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Euclidean tilings by convex regular polygons

  (Redirected from Tiling by regular polygons)
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).

Contents

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
     
 
36
(t=1, e=1)
 
63
(t=1, e=1)
 
44
(t=1, e=1)

Archimedean, uniform or semiregular tilingsEdit

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, only one of which is shown in the following table. All other regular and semiregular tilings are achiral.

Uniform tilings (8)
p6m, *632
 

 
3.122
(t=2, e=2)
 

 
3.4.6.4
(t=3, e=2)
 

 
4.6.12
(t=3, e=3)
 

 
(3.6)2
(t=2, e=1)
p4m, *442 p4g, 4*2 cmm, 2*22 p6, 632
 

 
4.82
(t=2, e=2)
 

 
32.4.3.4
(t=2, e=2)
 

 
33.42
(t=2, e=3)
 

 
34.6
(t=3, e=3)

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 tilingsEdit

3-uniform tiling #57 of 61 colored
 
by sides, yellow triangles, red squares
 
by 4-isohedral positions, 3 shaded colors of triangles

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

k-uniform, m-Archimedean tiling counts
m Total
1 2 3 4 5 6 7 8 9
k 1 11 0 0 0 0 0 0 0 0 11
2 0 20 0 0 0 0 0 0 0 20
3 0 22 39 0 0 0 0 0 0 61
4 0 33 85 33 0 0 0 0 0 151
5 0 74 149 94 15 0 0 0 0 332
6 0 100 284 187 92 10 0 0 0 673
7 0 ? ? ? ? ? 7 0 0 ?
8 0 ? ? ? ? ? 20 0 0 ?
9 0 ? ? ? ? ? ? 8 0 ?
10 0 ? ? ? ? ?  ? 27 0 ?
11 0 ? ? ? ? ?  ? ? 1 ?

Other types of vertices in Euclidean plane tilingsEdit

For edge-to-edge Euclidean tilings, the internal angles of the polygons meeting at a vertex must add to 360 degrees. A regular  -gon has internal angle   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 (unusable)
 
3.7.42
 
3.8.24
 
3.9.18
 
3.10.15
 
4.5.20
 
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
 
32.4.12
 
3.4.3.12
 
32.62
 
3.42.6
Example
2-uniform
tilings
 
with 36
 
with 3.122
 
with (3.6)2
 
with (3.6)2

Dissected regular polygonsEdit

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

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:

Dissected with 1 or 2 middle vertex
               
Triangle Square Hexagon


2-uniform tilingsEdit

There are twenty 2-uniform tilings of the Euclidean plane. (also called 2-isogonal tilings or demiregular tilings)[3][4][5] 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]
(t=3, e=3)
 
[3.4.6.4; 32.4.3.4]
(t=4, e=4)
 
[3.4.6.4; 33.42]
(t=4, e=4)
 
[3.4.6.4; 3.42.6]
(t=5, e=5)
 
[4.6.12; 3.4.6.4]
(t=4, e=4)
 
[36; 32.4.12]
(t=4, e=4)
 
[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
 
[36; 32.62]
(t=2, e=3)
 
[36; 34.6]1
(t=3, e=3)
 
[36; 34.6]2
(t=5, e=7)
 
[32.62; 34.6]
(t=2, e=4)
 
[3.6.3.6; 32.62]
(t=2, e=3)
 
[3.42.6; 3.6.3.6]2
(t=3, e=4)
 
[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
 
[33.42; 32.4.3.4]1
(t=4, e=5)
 
[33.42; 32.4.3.4]2
(t=3, e=6)
 
[44; 33.42]1
(t=2, e=4)
 
[44; 33.42]2
(t=3, e=5)
 
[36; 33.42]1
(t=3, e=4)
 
[36; 33.42]2
(t=4, e=5)

3-uniform tilingsEdit

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

3-uniform tilings, 3 vertex typesEdit

3-uniform tilings with 3 vertex types (39)
 
