Tetromino
From Wikipedia, the free encyclopedia
A tetromino, also spelled tetramino or tetrimino, is a geometric shape composed of four squares, connected orthogonally. This is a particular type of polyomino, like dominoes and pentominoes are. The corresponding polycube, called a tetracube, is a geometric shape composed of four cubes connected orthogonally.
A popular use of tetrominoes is in the video game Tetris. However, the spelling of the word used by The Tetris Company differs slightly by replacing the first 'o' with an 'i' to make the word Tetrimino.
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[edit] The seven tetrominoes
Counting rotations in two dimensions as equivalent, there are seven distinct one-sided shapes:
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I (also called "stick", "straight", "long"): four blocks in a straight line
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T: a row of three blocks with one added below the center. A common Tetris move with the T piece is to spin it in place to fill a line.
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J (also called "inverted L" or "Gamma"): a row of three blocks with one added below the right side.
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L: a row of three blocks with one added below the left side. This piece is a reflection of J but cannot be rotated into J in two dimensions; this is an example of chirality. However, in three dimensions, this piece is identical to J.
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O (also called "square", "package", "block"): four blocks in a 2×2 square
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S (also called "N"): bent trimino with block placed on outside of clockwise side
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Z (also called "inverted N"): bent trimino with block added on outside of anticlockwise side. The same symmetry properties as with L and J apply with S and Z.
[edit] Tiling the rectangle and filling the box with 2D pieces
Although a complete set of 2D tetrominoes has a total of 20 (or 28, when mirror images count) squares, it is not possible to pack them into a rectangle, unlike pentominoes. A parity argument. which is essentially same as hexomino, can prove this.
Two sets of pieces (two of each piece), which have total area of 40, can fit in 4×10 and 5×8 rectangles. The corresponding tetracubes can also fit in 2×4×5 and 2×2×10 box.
5×8 rectangle
L Z I I I I O O L Z Z T T T O O L L Z t T l l l o o t t t z z l o o i i i i z z
4×10 rectangle
L L L Z Z I I I I i L T T T Z Z l l l i o o T z z t l O O i o o z z t t t O O i
2×4×5 box
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
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
As a puzzle, these are relatively easy.
[edit] Tetracubes
Each tetromino has a corresponding tetracube, which is the tetromino extruded by one unit. Three more tetracubes are possible, all created by placing a unit cube on the bent tricube:
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Left screw: unit cube placed on top of anticlockwise side. Chiral in 3D.
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Right screw: unit cube placed on top of clockwise side. Chiral in 3D.
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Branch: unit cube placed on bend. Not chiral in 3D.
However, going to three dimensions means that rotation is allowed in three dimensions. Thus, the two L-shaped and S-shaped pieces are now equivalent.
[edit] 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.)
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
[edit] See also
[edit] External links
- Gerasimov, Vadim. "Tetris: the story."; The story of Tetris
- The Father of Tetrisfr:Tétromino
