Tetrakis square tiling

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Tetrakis square tiling
Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg
Type Dual semiregular tiling
Coxeter diagram CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node f1.png
Faces 45-45-90 triangle
Face configuration V4.8.8
Symmetry group p4m, [4,4], *442
Rotation group p4, [4,4]+, (442)
Dual Truncated square tiling
Properties face-transitive

In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a 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]

It is labeled V4.8.8 because each isosceles triangle face has two types of vertices: one with 4 triangles, and two with 8 triangles.

Dual tiling[edit]

It is the dual tessellation of the truncated square tiling which has one square and two octagons at each vertex.[3]

P1 dual.png

Applications[edit]

A 5 × 9 portion of the tetrakis square tiling is used to form the board for the Malagasy board game Fanorona. In this game, pieces are placed on the vertices of the tiling, and move along the edges, capturing pieces of the other color until one side has captured all of the other side's pieces. In this game, the degree-4 and degree-8 vertices of the tiling are called respectively weak intersections and strong intersections, a distinction that plays an important role in the strategy of the game.[4] A similar board is also used for the Brazilian game Adugo, and for the game of Hare and Hounds.

The tetrakis square tiling was used for a set of commemorative postage stamps issued by the United States Postal Service in 1997, with an alternating pattern of two different stamps. Compared to the simpler pattern for triangular stamps in which all diagonal perforations are parallel to each other, the tetrakis pattern has the advantage that, when folded along any of its perforations, the other perforations line up with each other, making repeated folding possible.[5]

This tiling also forms the basis for a commonly used "pinwheel", "windmill", and "broken dishes" patterns in quilting.[6][7][8]

Symmetry[edit]

The symmetry type is:

  • with the coloring: cmm; a primitive cell is 8 triangles, a fundamental domain 2 triangles (1/2 for each color)
  • with the dark triangles in black and the light ones in white: p4g; a primitive cell is 8 triangles, a fundamental domain 1 triangle (1/2 each for black and white)
  • with the edges in black and the interiors in white: p4m; a primitive cell is 2 triangles, a fundamental domain 1/2

The edges of the tetrakis square tiling form a simplicial arrangement of lines, a property it shares with the triangular tiling and the kisrhombille tiling.

These lines form the axes of symmetry of a reflection group (the wallpaper group [4,4], (*442) or p4m), which has the triangles of the tiling as its fundamental domains. This group is isomorphic to, but not the same as, the group of automorphisms of the tiling, which has additional axes of symmetry bisecting the triangles and which has half-triangles as its fundamental domains.

There are many small index subgroups of p4m, [4,4] symmetry (*442 orbifold notation), that can be seen in relation to the Coxeter diagram, with nodes colored to correspond to reflection lines, and gyration points labeled numerically. Rotational symmetry is shown by alternately white and blue colored areas with a single fundamental domain for each subgroup is filled in yellow. Glide reflections are given with dashed lines.

Subgroups can be expressed as Coxeter diagrams, along with fundamental domain diagrams.