[3.426; 3.6.3.6; 4.6.12]
(t=6, e=7)
 
[36; 324.12; 4.6.12]
(t=5, e=6)
 
[324.12; 3.4.6.4; 3.122]
(t=5, e=6)
 
[3.4.3.12; 3.4.6.4; 3.122]
(t=5, e=6)
 
[3342; 324.12; 3.4.6.4]
(t=6, e=8)
 
[36; 3342; 324.12]
(t=6, e=7)
 
[36; 324.3.4; 324.12]
(t=5, e=6)
 
[346; 3342; 324.3.4]
(t=5, e=6)
 
[36; 324.3.4; 3.426]
(t=5, e=6)
 
[36; 324.3.4; 3.4.6.4]
(t=5, e=6)
 
[36; 3342; 3.4.6.4]
(t=6, e=6)
 
[36; 324.3.4; 3.4.6.4]
(t=6, e=6)
 
[36; 3342; 324.3.4]
(t=4, e=5)
 
[324.12; 3.4.3.12; 3.122]
(t=4, e=7)
 
[3.4.6.4; 3.426; 44]
(t=3, e=4)
 
[324.3.4; 3.4.6.4; 3.426]
(t=4, e=6)
 
[3342; 324.3.4; 44]
(t=4, e=6)
 
[3.426; 3.6.3.6; 44]
(t=5, e=7)
 
[3.426; 3.6.3.6; 44]
(t=6, e=7)
 
[3.426; 3.6.3.6; 44]
(t=4, e=5)
 
[3.426; 3.6.3.6; 44]
(t=5, e=6)
 
[3342; 3262; 3.426]
(t=5, e=8)
 
[3262; 3.426; 3.6.3.6]
(t=4, e=7)
 
[3262; 3.426; 3.6.3.6]
(t=5, e=7)
 
[346; 3342; 3.426]
(t=5, e=7)
 
[3262; 3.6.3.6; 63]
(t=4, e=5)
 
[3262; 3.6.3.6; 63]
(t=2, e=4)
 
[346; 3262; 63]
(t=2, e=5)
 
[36; 3262; 63]
(t=2, e=3)
 
[36; 346; 3262]
(t=5, e=8)
 
[36; 346; 3262]
(t=3, e=5)
 
[36; 346; 3262]
(t=3, e=6)
 
[36; 346; 3.6.3.6]
(t=5, e=6)
 
[36; 346; 3.6.3.6]
(t=4, e=4)
 
[36; 346; 3.6.3.6]
(t=3, e=3)
 
[36; 3342; 44]
(t=4, e=6)
 
[36; 3342; 44]
(t=5, e=7)
 
[36; 3342; 44]
(t=3, e=5)
 
[36; 3342; 44]
(t=4, e=6)

3-uniform tilings, 2 vertex types (2:1)Edit

3-uniform tilings (2:1) (22)
 
[(3.4.6.4)2; 3.426]
(t=6, e=6)
 
[(36)2; 346]
(t=3, e=4)
 
[(36)2; 346]
(t=5, e=5)
 
[(36)2; 346]
(t=7, e=9)
 
[36; (346)2]
(t=4, e=6)
 
[36; (324.3.4)2]
(t=4, e=5)
 
[(3.426)2; 3.6.3.6]
(t=6, e=8)
 
[3.426; (3.6.3.6)2]
(t=4, e=6)
 
[3.426; (3.6.3.6)2]
(t=5, e=6)
 
[3262; (3.6.3.6)2]
(t=3, e=5)
 
[(346)2; 3.6.3.6]
(t=4, e=7)
 
[(346)2; 3.6.3.6]
(t=4, e=7)
 
[3342; (44)2]
(t=4, e=7)
 
[(3342)2; 44]
(t=5, e=7)
 
[3342; (44)2]
(t=3, e=6)
 
[(3342)2; 44]
(t=4, e=6)
 
[(3342)2; 324.3.4]
(t=5, e=8)
 
[3342; (324.3.4)2]
(t=6, e=9)
 
[36; (3342)2]
(t=5, e=7)
 
[36; (3342)2]
(t=4, e=6)
 
[(36)2; 3342]
(t=6, e=7)
 
[(36)2; 3342]
(t=5, e=6)

4-uniform tilingsEdit

There are 151 4-uniform tilings of the Euclidean plane. 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.

