]
4-uniform 87.svg
[3446; (3636)3]
4-uniform 90.svg
[3446; (3636)3]
4-uniform 114.svg
[3366; (3636)3]
4-uniform 117.svg
[3366; (3636)3]
4-uniform 38.svg
[33344; (4444)3]
4-uniform 58.svg
[33344; (4444)3]
4-uniform 53.svg
[(33344)3; 4444]
4-uniform 72.svg
[(33344)3; 4444]
4-uniform 76.svg
[(33344)3; 4444]
4-uniform 133.svg
[333333; (33344)3]
4-uniform 134.svg
[333333; (33344)3]
4-uniform 135.svg
[333333; (33344)3]
4-uniform 136.svg
[(333333)3; 33344]
4-uniform 137.svg
[(333333)3; 33344]
4-uniform tilings with 3 vertex types (85)
4-uniform 25.svg
[3464; (3446)2; 46.12]
4-uniform 28.svg
[3464; 3446; (46.12)2]
4-uniform 31.svg
[334.12; 3464; (3.12.12)2]
4-uniform 32.svg
[343.12; 3464; (3.12.12)2]
4-uniform 108.svg
[33434; 343.12; (3464)2]
4-uniform 130.svg
[(333333)2; 33344; 334.12]
4-uniform 94.svg
[(3464)2; 3446; 3636]
4-uniform 95.svg
[3464; 3446; (3636)2]
4-uniform 83.svg
[3464; (3446)2; 3636]
4-uniform 146.svg
[(333333)2; 33344; 33434]
4-uniform 138.svg
[(333333)2; 33344; 33434]
4-uniform 1.svg
[333333; 3366; (666)2]
4-uniform 2.svg
[333333; 3366; (666)2]
4-uniform 7.svg
[333333; (3366)2; 666]
4-uniform 8.svg
[333333; (3366)2; 666]
4-uniform 14.svg
[333333; 3366; (666)2]
4-uniform 17.svg
[333333; 3366; (666)2]
4-uniform 110.svg
[333333; (33336)2; 3366]
4-uniform 111.svg
[333333; (3366)2; 3636]
4-uniform 10.svg
[(33336)2; 3366; 666]
4-uniform 24.svg
[(33336)2; 3366; 666]
4-uniform 118.svg
[33336; 3366; (3636)2]
4-uniform 119.svg
[33336; 3366; (3636)2]
4-uniform 102.svg
[33344; 33434; (3464)2]
4-uniform 105.svg
[333333; 33434; (3464)2]
4-uniform 104.svg
[333333; (33434)2; 3464]
4-uniform 100.svg
[333333; (33344)2; 3464]
4-uniform 93.svg
[(3464)2; 3446; 3636]
4-uniform 97.svg
[33336; (33434)2; 3446]
4-uniform 145.svg
[333333; 33344; (33434)2]
4-uniform 147.svg
[333333; 33344; (33434)2]
4-uniform 57.svg
[(33344)2; 33434; 4444]
4-uniform 79.svg
[(33344)2; 33434; 4444]
4-uniform 80.svg
[3464; (3446)2; 4444]
4-uniform 132.svg
[33434; (334.12)2; 343.12]
4-uniform 19.svg
[333333; (3366)2; 666]
4-uniform 4.svg
[333333; (3366)2; 666]
4-uniform 109.svg
[333333; 33336; (3366)2]
4-uniform 122.svg
[(333333)2; 33336; 3366]
4-uniform 123.svg
[(333333)2; 33336; 3366]
4-uniform 128.svg
[(333333)2; 33336; 3636]
4-uniform 112.svg
[33336; (3366)2; 3636]
4-uniform 113.svg
[33336; (3366)2; 3636]
4-uniform 120.svg
[(33336)2; 3366; 3636]
4-uniform 116.svg
[(33336)2; 3366; 3636]
4-uniform 124.svg
[333333; 33336; (3636)2]
4-uniform 21.svg
[3366; (3636)2; 666]
4-uniform 22.svg
[3366; (3636)2; 666]
4-uniform 11.svg
[(3366)2; 3636; 666]
4-uniform 15.svg
[3366; 3636; (666)2]
4-uniform 16.svg
[33336; 3366; (666)2]
4-uniform 121.svg
[33336; (3366)2; 3636]
4-uniform 86.svg
[3366; 3446; (3636)2]
4-uniform 89.svg
[3366; 3446; (3636)2]
4-uniform 126.svg
[33336; (33344)2; 3636]
4-uniform 127.svg
[33336; (33344)2; 3636]
4-uniform 99.svg
[33336; 33344; (3446)2]
4-uniform 39.svg
[3446; 3636; (4444)2]
4-uniform 40.svg
[3446; 3636; (4444)2]
4-uniform 59.svg
[3446; 3636; (4444)2]
4-uniform 60.svg
[3446; 3636; (4444)2]
4-uniform 44.svg
[(3446)2; 3636; 4444]
4-uniform 45.svg
[(3446)2; 3636; 4444]
4-uniform 48.svg
[(3446)2; 3636; 4444]
4-uniform 49.svg
[(3446)2; 3636; 4444]
4-uniform 68.svg
[(3446)2; 3636; 4444]
4-uniform 69.svg
[(3446)2; 3636; 4444]
4-uniform 64.svg
[(3446)2; 3636; 4444]
4-uniform 65.svg
[(3446)2; 3636; 4444]
4-uniform 47.svg
[3446; (3636)2; 4444]
4-uniform 51.svg
[3446; (3636)2; 4444]
4-uniform 67.svg
[3446; (3636)2; 4444]
4-uniform 71.svg
[3446; (3636)2; 4444]
4-uniform 43.svg
[333333; 33344; (4444)2]
4-uniform 63.svg
[333333; 33344; (4444)2]
4-uniform 54.svg
[333333; (33344)2; 4444]
4-uniform 42.svg
[333333; 33344; (4444)2]
4-uniform 62.svg
[333333; 33344; (4444)2]
4-uniform 77.svg
[333333; (33344)2; 4444]
4-uniform 78.svg
[333333; (33344)2; 4444]
4-uniform 73.svg
[333333; (33344)2; 4444]
4-uniform 55.svg
[(333333)2; 33344; 4444]
4-uniform 56.svg
[(333333)2; 33344; 4444]
4-uniform 74.svg
[(333333)2; 33344; 4444]
4-uniform 75.svg
[(333333)2; 33344; 4444]
4-uniform tilings with 4 vertex types (33)
4-uniform 6.svg
[33434; 3366; 3446; 666]
4-uniform 26.svg
[33344; 3366; 3446; 46.12]
4-uniform 27.svg
[33434; 3366; 3446; 46.12]
4-uniform 131.svg
[333333; 33344; 33434; 334.12]
4-uniform 34.svg
[333333; 33434; 334.12; 3.12.12]
4-uniform 35.svg
[333333; 33434; 343.12; 3.12.12]
4-uniform 101.svg
[333333; 33344; 33434; 3464]
4-uniform 103.svg
[333333; 33344; 33434; 3464]
4-uniform 84.svg
[333333; 33434; 3464; 3446]
4-uniform 9.svg
[33336; 3366; 3636; 666]
4-uniform 23.svg
[33336; 3366; 3636; 666]
4-uniform 30.svg
[334.12; 343.12; 3464; 46.12]
4-uniform 37.svg
[33344; 334.12; 343.12; 3.12.12]
4-uniform 81.svg
[33344; 334.12; 343.12; 4444]
4-uniform 36.svg
[33344; 334.12; 343.12; 3.12.12]
4-uniform 82.svg
[333333; 33344; 33434; 4444]
4-uniform 85.svg
[33434; 3366; 3464; 3446]
4-uniform 92.svg
[333333; 33344; 3446; 3636]
4-uniform 88.svg
[333333; 33336; 3446; 3636]
4-uniform 91.svg
[333333; 33336; 3446; 3636]
4-uniform 96.svg
[333333; 33336; 33344; 3446]
4-uniform 98.svg
[333333; 33336; 33344; 3446]
4-uniform 5.svg
[333333; 33336; 3366; 666]
4-uniform 20.svg
[333333; 33336; 3366; 666]
4-uniform 12.svg
[333333; 33336; 3366; 666]
4-uniform 13.svg
[333333; 33336; 3366; 666]
4-uniform 115.svg
[333333; 33336; 3366; 3636]
4-uniform 3.svg
[33344; 3366; 3446; 666]
4-uniform 18.svg
[33344; 3366; 3446; 666]
4-uniform 66.svg
[3366; 3446; 3636; 4444]
4-uniform 70.svg
[3366; 3446; 3636; 4444]
4-uniform 46.svg
[3366; 3446; 3636; 4444]
4-uniform 50.svg
[3366; 3446; 3636; 4444]

