Truncated square tiling

Truncated square tiling

TypeSemiregular tiling
Vertex configuration4.8.8
Schläfli symboltr{4,4}
Wythoff symbol2 | 4 4
4 4 2 |
Coxeter diagram
Symmetryp4m, [4,4], (*442)
Rotation symmetryp4, [4,4]+, (442)
Bowers acronymTosquat
DualTetrakis square tiling
PropertiesVertex-transitive

Vertex figure: 4.8.8

In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon. It has Schläfli symbol of t0,1{4,4}.

Conway calls it a truncated quadrille, constructed as a truncation operation applied to a square tiling (quadrille).

Other names used for this pattern include Mediterranean tiling and octagonal tiling, which is often represented by smaller squares, and nonregular octagons which alternate long and short edges.

There are 3 regular and 8 semiregular tilings in the plane.

Uniform colorings

There are two distinct uniform colorings of a truncated square tiling. (Naming the colors by indices around a vertex (4.8.8): 122, 123.)


2 colors: 122
3 colors: 123

Circle packing

The truncated square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing number). Since there is an even number of sides of all the polygons, the circles can be alternately colored as shown below.

Related polyhedra and tilings

The truncated square tiling is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures 4.2n.2n, extending into the hyperbolic plane:

Dimensional family of truncated polyhedra and tilings: 4.2n.2n
Symmetry
*n42
[n,4]		 Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
D4h		 *342
[3,4]
Oh		 *442
[4,4]
P4m		 *542
[5,4]		 *642
[6,4]		 *742
[7,4]		 *842
[8,4]...		 *∞42
[∞,4]
Truncated
figures
4.4.4
4.6.6
4.8.8
4.10.10
4.12.12
4.14.14
4.16.16
4.∞.∞
Coxeter
Schläfli
t{2,4}
t{3,4}
t{4,4}
t{5,4}
t{6,4}
t{7,4}
t{8,4}
t{4,∞}
Uniform dual figures
n-kis
figures
V4.4.4
V4.6.6
V4.8.8
V4.10.10
V4.12.12
V4.14.14
V4.16.16
V4.∞.∞
Coxeter

The Pythagorean tiling alternates large and small squares, and may be seen as topologically identical to the truncated square tiling. The squares are rotated 45 degrees and octagons are distorted into squares with mid-edge vertices.

Variations on this pattern are often called Mediterranean patterns, shown in stone tiles like this one with smaller squares and diagonally aligned with the borders. Pythagorean tilings This weaving pattern has the same topology as well, with octagons flattened into 3 by 1 rectangles

The 3-dimensional bitruncated cubic honeycomb projected into the plane shows two copies of a truncated tiling. In the plane it can be represented by a compound tiling:

+

Wythoff constructions from square tiling

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three unique topologically forms: square tiling, truncated square tiling, snub square tiling.

Uniform tilings based on square tiling symmetry
Symmetry: [4,4], (*442) [4,4]+, (442) [4,4+], (4*2)
{4,4} t{4,4} r{4,4} t{4,4} {4,4} rr{4,4} tr{4,4} sr{4,4} s{4,4}
Uniform duals
V4.4.4.4 V4.8.8 V4.4.4.4 V4.8.8 V4.4.4.4 V4.4.4.4 V4.8.8 V3.3.4.3.4
Dimensional family of omnitruncated polyhedra and tilings: 4.8.2n
Symmetry
*n42
[n,4]		 Spherical Euclidean Compact hyperbolic Paracompact
*242
[2,4]
D4h		 *342
[3,4]
Oh		 *442
[4,4]
P4m		 *542
[5,4]		 *642
[6,4]		 *742
[7,4]		 *842
[8,4]...		 *∞42
[∞,4]
Omnitruncated
figure
4.8.4
4.8.6
4.8.8
4.8.10
4.8.12
4.8.14
4.8.16
4.8.∞
Coxeter
Schläfli
tr{2,4}
tr{3,4}
tr{4,4}
tr{5,4}
tr{6,4}
tr{7,4}
tr{8,4}
tr{∞,4}
Omnitruncated
duals
V4.8.4
V4.8.6
V4.8.8
V4.8.10
V4.8.12
V4.8.14
V4.8.16
V4.8.∞
Coxeter
Dimensional family of omnitruncated polyhedra and tilings: 4.2n.2n
Symmetry
*nn2
[n,n]		 Spherical Euclidean Compact hyperbolic Paracompact
*222
[2,2]
D2h		 *332
[3,3]
Td		 *442
[4,4]
P4m		 *552
[5,5]		 *662
[6,6]		 *772
[7,7]		 *882
[8,8]...		 *∞∞2
[∞,∞]
Figure
4.4.4
4.6.6
4.8.8
4.10.10
4.12.12
4.14.14
4.16.16
4.∞.∞
Coxeter
Schläfli
tr{2,2}
tr{3,3}
tr{4,4}
tr{5,5}
tr{6,6}
tr{7,7}
tr{8,8}
tr{∞,∞}
Dual
V4.4.4
V4.6.6
V4.8.8
V4.10.10
V4.12.12
V4.14.14
V4.16.16
V4.∞.∞
Coxeter

Tetrakis square tiling

The tetrakis square tiling

The tetrakis square tiling is the tiling of the Euclidean plane dual to the truncated square tiling. It can be constructed square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of √2.

Conway calls it a kisquadrille,[1] represented by a kis operation that adds a center point and triangles to replace the faces of a square tiling (quadrille). It is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices.[2]

See also

Wikimedia Commons has media related to Uniform tiling 4-8-8.

References

  1. ↑ John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
  2. ↑ Stephenson, John, "Ising Model with Antiferromagnetic Next-Nearest-Neighbor Coupling: Spin Correlations and Disorder Points", Phys. Rev. B 1 (11): 4405–4409, doi:10.1103/PhysRevB.1.4405.

External links