4-uniform tilings, 4 vertex typesEdit

There are 34 with 4 types of vertices.

4-uniform tilings with 4 vertex types (33)
 
[33434; 3262; 3446; 63]
 
[3342; 3262; 3446; 46.12]
 
[33434; 3262; 3446; 46.12]
 
[36; 3342; 33434; 334.12]
 
[36; 33434; 334.12; 3.122]
 
[36; 33434; 343.12; 3.122]
 
[36; 3342; 33434; 3464]
 
[36; 3342; 33434; 3464]
 
[36; 33434; 3464; 3446]
 
[346; 3262; 3636; 63]
 
[346; 3262; 3636; 63]
 
[334.12; 343.12; 3464; 46.12]
 
[3342; 334.12; 343.12; 3.122]
 
[3342; 334.12; 343.12; 44]
 
[3342; 334.12; 343.12; 3.122]
 
[36; 3342; 33434; 44]
 
[33434; 3262; 3464; 3446]
 
[36; 3342; 3446; 3636]
 
[36; 346; 3446; 3636]
 
[36; 346; 3446; 3636]
 
[36; 346; 3342; 3446]
 
[36; 346; 3342; 3446]
 
[36; 346; 3262; 63]
 
[36; 346; 3262; 63]
 
[36; 346; 3262; 63]
 
[36; 346; 3262; 63]
 
[36; 346; 3262; 3636]
 
[3342; 3262; 3446; 63]
 
[3342; 3262; 3446; 63]
 
[3262; 3446; 3636; 44]
 
[3262; 3446; 3636; 44]
 
[3262; 3446; 3636; 44]
 
[3262; 3446; 3636; 44]

4-uniform tilings, 3 vertex types (2:1:1)Edit

There are 85 with 3 types of vertices.

4-uniform tilings (2:1:1)
 
[3464; (3446)2; 46.12]
 
[3464; 3446; (46.12)2]
 
[334.12; 3464; (3.122)2]
 
[343.12; 3464; (3.122)2]
 
[33434; 343.12; (3464)2]
 
[(36)2; 3342; 334.12]
 
[(3464)2; 3446; 3636]
 
[3464; 3446; (3636)2]
 
[3464; (3446)2; 3636]
 
[(36)2; 3342; 33434]
 
[(36)2; 3342; 33434]
 
[36; 3262; (63)2]
 
[36; 3262; (63)2]
 
[36; (3262)2; 63]
 
[36; (3262)2; 63]
 
[36; 3262; (63)2]
 
[36; 3262; (63)2]
 
[36; (346)2; 3262]
 
[36; (3262)2; 3636]
 
[(346)2; 3262; 63]
 
[(346)2; 3262; 63]
 
[346; 3262; (3636)2]
 
[346; 3262; (3636)2]
 
[3342; 33434; (3464)2]
 
[36; 33434; (3464)2]
 
[36; (33434)2; 3464]
 
[36; (3342)2; 3464]
 
[(3464)2; 3446; 3636]
 
[346; (33434)2; 3446]
 
[36; 3342; (33434)2]
 
[36; 3342; (33434)2]
 
[(3342)2; 33434; 44]
 
[(3342)2; 33434; 44]
 
[3464; (3446)2; 44]
 
[33434; (334.12)2; 343.12]
 
[36; (3262)2; 63]
 
[36; (3262)2; 63]
 
[36; 346; (3262)2]
 
[(36)2; 346; 3262]
 
[(36)2; 346; 3262]
 
[(36)2; 346; 3636]
 
[346; (3262)2; 3636]
 
[346; (3262)2; 3636]
 
[(346)2; 3262; 3636]
 
