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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.

Contents 1 The seven tetrominoes 2 Tiling the rectangle and filling the box with 2D pieces 3 Etymology 4 Tetracubes 5 Filling the box with 3D pieces 6 See also 7 External links

Ordinarily, polyominoes are discussed in their free forms, which treat rotations and reflections in two dimensions as congruent. But due to the overwhelming association of tetrominoes with Tetris, which uses one-sided tetrominoes, 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.
• L (also called "gun"): 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.
• O (also called "square", "package", "block"): four blocks in a 2×2 square
• S (also called "N", "skew", "snake"): bent trimino with block placed on outside of clockwise side
• 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.
• 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.

The free tetrominoes additionally treat reflection (rotation in the third dimension) as equivalent. There are only 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.

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. 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.

The name "Tetromino" is derived from a combination of the Greek prefix "tetra-," meaning four, and "domino".

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.

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

• Tetris
• Tetrinet
• Domino
• Pentomino
• Polyomino
• polycube
• Soma cube

• Gerasimov, Vadim. "Tetris: the story."; The story of Tetris
• The Father of Tetris