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 Harmonices Mundi.

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.

Tiling Regular 3-6 Triangular.svg
36
Triangular tiling
Tiling Regular 4-4 Square.svg
44
Square tiling
Tiling Regular 6-3 Hexagonal.svg
63
Hexagonal tiling

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.

Tiling Semiregular 3-3-3-3-6 Snub Hexagonal.svg
34.6
Snub hexagonal tiling
Tiling Semiregular 3-3-3-3-6 Snub Hexagonal Mirror.svg
34.6
Snub hexagonal tiling reflection
Tiling Semiregular 3-6-3-6 Trihexagonal.svg
3.6.3.6
Trihexagonal tiling
Tiling Semiregular 3-3-3-4-4 Elongated Triangular.svg
33.42
Elongated triangular tiling
Tiling Semiregular 3-3-4-3-4 Snub Square.svg
32.4.3.4
Snub square tiling
Tiling Semiregular 3-4-6-4 Small Rhombitrihexagonal.svg
3.4.6.4
Rhombitrihexagonal tiling
Tiling Semiregular 4-8-8 Truncated Square.svg
4.82
Truncated square tiling
Tiling Semiregular 3-12-12 Truncated Hexagonal.svg
3.122
Truncated hexagonal tiling
Tiling Semiregular 4-6-12 Great Rhombitrihexagonal.svg
4.6.12
Truncated trihexagonal tiling

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 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 - not uniform, has two different types of vertices 32.4.12 and 36
  • 3.4.3.12 - not uniform, has two different types of vertices 3.4.3.12 and 3.3.4.3.4
  • 32.62 - not uniform, occurs in two patterns with vertices 32.62/36 and 32.62/3.6.3.6.
  • 3.6.3.6 - semi-regular, trihexagonal tiling
  • 44 - regular, square tiling
  • 3.42.6 - not uniform, has vertices 3.42.6 and 3.6.3.6.
  • 3.4.6.4 - 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]

Any number of non-uniform (sometimes called demiregular) edge-to-edge tilings by regular polygons may be drawn. Here are four examples:

Dem3366bc.png
32.62 and 36
Dem3366rbc.gif
32.62 and 3.6.3.6
Academ Periodic tiling by dodecagons and other convex regular polygons.svg
32.4.12 and 36
Academ Periodic tiling by hexagons and other regular polygons.svg
3.42.6 and 3.6.3.6

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. The examples above are four of 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.

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
Academ Regular hexagons and equilateral triangles in periodic tiling.svg
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.
A Pythagorean tiling View 7.svg
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]

References[edit]

External links[edit]

Euclidean and general tiling links:

Hyperbolic tiling links: