Tetrakis square tiling

Tetrakis square tiling
Tetrakis square tiling
Type	Dual semiregular tiling
Faces	45-45-90 triangle
Coxeter diagram	;
Symmetry group	p4m, [4,4], *442
Rotation group	p4, [4,4]+, (442)
Dual polyhedron	Truncated square tiling
Face configuration	V4.8.8
Properties	face-transitive

Last updated October 21, 2021

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, Burgiel, and Goodman-Strauss call 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.

As a dual uniform tiling

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

Applications

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

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
Coxeter notation Coxeter diagram	[1,4,1,4,1] = [4,4]	[1⁺,4,4] =	[4,4,1⁺] =	[4,1⁺,4] =	[1⁺,4,4,1⁺] =	[4⁺,4⁺] = [(4,4⁺,2⁺)]
Orbifold	*442			*2222		22×
Semidirect subgroups
index		2			4
Diagram
Coxeter		[4,4⁺]	[4⁺,4]	[(4,4,2⁺)]	[1⁺,4,1⁺,4]=[(2⁺,4,4)] = =	[4,1⁺,4,1⁺]=[(4,4,2⁺)] = =
Orbifold		4*2		2*22
Direct subgroups
Index	2	4			8
Diagram
Coxeter	[4,4]⁺	[1⁺,4,4⁺] = [4,4⁺]⁺ =	[4⁺,4,1⁺] = [4⁺,4]⁺ =	[(4,1⁺,4,2⁺)] = [(4,4,2⁺)]⁺ =	[1⁺,4,1⁺,4,1⁺] = [(4⁺,4⁺,2⁺)] = [4⁺,4⁺]⁺ =
Orbifold	442			2222

Notes

↑ Conway, John; Burgiel, Heidi; Goodman-Strauss, Chaim (2008), "Chapter 21: Naming Archimedean and Catalan polyhedra and tilings", The Symmetries of Things, AK Peters, p. 288, ISBN 978-1-56881-220-5
↑ 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 .
↑ Weisstein, Eric W. "Dual tessellation". MathWorld .
↑ Bell, R. C. (1983), "Fanorona", The Boardgame Book, Exeter Books, pp. 150–151, ISBN 0-671-06030-9
↑ Frederickson, Greg N. (2006), Piano-Hinged Dissections, A K Peters, p. 144.
↑ The Quilting Bible, Creative Publishing International, 1997, p. 55, ISBN 9780865732001 .
↑ Zieman, Nancy (2011), Quilt With Confidence, Krause Publications, p. 66, ISBN 9781440223556 .
↑ Fassett, Kaffe (2007), Kaffe Fassett's Kaleidoscope of Quilts: Twenty Designs from Rowan for Patchwork and Quilting, Taunton Press, p. 96, ISBN 9781561589388 .

Related Research Articles

In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

In geometry, an octagon is an eight-sided polygon or 8-gon.

A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles and tiles as well as wallpaper.

In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid.

In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.

In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{3,6}.

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 t{4,4}.

In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}.

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway calls this honeycomb a cubille.

The bitruncated cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.

The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation in Euclidean 3-space made up of identical tetragonal disphenoidal cells. Cells are face-transitive with 4 identical isosceles triangle faces. John Horton Conway calls it an oblate tetrahedrille or shortened to obtetrahedrille.

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.

In hyperbolic geometry, a uniform hyperbolic tiling is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry.

In geometry, the 3-7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 6, and 14 triangles meeting at each vertex.

In geometry, the truncated trioctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one hexadecagon (16-sides) on each vertex. It has Schläfli symbol of tr{8,3}.

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{6,4}.

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular triangles around each vertex.

In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,4}.

In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}.

References

Grünbaum, Branko & Shephard, G. C. (1987). Tilings and Patterns . New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65)
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 40. ISBN 0-486-23729-X.
Keith Critchlow, Order in Space: A design source book, 1970, p. 77-76, pattern 8

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[1] Conway, John; Burgiel, Heidi; Goodman-Strauss, Chaim (2008), "Chapter 21: Naming Archimedean and Catalan polyhedra and tilings", The Symmetries of Things, AK Peters, p. 288, ISBN 978-1-56881-220-5

[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 .

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