Tiling by regular polygons

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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.

p6m, *632 p4m, *442
1-uniform n11.svg
1-uniform n1.svg
1-uniform n5.svg

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.

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.

p4m, *442
1-uniform n2.svg
p6m, *632
1-uniform n4.svg
p6m, *632
1-uniform n7.svg
cmm, 2*22
1-uniform n8.svg
p6m, *632
1-uniform n6.svg
p6m, *632
1-uniform n3.svg
p6, 632
1-uniform n10.svg
p4, 442
1-uniform n9.svg

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.

Combinations of regular polygons that can meet at a vertex[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. 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.

With 3 polygons at a vertex:

  • 3.7.42 (cannot appear in any tiling of regular polygons)
  • 3.8.24 (cannot appear in any tiling of regular polygons)
  • 3.9.18 (cannot appear in any tiling of regular polygons)
  • 3.10.15 (cannot appear in any tiling of regular polygons)
  • 3.122 - semi-regular, truncated hexagonal tiling
  • 4.5.20 (cannot appear in any tiling of regular polygons)
  • 4.6.12 - semi-regular, truncated trihexagonal tiling
  • 4.82 - semi-regular, truncated square tiling
  • 52.10 (cannot appear in any tiling of regular polygons)
  • 63 - regular, hexagonal tiling

Below are diagrams of such vertices:

With 4 polygons at a vertex:

  • 32.4.12 - does not generate a uniform tiling; can generate a 2-uniform tiling when used with the vertex type 36 (shown below).
  • - does not generate a uniform tiling; can generate a 2-uniform tiling when used with the vertex type 3.122
  • 32.62 - does not generate a uniform tiling; can generate 2-uniform tilings when used with any one of the vertex types 36, 34.6 or (two of these are shown below).
  • - semi-regular, trihexagonal tiling
  • 44 - regular, square tiling
  • 3.42.6 - does not generate a uniform tiling; can generate a 2-uniform tiling when used with either of the vertex types or In the later case, there are two inequivalent 2-uniform tilings that can be generated. (One of these later two are shown below).
  • - semi-regular, rhombitrihexagonal tiling

Below are diagrams of such vertices:

With 5 polygons at a vertex:

Below are diagrams of such vertices:

With 6 polygons at a vertex:

Below is a diagram of such a vertex:

Other edge-to-edge 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.

Shown below are the twenty 2-uniform tilings. Chavey (1989) lists all those edge-to-edge tilings by regular polygons which are at most 3-uniform, 3-isohedral or 3-isotoxal.

2-uniform tilings
cmm, 2*22
2-uniform 4.png
cmm, 2*22
2-uniform 3.png
pmm, *2222
2-uniform 14.png
cmm, 2*22
2-uniform 15.png
cmm, 2*22
2-uniform 6.png
pmm, *2222
2-uniform 7.png
pmm, *2222
2-uniform 11.png
p4m, *442
2-uniform 2.png
p4g, 4*2
2-uniform 16.png
p2, 2222
2-uniform 17.png
p6m, *632
2-uniform 10.png
p6m, *632
2-uniform 19.png
p6, 632
2-uniform 20.png
cmm, 2*22
2-uniform 12.png
p6m, *632
2-uniform 18.png
p6m, *632
2-uniform 9.png
p6m, *632
2-uniform 1.png
p6m, *632
2-uniform 5.png
p6m, *632
2-uniform 8.png
p6m, *632
2-uniform 13.png

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

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 convex 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.

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; 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.

The hyperbolic plane[edit]

These tessellations are also related to regular and semiregular polyhedra and tessellations of the hyperbolic plane. Semiregular polyhedra are made from regular polygon faces, but their angles at a point add to less than 360 degrees. Regular polygons in hyperbolic geometry have angles smaller than they do in the plane. In both these cases, that the arrangement of polygons is the same at each vertex does not mean that the polyhedron or tiling is vertex-transitive.

Some regular tilings of the hyperbolic plane (Using Poincaré disc model projection)

Hyperspace tiling 4-5.png Hyperspace tiling 5-4.png Hyperbolic tiling 3-7.png Hyperbolic tiling 7-3.png Hyperbolic tiling truncated 3-7.png
Hyperbolic tiling rectified 3-7.png Hyperbolic tiling truncated 7-3.png Hyperbolic tiling cantellated 3-7.png Hyperbolic tiling omnitruncated 3-7.png Hyperbolic tiling snub 3-7.png

See also[edit]


External links[edit]

Euclidean and general tiling links:

Hyperbolic tiling links: