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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the tetrakis square tiling is a tiling of the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
. It is a
square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of th ...
with each square divided into four
isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
right triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
s 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, represented by a kis operation that adds a center point and triangles to replace the faces of a
square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of th ...
(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. 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. :


Applications

A 5 × 9 portion of the tetrakis square tiling is used to form the board for the Malagasy
board game Board games are tabletop games that typically use . These pieces are moved or placed on a pre-marked board (playing surface) and often include elements of table, card, role-playing, and miniatures games as well. Many board games feature a co ...
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. A similar board is also used for the
Brazil Brazil ( pt, Brasil; ), officially the Federative Republic of Brazil (Portuguese: ), is the largest country in both South America and Latin America. At and with over 217 million people, Brazil is the world's fifth-largest country by area ...
ian game
Adugo Adugo is a two-player abstract strategy game from the Bororo tribe in the Pantanal region of Brazil. It is a hunting game similar to those in Southeast Asia and the Indian subcontinent. It is especially similar to Komikan, Rimau, Rimau-rimau, ...
, and for the game of Hare and Hounds. The tetrakis square tiling was used for a set of commemorative
postage stamp A postage stamp is a small piece of paper issued by a post office, postal administration, or other authorized vendors to customers who pay postage (the cost involved in moving, insuring, or registering mail), who then affix the stamp to the f ...
s issued by the
United States Postal Service The United States Postal Service (USPS), also known as the Post Office, U.S. Mail, or Postal Service, is an independent agency of the executive branch of the United States federal government responsible for providing postal service in the ...
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. This tiling also forms the basis for a commonly used "pinwheel", "windmill", and "broken dishes" patterns in quilting..


Symmetry

The symmetry type is: *with the coloring: cmm; a
primitive cell In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessari ...
is 8 triangles, a
fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
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 In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
and the kisrhombille tiling. These lines form the axes of symmetry of a reflection group (the
wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
,4 (*442) or p4m), which has the triangles of the tiling as its
fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
s. This group is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to, but not the same as, the group of
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
s 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, ,4symmetry (*442
orbifold notation In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The adva ...
), that can be seen in relation to the
Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
, 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 reflection In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflecti ...
s are given with dashed lines. Subgroups can be expressed as
Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
s, along with fundamental domain diagrams.


See also

*
Tilings of regular polygons Euclidean Plane (mathematics), plane Tessellation, tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler, Kepler in his ''Harmonices Mundi'' (Latin langua ...
* List of uniform tilings *
Percolation threshold The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a ...


Notes


References

* (Chapter 2.1: ''Regular and uniform tilings'', p. 58-65) * * Keith Critchlow, ''Order in Space: A design source book'', 1970, p. 77-76, pattern 8 {{Tessellation Euclidean tilings Isohedral tilings