Tetromino
Encyclopedia
A tetromino is a geometric shape composed of four square
s, connected orthogonally
. This, like dominoes
and pentomino
es, is a particular type of polyomino
. The corresponding polycube
, called a tetracube, is a geometric shape composed of four cube
s connected orthogonally.
A popular use of tetrominos is in the video game Tetris
, where they are often called Tetriminos.
. That is, two free polyominos are the same if there is a combination of translation
s, rotation
s, and reflection
s that turns one into the other.
A free tetromino is a free polyomino made from four squares. There are five free tetrominos (see figure).
The remaining four tetrominos exhibit a phenomenon called chirality
. These four come in two sets of two. Each of the members of these sets is the reflection of the other:
As free tetrominos, 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.
es and unlike pentomino
es.
The proof is that a rectangle covered with a checkerboard pattern will have 10 or 14 each of light and dark squares, while a complete set of free tetrominos (pictured) has 11 light squares and 9 dark squares, and a complete set of one-sided tetrominos has 15 light squares and 13 dark squares.
A bag
including two of each free tetromino, which has a total area of 40 squares, can fit in 4×10 and 5×8 cell rectangles. Likewise, two sets of one-sided tetrominos can be fit to a rectangle in more than one way. The corresponding tetracubes can also fit in 2×4×5 and 2×2×10 boxes.
5×8 rectangle
4×10 rectangle
2×4×5 box
2×2×10 box
tetra- "four" (from Ancient Greek
), and "domino".
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
:
4×4×2 box
8×2×2 box
If chiral pairs (D and S) are considered as identical, remaining 7 pieces can fill 7×2×2 box. (C represents D or S.)
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
s, connected orthogonally
Orthogonality
Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...
. This, like dominoes
Domino (mathematics)
In mathematics, a domino is a polyomino of order 2, that is, a polygon in the plane made of two equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there is only one free domino....
and pentomino
Pentomino
A pentomino is a polyomino composed of five congruent squares, connected along their edges ....
es, is a particular type of polyomino
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 with a connected interior....
. The corresponding polycube
Polycube
thumb|200px|right|The seven free tetracubesthumb|200px|right|A [[Chirality |chiral]] pentacubethumb|200px|right|Puzzle with a unique solution...
, called a tetracube, is a geometric shape composed of four cube
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. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
s connected orthogonally.
A popular use of tetrominos is in the video game Tetris
Tetris
Tetris is a puzzle video game originally designed and programmed by Alexey Pajitnov in the Soviet Union. It was released on June 6, 1984, while he was working for the Dorodnicyn Computing Centre of the Academy of Science of the USSR in Moscow, Russian Soviet Federative Socialist Republic...
, where they are often called Tetriminos.
The tetrominos
Free tetrominos
Polyominos are formed by joining unit squares along their edges. A free polyomino is a polyomino considered up to congruenceCongruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
. That is, two free polyominos are the same if there is a combination of translation
Translation (geometry)
In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
s, rotation
Rotation (mathematics)
In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming...
s, and reflection
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
s that turns one into the other.
A free tetromino is a free polyomino made from four squares. There are five free tetrominos (see figure).
One-sided tetrominos
One-sided tetrominos are tetrominos that may be translated and rotated but not reflected. They are used by, and are overwhelmingly associated with, the game Tetris. There are seven distinct one-sided tetrominos. Of these seven, three have reflectional symmetry, so it does not matter whether they are considered as free tetrominos or one-sided tetrominos. These tetrominos are:- I (also called "stick", "straight", "long", "line"): four blocks in a straight line
- O (also called "square", "package", "block"): four blocks in a 2×2 square.
- T: a row of three blocks with one added below the center.
The remaining four tetrominos exhibit a phenomenon called chirality
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...
. These four come in two sets of two. Each of the members of these sets is the reflection of the other:
- J (also called "inverted L" or "Gamma"): a row of three blocks with one added below the right side.
- L (also called "gun"): a row of three blocks with one added below the left side.
- S (also called "inverted N", "reverse squiggly", "s-zigzag"): two stacked horizontal dominoes with the top one offset to the right
- Z (also called "N", "skew", "snake", "squiggly", "z-zigzag"): two stacked horizontal dominoes with the top one offset to the left.
As free tetrominos, 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 tetrominos
The fixed tetrominos allow only translation, not rotation or reflection. There are 2 distinct fixed I-tetrominos, four J, four L, one O, two S, four T, and two Z, for a total of 19 fixed tetrominos.Tiling the rectangle and filling the box with 2D pieces
Although a complete set of free tetrominos has a total of 20 squares, and a complete set of one-sided tetrominos has 28 squares, it is not possible to pack them into a rectangle, like hexominoHexomino
A hexomino is a polyomino of order 6, that is, a polygon in the plane made of 6 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix hex-. When rotations and reflections are not considered to be distinct shapes, there are 35 different free hexominoes...
es and unlike pentomino
Pentomino
A pentomino is a polyomino composed of five congruent squares, connected along their edges ....
es.
The proof is that a rectangle covered with a checkerboard pattern will have 10 or 14 each of light and dark squares, while a complete set of free tetrominos (pictured) has 11 light squares and 9 dark squares, and a complete set of one-sided tetrominos has 15 light squares and 13 dark squares.
A bag
Multiset
In mathematics, the notion of multiset is a generalization of the notion of set in which members are allowed to appear more than once...
including two of each free tetromino, which has a total area of 40 squares, can fit in 4×10 and 5×8 cell rectangles. Likewise, two sets of one-sided tetrominos can be fit to a rectangle in more than one way. The corresponding tetracubes can also fit in 2×4×5 and 2×2×10 boxes.
5×8 rectangle
4×10 rectangle
2×4×5 box
layer 1 : layer 2
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
2×2×10 box
layer 1 : layer 2
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
Etymology
The name "tetromino" is a combination of the prefixPrefix
A prefix is an affix which is placed before the root of a word. Particularly in the study of languages,a prefix is also called a preformative, because it alters the form of the words to which it is affixed.Examples of prefixes:...
tetra- "four" (from Ancient Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...
), and "domino".
Tetracubes
Each of the five free tetrominos has a corresponding tetracube, which is the tetromino extrudedExtrusion
Extrusion is a process used to create objects of a fixed cross-sectional profile. A material is pushed or drawn through a die of the desired cross-section...
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
Tromino
A tromino is a polyomino of order 3, that is, a polygon in the plane made of three equal-sized squares connected edge-to-edge...
:
- Left screw: unit cube placed on top of anticlockwise side. ChiralChirality (mathematics)In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...
in 3D. - Right screw: unit cube placed on top of clockwise side. Chiral in 3D.
- Branch: unit cube placed on bend. Not chiral in 3D.
Filling the box with 3D pieces
In 3D, these eight tetracubes (suppose each piece consists of 4 cubes, L and J are the same, Z and S are the same) can fit in a 4×4×2 or 8×2×2 box. The following is one of the solutions. D, S and B represent right screw, left screw and branch point, respectively:4×4×2 box
layer 1 : layer 2
S T T T : S Z Z B
S S T B : Z Z B B
O O L D : L L L D
O O D D : I I I I
8×2×2 box
layer 1 : layer 2
D Z Z L O T T T : D L L L O B S S
D D Z Z O B T S : I I I I O B B S
If chiral pairs (D and S) are considered as identical, remaining 7 pieces can fill 7×2×2 box. (C represents D or S.)
layer 1 : layer 2
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
External links
- Vadim GerasimovVadim GerasimovVadim Gerasimov is an engineer at Google. In 1994-2003 Vadim worked and studied at the MIT Media Lab. At age 16 he was one of the original co-developers of the famous video game Tetris: he ported Alexey Pajitnov's original game to the PC architecture and the two later added features to the game....
, "Tetris: the story."; The story of Tetris - The Father of Tetris (Web Archive copy of the page here)
- Open-source tetrominoes game