A tetromino is a geometric shape composed of four squares, connected orthogonally. 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.
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 tetrominoes (see figure).
One-sided tetrominoes are tetrominoes 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 tetrominoes. Of these seven, three have reflectional symmetry, so it does not matter whether they are considered as free tetrominoes or one-sided tetrominoes. These tetrominoes are:
- I (also a "Straight Polyomino"): four blocks in a straight line.
- O (also a "Square Polyomino"): four blocks in a 2×2 square.
- T (also a "T-Polyomino"): a row of three blocks with one added below the center.
- J: a row of three blocks with one added below the right side.
- L: a row of three blocks with one added below the left side.
The "Skew Polyominos":
- S: two stacked horizontal dominoes with the top one offset to the right.
- Z: two stacked horizontal dominoes with the top one offset to the left.
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.
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 the rectangle and filling the box with 2D pieces
Although a complete set of free tetrominoes has a total of 20 squares, and a complete set of one-sided tetrominoes 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 tetrominoes (pictured) has 11 light squares and 9 dark squares, and a complete set of one-sided tetrominoes 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 tetrominoes 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.
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
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
Each of the five free tetrominoes 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:
- Right screw: unit cube placed on top of clockwise side. Chiral in 3D.(Letter D in the diagrams below)
- Left screw: unit cube placed on top of anticlockwise side. Chiral in 3D. (Letter S in the diagrams below)
- Branch: unit cube placed on bend. Not chiral in 3D. (Letter B in the diagrams below)
Filling the box with 3D pieces
In 3D, these eight tetracubes (suppose each piece consists of four 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:
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
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, the remaining seven pieces can fill a 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
- Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 0-691-02444-8.
- Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics 36: 191–203. doi:10.1016/0012-365X(81)90237-5.
- "About Tetris", Tetris.com. Retrieved 2014-04-19.
- Weisstein, Eric W. "Straight Polyomino." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/StraightPolyomino.html
- Weisstein, Eric W. "Square Polyomino." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SquarePolyomino.html
- Weisstein, Eric W. "T-Polyomino." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/T-Polyomino.html
- Weisstein, Eric W. "L-Polyomino." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/L-Polyomino.html
- Weisstein, Eric W. "Skew Polyomino." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SkewPolyomino.html
- Vadim Gerasimov, "Tetris: the story."; The story of Tetris
- The Father of Tetris (Web Archive copy of the page here)
- Open-source tetrominoes game