In mathematics, a domino is a
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 pop ...
of order 2, that is, a
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
in the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
made of two equal-sized
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
s connected edge-to-edge. When
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s and
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
s are not considered to be distinct shapes, there is only one
''free'' domino.
Since it has
reflection 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 ...
, it is also the only ''one-sided'' domino (with reflections considered distinct). When rotations are also considered distinct, there are two ''fixed'' dominoes: The second one can be created by rotating the one above by 90°.
In a wider sense, the term ''domino'' is sometimes understood to mean a
tile
Tiles are usually thin, square or rectangular coverings manufactured from hard-wearing material such as ceramic, stone, metal, baked clay, or even glass. They are generally fixed in place in an array to cover roofs, floors, walls, edges, or o ...
of any shape.
Packing and tiling
Dominos can tile the plane in a countably infinite number of ways. The number of tilings of a 2×''n'' rectangle with dominoes is
, the ''n''th
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
.
Domino tilings figure in several celebrated problems, including the
Aztec diamond
In combinatorial mathematics, an Aztec diamond of order ''n'' consists of all squares of a square lattice whose centers (''x'',''y'') satisfy , ''x'', + , ''y'', ≤ ''n''. Here ''n'' is a fixed integer, and the square lattice consists of unit s ...
problem in which large diamond-shaped regions have a number of tilings equal to a
power of two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negative ...
, with most tilings appearing random within a central circular region and having a more regular structure outside of this "arctic circle", and the
mutilated chessboard problem
The mutilated chessboard problem is a tiling puzzle posed by Max Black in 1946 that asks:
Suppose a standard 8×8 chessboard (or checkerboard) has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoe ...
, in which removing two opposite corners from a
chessboard
A chessboard is a used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During play, the bo ...
makes it impossible to tile with dominoes.
[.]
See also
*
Dominoes
Dominoes is a family of tile-based games played with gaming pieces, commonly known as dominoes. Each domino is a rectangular tile, usually with a line dividing its face into two square ''ends''. Each end is marked with a number of spots (also ca ...
, a set of domino-shaped gaming pieces
*
Tatami
A is a type of mat used as a flooring material in traditional Japanese-style rooms. Tatamis are made in standard sizes, twice as long as wide, about 0.9 m by 1.8 m depending on the region. In martial arts, tatami are the floor used for traini ...
, Japanese domino-shaped floor mats
References
{{polyforms
Polyforms