In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the square tiling, square tessellation or square grid is a
regular tiling of the
Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
consisting of four
squares
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
around every
vertex.
John Horton Conway called it a quadrille.
Structure and properties
The square tiling has a structure consisting of one type of congruent
prototile, the
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
, sharing two vertices with other identical ones. This is an example of monohedral tiling.
Each vertex at the tiling is surrounded by four squares, which denotes in a
vertex configuration
In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...
as
or
. The vertices of a square can be considered as the lattice, so the square tiling can be formed through the
square lattice.
This tiling is commonly familiar with the flooring and game boards.
It is
self-dual, meaning the center of each square connects to another of the adjacent tile, forming square tiling itself.
The square tiling
acts transitively on the ''flags'' of the tiling. In this case, the
flag
A flag is a piece of textile, fabric (most often rectangular) with distinctive colours and design. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design employed, and fla ...
consists of a mutually incident vertex, edge, and tile of the tiling. Simply put, every pair of flags has a symmetry operation mapping the first flag to the second: they are
vertex-transitive (mapping the vertex of a tile to another),
edge-transitive (mapping the edge to another), and
face-transitive (mapping square tile to another). By meeting these three properties, the square tiling is categorized as one of three
regular tilings; the remaining being
triangular tiling and
hexagonal tiling with its prototiles are
equilateral triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
s and
regular hexagons, respectively. The
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
of a square tiling is p4m: there is an
order-4 dihedral group of a tile and an order-2
dihedral group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
around the vertex surrounded by four squares lying on the line of reflection.
The square tiling is alternatively formed by the
assemblage of infinitely many circles arranged vertically and horizontally, wherein their equal diameter at the center of every point contact with four other circles.
Its densest packing is
.
Topologically equivalent tilings
Isohedral tilings have identical faces (
face-transitivity) and
vertex-transitivity. There are eighteen variations, with six identified as triangles that do not connect edge-to-edge, or as quadrilateral with two collinear edges. Symmetry given assumes all faces are the same color.
Related regular complex apeirogons
There are 3
regular complex apeirogons, sharing the vertices of the square tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons pr are constrained by: 1/''p'' + 2/''q'' + 1/''r'' = 1. Edges have ''p'' vertices, and vertex figures are ''r''-gonal.
[Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136.]
See also
*
Fenestrane
*
Langton's ant
*
Polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
Polyominoes have been used in popu ...
References
*
Coxeter, H.S.M. ''
Regular Polytopes'', (3rd edition, 1973), Dover edition, p. 296, Table II: Regular honeycombs
External links
*
*
*
{{Tessellation
Euclidean tilings
Isohedral tilings
Isogonal tilings
Polyhedra
Regular tilings
Self-dual tilings
Regular tessellations