[(346)2; 3262; 3636]
 
[36; 346; (3636)2]
 
[3262; (3636)2; 63]
 
[3262; (3636)2; 63]
 
[(3262)2; 3636; 63]
 
[3262; 3636; (63)2]
 
[346; 3262; (63)2]
 
[346; (3262)2; 3636]
 
[3262; 3446; (3636)2]
 
[3262; 3446; (3636)2]
 
[346; (3342)2; 3636]
 
[346; (3342)2; 3636]
 
[346; 3342; (3446)2]
 
[3446; 3636; (44)2]
 
[3446; 3636; (44)2]
 
[3446; 3636; (44)2]
 
[3446; 3636; (44)2]
 
[(3446)2; 3636; 44]
 
[(3446)2; 3636; 44]
 
[(3446)2; 3636; 44]
 
[(3446)2; 3636; 44]
 
[(3446)2; 3636; 44]
 
[(3446)2; 3636; 44]
 
[(3446)2; 3636; 44]
 
[(3446)2; 3636; 44]
 
[3446; (3636)2; 44]
 
[3446; (3636)2; 44]
 
[3446; (3636)2; 44]
 
[3446; (3636)2; 44]
 
[36; 3342; (44)2]
 
[36; 3342; (44)2]
 
[36; (3342)2; 44]
 
[36; 3342; (44)2]
 
[36; 3342; (44)2]
 
[36; (3342)2; 44]
 
[36; (3342)2; 44]
 
[36; (3342)2; 44]
 
[(36)2; 3342; 44]
 
[(36)2; 3342; 44]
 
[(36)2; 3342; 44]
 
[(36)2; 3342; 44]

4-uniform tilings, 2 vertex types (2:2) and (3:1)Edit

There are 33 with 2 types of vertices, 12 with two pairs of types, and 21 with 3:1 ratio of types.

4-uniform tilings (2:2)
 
[(3464)2; (46.12)2]
 
[(33434)2; (3464)2]
 
[(33434)2; (3464)2]
 
[(346)2; (3636)2]
 
[(36)2; (346)2]
 
[(3342)2; (33434)2]
 
[(3342)2; (44)2]
 
[(3342)2; (44)2]
 
[(3342)2; (44)2]
 
[(36)2; (3342)2]
 
[(36)2; (3342)2]
 
[(36)2; (3342)2]
4-uniform tilings (3:1)
 
[343.12; (3.122)3]
 
[(346)3; 3636]
 
[36; (346)3]
 
[(36)3; 346]
 
[(36)3; 346]
 
[(3342)3; 33434]
 
[3342; (33434)3]
 
[3446; (3636)3]
 
[3446; (3636)3]
 
[3262; (3636)3]
 
[3262; (3636)3]
 
[3342; (44)3]
 
[3342; (44)3]
 
[(3342)3; 44]
 
[(3342)3; 44]
 
[(3342)3; 44]
 
[36; (3342)3]
 
[36; (3342)3]
 
[36; (3342)3]
 
[(36)3; 3342]
 
[(36)3; 3342]

5-uniform tilingsEdit

There are 332 5-uniform tilings of the Euclidean plane. Brian Galebach's search identified 332 5-uniform tilings, with 2 to 5 types of vertices. There are 74 with 2 vertex types, 149 with 3 vertex types, 94 with 4 vertex types, and 15 with 5 vertex types.

5-uniform tilings, 5 vertex typesEdit

There are 15 5-uniform tilings with 5 unique vertex figure types.

