Tetromino

The 5 free tetrominoes

A tetromino is a geometric shape composed of four squares, connected orthogonally.[1][2] 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.

A popular use of tetrominoes is in the video game Tetris, where they have been called Tetriminos (spelled with an "i" as opposed to the "o" in "tetromino") since 2001.[3]

The tetrominoes

The five free tetrominoes, top to bottom I, O, Z, T, L, marked with light and dark squares. As there are a total of 11 light squares and 9 dark squares, it is not possible to pack them into a rectangle (such as ones with 4×5 or 2×10 squares) as any such rectangle has the same number of light and dark squares.

Free tetrominoes

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

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:

The remaining four tetrominoes exhibit a phenomenon called chirality. These four come in two sets of two. Each of the members of these sets is the reflection of the other. The "L-Polyominos":[7]

The "Skew Polyominos":[8]

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.

Fixed tetrominoes

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.

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

Etymology

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

Tetracubes

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:

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:

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, 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

See also

References

  1. ↑ Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 0-691-02444-8.
  2. ↑ Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics 36: 191–203. doi:10.1016/0012-365X(81)90237-5.
  3. ↑ "About Tetris", Tetris.com. Retrieved 2014-04-19.
  4. ↑ Weisstein, Eric W. "Straight Polyomino." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/StraightPolyomino.html
  5. ↑ Weisstein, Eric W. "Square Polyomino." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SquarePolyomino.html
  6. ↑ Weisstein, Eric W. "T-Polyomino." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/T-Polyomino.html
  7. ↑ Weisstein, Eric W. "L-Polyomino." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/L-Polyomino.html
  8. ↑ Weisstein, Eric W. "Skew Polyomino." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SkewPolyomino.html

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