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 of ...
. 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 an ...
s from the center point, forming an infinite
arrangement of lines
In music, an arrangement is a musical adaptation of an existing composition. Differences from the original composition may include reharmonization, melodic paraphrasing, orchestration, or formal development. Arranging differs from orchestr ...
. 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
Kis or KIS may refer to:
Places
* Kiş, Khojavend, Azerbaijan
* Kiş, Shaki, Azerbaijan
* Kish (Sumer) (Sumerian: Kiš), an ancient city in Sumer
* Kis, Babol Kenar, a village in Mazandaran Province, Iran
* Kis, Bandpey-ye Gharbi, a village in ...
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
The national flag of the United Kingdom is the Union Jack, also known as the Union Flag.
The design of the Union Jack dates back to the Act of Union 1801 which united the Kingdom of Great Britain and the Kingdom of Ireland (previously in per ...
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
In geometry, every polyhedron is associated with a second dual structure, where the Vertex (geometry), vertices of one correspond to the Face (geometry), faces of the other, and the edges between pairs of vertices of one correspond to the edges b ...
of the
truncated square tiling
In geometry, the truncated square tiling is a semiregular tiling, semiregular tiling by regular polygons of the Euclidean plane with one square (geometry), square and two octagons on each vertex (geometry), vertex. This is the only edge-to-edge ti ...
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 comp ...
Fanorona
Fanorona () is a strategy board game for two players. The game is indigenous to Madagascar.
Rules
Fanorona has three standard versions: Fanoron-Telo, Fanoron-Dimy, and Fanoron-Tsivy. The difference between these variants is the size of board play ...
. 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 fa ...
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 U ...
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
Quilting is the term given to the process of joining a minimum of three layers of fabric together either through stitching manually using a needle and thread, or mechanically with a sewing machine or specialised longarm quilting system. A ...
.
[.]
Symmetry
The symmetry type is:
*with the coloring: cmm; a
primitive cell 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 equilate ...
and the
kisrhombille tiling
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''.
Names
Uniform colorings
The ...
.
These lines form the
axes of symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D ther ...
of a
reflection group In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent cop ...
(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 transformatio ...
,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 is ...
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 automorphisms ...
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 advanta ...
), 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 to ...
, 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 reflection ...
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 to ...
s, along with fundamental domain diagrams.
See also
*
Tilings of regular polygons
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his ''Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619).
Notation of Eucli ...
*
List of uniform tilings
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dual ...
*
Percolation threshold
The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in Randomness, random systems. Below the threshold a giant connected component (graph theory), connected component ...
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