Truncated square tiling

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Truncated square tiling
Truncated square tiling
Type Semiregular tiling
Vertex configuration 4.8.8
Schläfli symbol tr{4,4}
Wythoff symbol 2 | 4 4
4 4 2 |
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Symmetry p4m, [4,4], (*442)
Rotation symmetry p4, [4,4]+, (442)
Bowers acronym Tosquat
Dual Tetrakis square tiling
Properties Vertex-transitive
Truncated square tiling
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[edit]

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

Uniform tiling 44-t12.png
2 colors: 122
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t012.png
3 colors: 123
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png

Circle packing[edit]

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.

Truncated square tiling circle packing.png Truncated square tiling circle packing2.png

Related polyhedra and tilings[edit]

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
Spherical square prism.png
4.4.4
Uniform tiling 432-t12.png
4.6.6
Uniform tiling 44-t01.png
4.8.8
Uniform tiling 54-t01.png
4.10.10
Uniform tiling 64-t01.png
4.12.12
Uniform tiling 74-t01.png
4.14.14
Uniform tiling 84-t01.png
4.16.16
H2 tiling 24i-3.png
4.∞.∞
Coxeter
Schläfli
CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
t{2,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t{3,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
t{4,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 5.pngCDel node 1.png
t{5,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 6.pngCDel node 1.png
t{6,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 7.pngCDel node 1.png
t{7,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node 1.png
t{8,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel infin.pngCDel node 1.png
t{4,∞}
Uniform dual figures
n-kis
figures
Spherical square bipyramid.png
V4.4.4
Tetrakishexahedron.jpg
V4.6.6
Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg
V4.8.8
Order-5 tetrakis square tiling.png
V4.10.10
Order-6 tetrakis square tiling.png
V4.12.12
Hyperbolic domains 772.png
V4.14.14
Order-8 tetrakis square tiling.png
V4.16.16
H2checkers 2ii.png
V4.∞.∞
Coxeter CDel node.pngCDel 4.pngCDel node f1.pngCDel 2.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 6.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 7.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 8.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node f1.pngCDel infin.pngCDel node f1.png

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.

Mediterranean tiling.png Distorted truncated square tiling.png Distorted truncated square tiling2.png Weaved truncated square tiling.png
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:

Uniform tiling 44-t01.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
Uniform tiling 44-t12.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Uniform tiling 44-t01 and t12.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png + CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png

Wythoff constructions from square tiling[edit]

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)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 44-t0.png Uniform tiling 44-t01.png Uniform tiling 44-t1.png Uniform tiling 44-t12.png Uniform tiling 44-t2.png Uniform tiling 44-t02.png Uniform tiling 44-t012.png Uniform tiling 44-snub.png Uniform tiling 44-h01.png
{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
CDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node fh.png
Uniform tiling 44-t0.png Tetrakis square tiling.png Uniform tiling 44-t0.png Tetrakis square tiling.png Uniform tiling 44-t0.png Uniform tiling 44-t0.png Tetrakis square tiling.png Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg
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
Spherical octagonal prism2.png
4.8.4
Uniform tiling 432-t012.png
4.8.6
Uniform tiling 44-t012.png
4.8.8
H2 tiling 245-7.png
4.8.10
H2 tiling 246-7.png
4.8.12
H2 tiling 247-7.png
4.8.14
H2 tiling 248-7.png
4.8.16
H2 tiling 24i-7.png
4.8.∞
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{2,4}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{3,4}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{4,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{5,4}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{6,4}
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{7,4}
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{8,4}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{∞,4}
Omnitruncated
duals
Hexagonale bipiramide.png
V4.8.4
Disdyakisdodecahedron.jpg
V4.8.6
Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg
V4.8.8
Order-4 bisected pentagonal tiling.png
V4.8.10
Hyperbolic domains 642.png
V4.8.12
Hyperbolic domains 742.png
V4.8.14
Hyperbolic domains 842.png
V4.8.16
H2checkers 24i.png
V4.8.∞
Coxeter CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node f1.png
Dimensional family of omnitruncated polyhedra and tilings: 4.2n.2n
Symmetry
*nn2
[n,n]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*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
[∞,∞]
 
[∞,iπ/λ]
Figure Spherical square prism.png
4.4.4
Uniform tiling 332-t012.png
4.6.6
Uniform tiling 44-t012.png
4.8.8
H2 tiling 255-7.png
4.10.10
H2 tiling 266-7.png
4.12.12
H2 tiling 277-7.png
4.14.14
H2 tiling 288-7.png
4.16.16
H2 tiling 2ii-7.png
4.∞.∞
H2 tiling 2iu-7.png
4.∞.∞
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
tr{2,2}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{3,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tr{4,4}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 5.pngCDel node 1.png
tr{5,5}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node 1.png
tr{6,6}
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node 1.png
tr{7,7}
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png
tr{8,8}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.png
tr{∞,∞}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel ultra.pngCDel node 1.png
Dual Octahedron.png
V4.4.4
Tetrakishexahedron.jpg
V4.6.6
Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg
V4.8.8
Order-4 bisected pentagonal tiling.png
V4.10.10
Hyperbolic domains 642.png
V4.12.12
Hyperbolic domains 742.png
V4.14.14
Hyperbolic domains 842.png
V4.16.16
H2checkers 24i.png
V4.∞.∞
Coxeter CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 5.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 7.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel ultra.pngCDel node f1.png

Tetrakis square tiling[edit]

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[edit]

References[edit]

  1. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1] (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[edit]