Small index subgroups of p4m, [4,4], (*442)
index 1 2 4
Fundamental
domain
diagram
442 symmetry 000.png 442 symmetry a00.png 442 symmetry 00a.png 442 symmetry 0a0.png 442 symmetry a0b.png 442 symmetry xxx.png
Coxeter notation
Coxeter diagram
[1,4,1,4,1] = [4,4]
CDel node c5.pngCDel 4.pngCDel node c1.pngCDel 4.pngCDel node c3.png
[1+,4,4]
CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 4.pngCDel node c3.png = CDel nodeab c1.pngCDel split2-44.pngCDel node c3.png
[4,4,1+]
CDel node c5.pngCDel 4.pngCDel node c1.pngCDel 4.pngCDel node h0.png = CDel node c5.pngCDel split1-44.pngCDel nodeab c1.png
[4,1+,4]
CDel node c5.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node c3.png = CDel nodeab c5.pngCDel iaib.pngCDel nodeab c3.png
[1+,4,4,1+]
CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 4.pngCDel node h0.png = CDel labelinfin.pngCDel branch c1.pngCDel 2.pngCDel branch c1.pngCDel labelinfin.png
[4+,4+] = [(4,4+,2+)]
CDel node h2.pngCDel 4.pngCDel node h4.pngCDel 4.pngCDel node h2.png
Orbifold *442 *2222 22×
Semidirect subgroups
index 2 4
Diagram 442 symmetry 0aa.png 442 symmetry aa0.png 442 symmetry a0a.png 442 symmetry 0ab.png 442 symmetry ab0.png
Coxeter [4,4+]
CDel node c5.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node h2.png
[4+,4]
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node c3.png
[(4,4,2+)]
CDel node c1.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png
[1+,4,1+,4]=[(2+,4,4)]
CDel node h0.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node c3.png = CDel label2.pngCDel branch h2h2.pngCDel split2-44.pngCDel node c3.png = CDel label2.pngCDel branch h2h2.pngCDel iaib.pngCDel nodeab c3.png
[4,1+,4,1+]=[(4,4,2+)]
CDel node c5.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node h0.png = CDel node c5.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png = CDel nodeab c5.pngCDel iaib.pngCDel branch h2h2.pngCDel label2.png
Orbifold 4*2 2*22
Direct subgroups
Index 2 4 8
Diagram 442 symmetry aaa.png 442 symmetry abb.png 442 symmetry aab.png 442 symmetry aba.png 442 symmetry abc.png
Coxeter [4,4]+
CDel node h2.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png
[1+,4,4+] = [4,4+]+
CDel node h0.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node h2.png = CDel node h2.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png
[4+,4,1+] = [4+,4]+
CDel node h2.pngCDel 4.pngCDel node h2.pngCDel 4.pngCDel node h0.png = CDel node h2.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png
[(4,1+,4,2+)] = [(4,4,2+)]+
CDel labelh.pngCDel node.pngCDel split1-44.pngCDel branch h2h2.pngCDel label2.png = CDel label2.pngCDel branch h2h2.pngCDel iaib.pngCDel branch h2h2.pngCDel label2.png
[1+,4,1+,4,1+] = [(4+,4+,2+)] = [4+,4+]+
CDel node h0.pngCDel 4.pngCDel node h0.pngCDel 4.pngCDel node h0.png = CDel label2.pngCDel branch h2h2.pngCDel iaib.pngCDel branch h2h2.pngCDel label2.png
Orbifold 442 2222

Related polyhedra and tilings[edit]

It is topologically related to a series of polyhedra and tilings with face configuration Vn.6.6.

Dimensional family of truncated polyhedra and tilings: n.8.8
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
2.8.8 Uniform tiling 432-t01.png
3.8.8
Uniform tiling 44-t12.png
4.8.8
Uniform tiling 54-t12.png
5.8.8
Uniform tiling 64-t12.png
6.8.8
Uniform tiling 74-t12.png
7.8.8
Uniform tiling 84-t12.png
8.8.8
H2 tiling 24i-6.png
∞.8.8
Coxeter
Schläfli
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node.png
t{4,2}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t{4,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
t{4,4}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 5.pngCDel node.png
t{4,5}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 6.pngCDel node.png
t{4,6}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 7.pngCDel node.png
t{4,7}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png
t{4,8}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel infin.pngCDel node.png
t{4,∞}
Uniform dual figures
n-kis
figures
Spherical octagonal hosohedron.png
V2.8.8
Triakisoctahedron.jpg
V3.8.8
Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg
V4.8.8
Order-4 pentakis pentagonal tiling.png
V5.8.8
Order4 hexakis hexagonal til.png
V6.8.8
Order4 heptakis heptagonal til.png
V7.8.8
Uniform tiling 83-t2.png
V8.8.8
Ord4 apeirokis apeirogonal til.png
V∞.8.8
Coxeter CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 2.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 7.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel infin.pngCDel node.png
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

See also[edit]

Notes[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 .
  3. ^ Weisstein, Eric W., "Dual tessellation", MathWorld.
  4. ^ Bell, R. C. (1983), "Fanorona", The Boardgame Book, Exeter Books, pp. 150–151, ISBN 0-671-06030-9 
  5. ^ Frederickson, Greg N. (2006), Piano-Hinged Dissections, A K Peters, p. 144 .
  6. ^ The Quilting Bible, Creative Publishing International, 1997, p. 55, ISBN 9780865732001 .
  7. ^ Zieman, Nancy (2011), Quilt With Confidence, Krause Publications, p. 66, ISBN 9781440223556 .
  8. ^ Fassett, Kaffe (2007), Kaffe Fassett's Kaleidoscope of Quilts: Twenty Designs from Rowan for Patchwork and Quilting, Taunton Press, p. 96, ISBN 9781561589388 .

References[edit]