A tetromino is a geometric shape composed of four

The story of Tetris

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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 orthogonally (i.e. at the edges and not the corners). Tetrominoes, like 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 ...

and pentomino
Derived from the Greek word for ' 5', and "domino", a pentomino (or 5-omino) is a polyomino of order 5, that is, a polygon in the plane made of 5 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to ...

es, are a particular type of 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 ...

. The corresponding polycube
upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total
A puzzle involving arranging nine L tricubes into a 3×3 cube
A polycube is a solid figure formed by j ...

, called a tetracube, is a geometric shape composed of four cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...

s connected orthogonally.
A popular use of tetrominoes is in the video game ''Tetris
''Tetris'' (russian: link=no, Тетрис) is a puzzle video game created by Soviet software engineer Alexey Pajitnov in 1984. It has been published by several companies for multiple platforms, most prominently during a dispute over the approp ...

'' created by the Soviet game designer Alexey Pajitnov
Alexey Leonidovich Pajitnov. (born 16 April 1955) is a Russian-born American computer engineer and video game designer. He is best-known for designing and developing ''Tetris'' in 1984 while working at the Dorodnitsyn Computing Centre under the A ...

, which refers to them as tetriminos. The tetrominoes used in the game are specifically the one-sided tetrominoes.
The tetrominoes

Free tetrominoes

Polyominos are formed by joining unit squares along their edges. Afree 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 ...

is a polyomino considered up to congruence. That is, two free polyominos are the same if there is a combination of translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...

s, 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 that turns one into the other. A free tetromino is a free polyomino made from four squares. There are five free tetrominoes.
The free tetrominoes have the following symmetry:
* Straight: vertical and horizontal reflection symmetry, and two points of rotational symmetry
* Square: vertical and horizontal reflection symmetry, and four points of rotational symmetry
* T: vertical reflection symmetry only
* L: no symmetry
* S and Z: two points of rotational symmetry only
One-sided tetrominoes

One-sided tetrominoes are tetrominoes that may be translated and rotated but not reflected. They are used by, and are overwhelmingly associated with, ''Tetris
''Tetris'' (russian: link=no, Тетрис) is a puzzle video game created by Soviet software engineer Alexey Pajitnov in 1984. It has been published by several companies for multiple platforms, most prominently during a dispute over the approp ...

''. There are seven distinct one-sided tetrominoes. These tetrominoes are named by the letter of the alphabet they most closely resemble. The "I", "O", and "T" tetrominoes have reflectional symmetry, so it does not matter whether they are considered as free tetrominoes or one-sided tetrominoes. The remaining four tetrominoes, "J", "L", "S", and "Z", exhibit a phenomenon called chirality
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from ...

. J and L are reflections of each other, and S and Z are reflections of each other.
As free tetrominoes, J is equivalent to L, and S is equivalent to Z. But in two dimensions and without reflections, it is not possible to transform J into L or S into Z.
Fixed tetrominoes

The fixed tetrominoes allow only translation, not rotation or reflection. There are two distinct fixed I-tetrominoes, four J, four L, one O, two S, four T, and two Z, for a total of 19 fixed tetrominoes:Tiling a rectangle

Filling a rectangle with one set of tetrominoes

A single set of free tetrominoes or one-sided tetrominoes cannot fit in a rectangle. This can be shown with a proof similar to the mutilated chessboard argument. A 5×4 rectangle with a checkerboard pattern has 20 squares, containing 10 light squares and 10 dark squares, but a complete set of free tetrominoes has either 11 dark squares and 9 light squares, or 11 light squares and 9 dark squares. This is due to the T tetromino having either 3 dark squares and one light square, or 3 light squares and one dark square, while all other tetrominoes each have 2 dark squares and 2 light squares. Similarly, a 7×4 rectangle has 28 squares, containing 14 squares of each shade, but the set of one-sided tetrominoes has either 15 dark squares and 13 light squares, or 15 light squares and 13 dark squares. By extension, any odd number of sets for either type cannot fit in a rectangle. Additionally, the 19 fixed tetrominoes cannot fit in a 4×19 rectangle. This was discovered by exhausting all possibilities in a computer search.Filling a modified rectangle with one set of tetrominoes

However, all three sets of tetrominoes can fit rectangles with holes:- All 5 free tetrominoes fit a 7×3 rectangle with a hole.
- All 7 one-sided tetrominoes fit a 6×5 rectangle with two holes of the same "checkerboard color".
- All 19 fixed tetrominoes fit a 11×7 rectangle with a hole.