5-uniform tilings, 5 types
 
[33434; 3262; 3464; 3446; 63]
 
[36; 346; 3262; 3636; 63]
 
[36; 346; 3342; 3446; 46.12]
 
[346; 3342; 33434; 3446; 44]
 
[36; 33434; 3464; 3446; 3636]
 
[36; 346; 3464; 3446; 3636]
 
[33434; 334.12; 3464; 3.12.12; 46.12]
 
[36; 346; 3446; 3636; 44]
 
[36; 346; 3446; 3636; 44]
 
[36; 346; 3446; 3636; 44]
 
[36; 346; 3446; 3636; 44]
 
[36; 3342; 3446; 3636; 44]
 
[36; 346; 3342; 3446; 44]
 
[36; 3342; 3262; 3446; 3636]
 
[36; 346; 3342; 3262; 3446]

5-uniform tilings, 4 vertex types (2:1:1:1)Edit

There are 94 5-uniform tilings with 4 vertex types.

5-uniform tilings (2:1:1:1)
 
[36; 33434; (3446)2; 46.12]
 
[36; 33434; 3446; (46.12)2]
 
[36; 33434; 3464; (46.12)2]
 
[36; 3342; (334.12)2; 3464]
 
[36; (3342)2; 334.12; 3464]
 
[36; 33434; (334.12)2; 3464]
 
[36; 33434; 334.12; (3.12.12)2]
 
[36; 346; (3342)2; 334.12]
 
[36; 33434; 343.12; (3.12.12)2]
 
[(3342)2; 334.12; 343.12; 3.12.12]
 
[(3342)2; 334.12; 343.12; 3.12.12]
 
[(3342)2; 334.12; 343.12; 44]
 
[33434; 3262; (3446)2; 44]
 
[36; (3342)2; 33434; 44]
 
[346; (3342)2; 33434; 44]
 
[36; 3342; (3464)2; 3446]
 
[3342; 3262; 3464; (3446)2]
 
[33434; 3262; 3464; (3446)2]
 
[36; 33434; (3446)2; 3636]
 
[3342; 33434; 3464; (3446)2]
 
[36; 33434; (3262)2; 3446]
 
[3342; 3262; (3464)2; 3446]
 
[33434; 3262; (3464)2; 3446]
 
[346; 3342; (3464)2; 3446]
 
[36; (3342)2; 33434; 3464]
 
[36; (3342)2; 33434; 3464]
 
[36; 3342; (33434)2; 3464]
 
[(36)2; 3342; 33434; 3464]
 
[36; 3342; (33434)2; 3464]
 
[(36)2; 3342; 33434; 334.12]
 
[36; 33434; (334.12)2; 343.12]
 
[(36)2; 346; 3342; 33434]
 
[(36)2; 346; 3262; 63]
 
[36; (346)2; 3262; 63]
 
[(36)2; 346; 3262; 3636]
 
[36; 346; (3262)2; 3636]
 
[36; (346)2; 3262; 3636]
 
[(36)2; 346; 3262; 3636]
 
[36; 346; 3262; (3636)2]
 
[36; (346)2; 3262; 3636]
 
[36; (346)2; 3262; 3636]
 
[36; (346)2; 3262; 3636]
 
[36; 346; (3262)2; 3636]
 
[36; 346; (3262)2; 3636]
 
[36; 346; 3262; (63)2]
 
[36; 346; (3262)2; 63]
 
[346; (3262)2; 3636; 63]
 
[(346)2; 3262; 3636; 63]
 
[(36)2; 346; 3262; 63]
 
[(36)2; 346; 3262; 63]
 
[36; 346; 3262; (63)2]
 
[36; 346; 3262; (63)2]
 
[36; 346; 3262; (63)2]
 
[36; 346; (3262)2; 63]
 
[346; (3262)2; 3636; 63]
 
[346; (3262)2; 3636; 63]
 
[346; (3262)2; 3636; 63]
 
[346; 3262; 3636; (63)2]
 
[346; (3262)2; 3636; 63]
 
[3342; 3262; 3446; (63)2]
 
[3342; 3262; 3446; (63)2]
 
[3262; 3446; 3636; (44)2]
 
[3262; 3446; 3636; (44)2]
 
[3262; 3446; (3636)2; 44]
 
[3262; 3446; (3636)2; 44]
 
[3342; 3262; 3446; (44)2]
 
[346; 3342; 3446; (44)2]
 
[3262; 3446; 3636; (44)2]
 
