- Tetromino
A tetromino, also spelled tetramino or tetrimino, is a geometric shape composed of four squares, connected orthogonally. This is a particular type of
polyomino , likedomino es andpentomino es are. The correspondingpolycube , called a tetracube, is a geometric shape composed of fourcube s 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.The seven tetrominoes
Ordinarily, polyominoes are discussed in their "free" forms, which treat rotations and
reflection s in twodimension s as congruent. In that case, there are five unique tetrominoes. However, due to the overwhelming association of tetrominoes with "Tetris ", which uses one-sided tetrominoes (making reflections distinct but all rotations congruent), people recognize seven distinct tetrominoes:
* I (also called "stick", "straight", "long"): four blocks in a straight line
* J (also called "inverted L" or "Gamma"): a row of three blocks with one added below the right side.
** s, this piece is identical to J.
* O (also called "square", "package", "block"): four blocks in a 2×2 square.
* S (also called "inverted N"): two stacked horizontal dominoes with the top one offset to the right
** Z (also called "N", "skew", "snake"): two stacked horizontal dominoes with the top one offset to the left. The same symmetry properties as with L and J apply with S and Z.
* 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.The "free tetrominoes" additionally treat
reflection (rotation in the third dimension) as equivalent.This eliminates J and Z, leaving five free tetrominoes: I, L, O, S, and T.The "fixed tetrominoes" do not allow rotation or reflection. There are 2 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
hexomino es and unlikepentomino 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 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. 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 boxZ Z T t I l T T T iL Z Z t I l l l t iL z z t I o o z z iL L O O I o o O O iL L L z z Z Z T O O o o z z Z Z T T T lL I I I I t t t O O o o i i i i t l l lAs a puzzle, these are relatively easy.
Etymology
The name "Tetromino" is derived from a combination of the Greek prefix "tetra-," meaning four, and "domino".
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:
* Left screw: unit cube placed on top of anticlockwise side. Chiral 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.However, going to three dimensions means that rotation is allowed in three dimensions. Thus, the two L-shaped pieces are now equivalent, as are the two S-shaped pieces.
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 2S T T T : S Z Z BS S T B : Z Z B BO O L D : L L L DO O D D : I I I I
8×2×2 box
layer 1 : layer 2D Z Z L O T T T : D L L L O B S SD 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 BC I I I I T B : C C O O T T Tee also
*
Tetris
*Domino
*Pentomino
*Polyomino
*Polycube
*Soma cube References
External links
*Gerasimov, Vadim. "Tetris: the story."; [http://vadim.oversigma.com/Tetris.htm The story of Tetris]
* [http://www.tetris-today.com/story/original-tetris0.shtml The Father of Tetris] ( [http://web.archive.org/web/20061202094148/http://www.tetris-today.com/story/original-tetris0.shtml Web Archive copy of the page here] )
* [http://code.google.com/p/tetrominoes/ Open-source Tetrominoes game]
Wikimedia Foundation. 2010.