Filling a rectangle with two sets of tetrominoes

Two sets of free or one-sided tetrominoes can fit into a rectangle in different ways, as shown below:Etymology

The name "tetromino" is a combination of theprefix
A prefix is an affix which is placed before the Word stem, stem of a word. Adding it to the beginning of one word changes it into another word. For example, when the prefix ''un-'' is added to the word ''happy'', it creates the word ''unhappy'' ...

''tetra-'' 'four' (from Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...

), and "domino
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 ...

". The name was introduced by Solomon W. Golomb in 1953 along with other nomenclature related to polyominos.
Filling a box with tetracubes

Each of the five free tetrominoes has a corresponding tetracube, which is the tetrominoextruded
Extrusion is a process used to create objects of a fixed cross-sectional profile by pushing material through a die of the desired cross-section. Its two main advantages over other manufacturing processes are its ability to create very complex c ...

by one unit.
J and L are the same tetracube, as are S and Z, because one may be rotated around an axis parallel to the tetromino's plane to form the other. Three more tetracubes are possible, all created by placing a unit cube on the bent tricube:
The tetracubes can be packed into two-layer 3D boxes in several different ways, based on the dimensions of the box and criteria for inclusion. They are shown in both a pictorial diagram and a text diagram. For boxes using two sets of the same pieces, the pictorial diagram depicts each set as a lighter or darker shade of the same color. The text diagram depicts each set as having a capital or lower-case letter. In the text diagram, the top layer is on the left, and the bottom layer is on the right.
1.) 2×4×5 box filled with two sets of free tetrominoes: Z Z T t I l T T T i L Z Z t I l l l t i L z z t I o o z z i L L O O I o o O O i 2.) 2×2×10 box filled with two sets of free tetrominoes: L L L z z Z Z T O O o o z z Z Z T T T l L I I I I t t t O O o o i i i i t l l l 3.) 2×4×4 box filled with one set of all tetrominoes: F T T T F Z Z B F F T B Z Z B B O O L D L L L D O O D D I I I I 4.) 2×2×8 box filled with one set of all tetrominoes: D Z Z L O T T T D L L L O B F F D D Z Z O B T F I I I I O B B F 5.) 2×2×7 box filled with tetrominoes, with mirror-image pieces removed: L L L Z Z B B L C O O Z Z B C I I I I T B C C O O T T T

See also

*Soma cube
The Soma cube is a solid dissection puzzle invented by Danish polymath Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg.
Seven pieces made out of unit cubes must be assembled into a 3×3×3 cube. The pie ...

Previous and next orders

*Tromino
A tromino or triomino is a polyomino of order 3, that is, a polygon in the plane made of three equal-sized squares connected edge-to-edge.
Symmetry and enumeration
When rotations and reflections are not considered to be distinct shapes, there a ...

*Pentomino
Derived from the Greek word for ' 5', and "domino", a pentomino (or 5-omino) is a polyomino of order 5, that is, a polygon in the plane made of 5 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to ...

References

External links

*Vadim Gerasimov Vadim Viktorovich Gerasimov (russian: Вадим Викторович Герасимов, born 15 June 1969) is an engineer at Google. From 1994 to 2003, Vadim worked and studied at the MIT Media Lab. Vadim earned a BS/MS in applied mathematics from ...

, "Tetris: the story."The story of Tetris

The Father of Tetris

Web Archive copy of the page here

{{Polyforms Polyforms Tetris Mathematical games