[3262; 3446; 3636; (44)2]
 
[3262; 3446; (3636)2; 44]
 
[3262; 3446; (3636)2; 44]
 
[3342; 3262; 3446; (44)2]
 
[346; 3342; 3446; (44)2]
 
[346; (3342)2; 3636; 44]
 
[36; 3342; (3446)2; 3636]
 
[346; (3342)2; 3446; 3636]
 
[346; (3342)2; 3446; 3636]
 
[(36)2; 346; 3446; 3636]
 
[36; 3342; (3446)2; 3636]
 
[346; (3342)2; 3446; 3636]
 
[346; (3342)2; 3446; 3636]
 
[(36)2; 346; 3446; 3636]
 
[(36)2; 3342; 3446; 3636]
 
[36; 3342; 3446; (3636)2]
 
[346; 3342; (3446)2; 3636]
 
[36; 346; (3342)2; 3446]
 
[346; (3342)2; 3262; 3636]
 
[346; (3342)2; 3262; 3636]
 
[36; (346)2; 3342; 3446]
 
[36; (346)2; 3342; 3446]
 
[36; (346)2; 3342; 3446]
 
[36; 346; (3342)2; 3262]
 
[(36)2; 346; 3342; 3636]
 
[(36)2; 346; 3342; 3636]

5-uniform tilings, 3 vertex types (3:1:1) and (2:2:1)Edit

There are 149 5-uniform tilings, with 60 having 3:1:1 copies, and 89 having 2:2:1 copies.

5-uniform tilings (3:1:1)
 
[36; 334.12; (46.12)3]
 
[3464; 3446; (46.12)3]
 
[36; (334.12)3; 46.12]
 
[334.12; 343.12; (3.12.12)3]
 
[36; (33434)3; 343.12]
 
[3262; 3636; (63)3]
 
[346; 3262; (63)3]
 
[36; (3262)3; 63]
 
[36; (3262)3; 63]
 
[3262; (3636)3; 63]
 
[3446; 3636; (44)3]
 
[3446; 3636; (44)3]
 
[36; 3342; (44)3]
 
[36; 3342; (44)3]
 
[3446; (3636)3; 44]
 
[3446; (3636)3; 44]
 
[36; (3342)3; 44]
 
[36; (3342)3; 44]
 
[36; (3342)3; 44]
 
[(36)3; 3342; 44]
 
[(36)3; 3342; 44]
 
[3446; 3636; (44)3]
 
[3446; 3636; (44)3]
 
[36; 3342; (44)3]
 
[36; 3342; (44)3]
 
[(3342)3; 3262; 3446]
 
[3262; 3446; (3636)3]
 
[3262; 3446; (3636)3]
 
[3262; 3446; (3636)3]
 
[3262; 3446; (3636)3]
 
[3446; (3636)3; 44]
 
[3446; (3636)3; 44]
 
[36; (3342)3; 44]
 
[36; (3342)3; 44]
 
[36; (3342)3; 44]
 
[(36)3; 3342; 44]
 
[(36)3; 3342; 44]
 
[36; (3342)3; 44]
 
[36; (3342)3; 44]
 
[36; (3342)3; 44]
 
[(3342)3; 3446; 3636]
 
[(3342)3; 3446; 3636]
 
[346; (3342)3; 3446]
 
[(36)3; 346; 3262]
 
[(36)3; 346; 3262]
 
[(36)3; 346; 3262]
 
[346; (3262)3; 3636]
 
[346; (3262)3; 3636]
 
[(346)3; 3262; 3636]
 
[(346)3; 3262; 3636]
 
[(36)3; 346; 3262]
 
[(36)3; 346; 3262]
 
[(346)3; 3262; 3636]
 
[36; 346; (3636)3]
 
[36; 346; (3636)3]
 
[36; 346; (3636)3]
 
[36; 346; (3636)3]
 
[(36)3; 346; 3636]
 
[(36)3; 346; 3636]
 