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

Regular polygons can also form plane tilings that are not edge-to-edge. Such tilings may also be known as uniform if they are vertex-transitive (isogonal); there are eight families of such uniform tilings, 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.

Periodic tilings with star polygons
Academ Squares and regular star octagons in a periodic tiling.svg
Octagrams and squares
Academ Periodic tiling by star dodecagons and equilateral triangles.svg
Dodecagrams and equilateral triangles

Two tilings by regular polygons of two kinds. Two elements of the same kind are congruent.
Every element which is not convex is a stellation of a regular polygon with stripes.

Periodic tilings by regular polygons
Gyrated hexagonal tiling2.png
Six triangles surround every hexagon.
No pair of triangles has a common boundary, if their sides have a length lower than
the side length of hexagons.
Distorted trihexagonal tiling.png
Two size triangles
Distorted truncated square tiling.png
A tiling by squares of two different sizes, manifestly periodic by overlaying an appropriate grid. The present grid divides every large tile into four congruent polygons: possible puzzle pieces to prove the Pythagorean theorem.
Hexagon hexagram tiling2.png
Regular octagrams with small and large squares

See also[edit]

References[edit]

  1. ^ Critchlow, p.60-61
  2. ^ Critchlow, p.62-67
  3. ^ Tilings and Patterns, Grünbaum and Shephard 1986, pp. 65-67
  4. ^ In Search of Demiregular Tilings
  5. ^ http://www2.math.uu.se/research/pub/Lenngren1.pdf k-uniform tilings by regular polygons] Nils Lenngren, 2009

External links[edit]

Euclidean and general tiling links: