A tetromino is a geometric shape composed of four squares, connected orthogonally.[1][2] This, like dominoes and pentominoes, is a particular type of polyomino. The corresponding polycube, called a tetracube, is a geometric shape composed of four cubes connected orthogonally.
A popular use of tetrominos is in the video game Tetris, where they are often called Tetriminos.[3]
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Polyominos are formed by joining unit squares along their edges. A free polyomino is a polyomino considered up to congruence. That is, two free polyominos are the same if there is a combination of translations, rotations, and reflections 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 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:
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.
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.
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 hexominoes and unlike pentominoes. 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
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
The name "tetromino" is a combination of the prefix tetra- "four" (from Ancient Greek τετρα-), and "domino".
Each of the five free tetrominos has a corresponding tetracube, which is the tetromino extruded 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:
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
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