[36; (346)3; 3636]
5-uniform tilings (2:2:1)
 
[(3446)2; (3636)2; 46.12]
 
[(36)2; (3342)2; 3464]
 
[(3342)2; 334.12; (3464)2]
 
[36; (33434)2; (3464)2]
 
[3342; (33434)2; (3464)2]
 
[3342; (33434)2; (3464)2]
 
[3342; (33434)2; (3464)2]
 
[(33434)2; 343.12; (3464)2]
 
[36; (3262)2; (63)2]
 
[(3262)2; (3636)2; 63]
 
[(36)2; (3342)2; 33434]
 
[(36)2; 3342; (33434)2]
 
[346; (3342)2; (33434)2]
 
[(36)2; 3342; (33434)2]
 
[(36)2; 3342; (33434)2]
 
[(3262)2; 3636; (63)2]
 
[(3446)2; 3636; (44)2]
 
[(3446)2; 3636; (44)2]
 
[3446; (3636)2; (44)2]
 
[(3446)2; 3636; (44)2]
 
[(3446)2; 3636; (44)2]
 
[3446; (3636)2; (44)2]
 
[36; (3342)2; (44)2]
 
[(36)2; 3342; (44)2]
 
[(36)2; 3342; (44)2]
 
[(3446)2; 3636; (44)2]
 
[(3446)2; 3636; (44)2]
 
[(3446)2; 3636; (44)2]
 
[(3446)2; 3636; (44)2]
 
[(3446)2; 3636; (44)2]
 
[36; (3342)2; (44)2]
 
[(36)2; (3342)2; 44]
 
[(3446)2; 3636; (44)2]
 
[(3446)2; 3636; (44)2]
 
[3446; (3636)2; (44)2]
 
[(3446)2; 3636; (44)2]
 
[(3446)2; 3636; (44)2]
 
[3446; (3636)2; (44)2]
 
[36; (3342)2; (44)2]
 
[(36)2; 3342; (44)2]
 
[(36)2; 3342; (44)2]
 
[36; (3342)2; (44)2]
 
[36; (3342)2; (44)2]
 
[(3446)2; 3636; (44)2]
 
[(36)2; (3342)2; 44]
 
[(36)2; (3342)2; 44]
 
[(36)2; (3342)2; 44]
 
[(36)2; (3342)2; 44]
 
[(33434)2; 3262; (3446)2]
 
[3342; (3262)2; (3446)2]
 
[3342; (3262)2; (3446)2]
 
[3262; (3446)2; (3636)2]
 
[(3262)2; 3446; (3636)2]
 
[(3262)2; 3446; (3636)2]
 
[(3464)2; (3446)2; 3636]
 
[3262; (3446)2; (3636)2]
 
[3262; (3446)2; (3636)2]
 
[(346)2; (3446)2; 3636]
 
[(346)2; (3446)2; 3636]
 
[(346)2; (3446)2; 3636]
 
[(346)2; (3446)2; 3636]
 
[(3342)2; (3446)2; 3636]
 
[(3342)2; (3446)2; 3636]
 
[(346)2; (3342)2; 3446]
 
[(346)2; 3342; (3446)2]
 
[(36)2; (346)2; 3262]
 
[36; (346)2; (3262)2]
 
[(36)2; 346; (3262)2]
 
[(346)2; (3262)2; 63]
 
[36; (3262)2; (63)2]
 
[36; (346)2; (3262)2]
 
[346; (3262)2; (3636)2]
 
[(346)2; (3262)2; 3636]
 
[36; (346)2; (3262)2]
 
[(346)2; 3262; (3636)2]
 
[(346)2; (3262)2; 3636]
 
[(36)2; (346)2; 3262]
 
[(36)2; (346)2; 3262]
 
[(36)2; (346)2; 3636]
 
[(36)2; (346)2; 3636]
 
[36; (346)2; (3342)2]
 
[(36)2; (346)2; 3262]
 
[36; (346)2; (3262)2]
 
[36; (346)2; (3262)2]
 
[346; (3342)2; (3636)2]
 
[346; (3342)2; (3636)2]
 
[(36)2; 346; (3636)2]
 
[(36)2; (346)2; 3636]
 
[(36)2; 3342; (33434)2]

5-uniform tilings, 2 vertex types (4:1) and (3:2)Edit

There are 74 5-uniform tilings with 2 types of vertices, 27 with 4:1 and 47 with 3:2 copies of each.

