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A tetromino, also called a tetramino or tetrimino, is a geometric shape composed of four squares, connected orthogonally.^[1]^[2] 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 word used by The Tetris Company differs slightly by replacing the first 'o' with an 'i' to make 'Tetrimino'.

The seven tetriminoes

Ordinarily, polyominoes are discussed in their free forms, which treat rotations and reflections in two dimensions 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", "line"): 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"^[3]): 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",^[3] "package", "block"): four blocks in a 2×2 square.
S (also called "inverted N", "reverse squiggly", "s-zigzag"): two stacked horizontal dominoes with the top one offset to the right
- Z (also called "N", "skew", "snake"^[3], "squiggly", "z-zigzag"): 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.

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 (also called N or Z), 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

The five free tetrominoes, top to bottom I, O, Z, T, L, marked with light and dark squares.

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 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

As a puzzle, these are relatively easy.

Etymology

The name "tetromino" is a combination of the prefix tetra- "four" (from Ancient Greek τετρα-), 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 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

References

^ Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 0–691–02444–8. {{cite book}}: Check |isbn= value: invalid character (help)
^ Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics. 36: 191–203. doi:10.1016/0012-365X(81)90237-5.
^ ^a ^b ^c Demaine, Hohenberger, and Liben-Nowell. Tetris Is Hard, Even to Approximate.

External links

Vadim Gerasimov, "Tetris: the story."; The story of Tetris
The Father of Tetris (Web Archive copy of the page here)
Open-source Tetrominoes game

[1] Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 0–691–02444–8. {{cite book}}: Check |isbn= value: invalid character (help)

[2] Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics. 36: 191–203. doi:10.1016/0012-365X(81)90237-5.

[tetris_is_hard-3] Demaine, Hohenberger, and Liben-Nowell. Tetris Is Hard, Even to Approximate.

[1]

[2]

[3]

@@ Line 10: / Line 10: @@
 ** [[File:Tetromino L.svg|56px]] L (also called "gun"<ref name="tetris_is_hard">Demaine, Hohenberger, and Liben-Nowell. [http://www.cs.carleton.edu/faculty/dlibenno/papers/tetris/tetris-short.pdf Tetris Is Hard, Even to Approximate].</ref>): 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 (mathematics)|chirality]]. However, in three [[dimension]]s, this piece is identical to J.
 * [[File:Tetromino O.svg|39px]] O (also called "square",<ref name="tetris_is_hard" /> "package", "block"): four blocks in a 2×2 square.
-* [[File:Tetromino S.svg|56px]] S (also called "inverted N", "reverse squiggly"): two stacked horizontal dominoes with the top one offset to the right
+* [[File:Tetromino S.svg|56px]] S (also called "inverted N", "reverse squiggly", "s-zigzag"): two stacked horizontal dominoes with the top one offset to the right
-** [[File:Tetromino Z.svg|53px]] Z (also called "N", "skew", "snake"<ref name="tetris_is_hard" />, "squiggly"): 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.
+** [[File:Tetromino Z.svg|53px]] Z (also called "N", "skew", "snake"<ref name="tetris_is_hard" />, "squiggly", "z-zigzag"): 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.
 * [[File:Tetromino T.svg|54px]] T: a row of three blocks with one added below the center.

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