5-uniform tilings (4:1)
 
[(3464)4; 46.12]
 
[343.12; (3.12.12)4]
 
[36; (33434)4]
 
[36; (33434)4]
 
[(36)4; 346]
 
[(36)4; 346]
 
[(36)4; 346]
 
[36; (346)4]
 
[3262; (3636)4]
 
[(346)4; 3262]
 
[(346)4; 3262]
 
[(346)4; 3636]
 
[3262; (3636)4]
 
[3446; (3636)4]
 
[3446; (3636)4]
 
[(3342)4; 33434]
 
[3342; (33434)4]
 
[3342; (44)4]
 
[3342; (44)4]
 
[(3342)4; 44]
 
[(3342)4; 44]
 
[(3342)4; 44]
 
[36; (3342)4]
 
[36; (3342)4]
 
[36; (3342)4]
 
[(36)4; 3342]
 
[(36)4; 3342]

There are 29 5-uniform tilings with 3 and 2 unique vertex figure types.

5-uniform tilings (3:2)
 
[(3464)2; (46.12)3]
 
[(3464)2; (46.12)3]
 
[(3464)3; (3446)2]
 
[(33434)2; (3464)3]
 
[(33434)3; (3464)2]
 
[(36)2; (346)3]
 
[(36)2; (346)3]
 
[(36)3; (346)2]
 
[(36)3; (346)2]
 
[(36)3; (346)2]
 
[(36)3; (346)2]
 
[(36)2; (346)3]
 
[(36)2; (346)3]
 
[(36)2; (346)3]
 
[(3262)2; (3636)3]
 
[(346)3; (3636)2]
 
[(346)3; (3636)2]
 
[(346)2; (3636)3]
 
[(3446)3; (3636)2]
 
[(3446)2; (3636)3]
 
[(3446)3; (3636)2]
 
[(3446)2; (3636)3]
 
[(3446)2; (3636)3]
 
[(3342)3; (33434)2]
 
[(3342)3; (33434)2]
 
[(3342)2; (33434)3]
 
[(3342)2; (33434)3]
 
[(3342)2; (44)3]
 
[(3342)2; (44)3]
 
[(3342)2; (44)3]
 
[(3342)3; (44)2]
 
[(3342)2; (44)3]
 
[(3342)3; (44)2]
 
[(3342)2; (44)3]
 
[(3342)2; (44)3]
 
[(3342)3; (44)2]
 
[(3342)3; (44)2]
 
[(36)2; (3342)3]
 
[(36)2; (3342)3]
 
[(36)2; (3342)3]
 
[(36)2; (3342)3]
 
[(36)3; (3342)2]
 
[(36)3; (3342)2]
 
[(36)3; (3342)2]
 
[(36)3; (3342)2]
 
[(36)3; (3342)2]
 
[(36)3; (3342)2]

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.

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 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.[6] 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
 
Rows of squares with horizontal offsets
 
Rows of triangles with horizontal offsets
 
A tiling by squares
 
Three hexagons surround each triangle
 
Six triangles surround every hexagon.
 
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 alsoEdit

ReferencesEdit

  1. ^ Critchlow, p.60-61
  2. ^ k-uniform tilings by regular polygons Nils Lenngren, 2009
  3. ^ Critchlow, p.62-67
  4. ^ Tilings and Patterns, Grünbaum and Shephard 1986, pp. 65-67
  5. ^ In Search of Demiregular Tilings
  6. ^ Tilings by regular polygons p.236

External linksEdit

Euclidean and general tiling links: