World Library  
Flag as Inappropriate
Email this Article

Tetromino

Article Id: WHEBN0000030411
Reproduction Date:

Title: Tetromino  
Author: World Heritage Encyclopedia
Language: English
Subject: Tetris, Tetris Party, Magical Tetris Challenge, L game, Up to
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

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:

  • I (also a "Straight Polyomino"[4]): four blocks in a straight line.
  • O (also a "Square Polyomino"[5]): four blocks in a 2×2 square.
  • T (also a "T-Polyomino"[6]): a row of three blocks with one added below the center.

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]

  • J: a row of three blocks with one added below the right side.
  • L: a row of three blocks with one added below the left side.

The "Skew Polyominos":[8]

  • S: two stacked horizontal dominoes with the top one offset to the right.
  • Z: two stacked horizontal dominoes with the top one offset to the left.

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:

  • Right screw: unit cube placed on top of clockwise side. Chiral in 3D.(Letter D in the diagrams below)
  • Left screw: unit cube placed on top of anticlockwise side. Chiral in 3D. (Letter S in the diagrams below)
  • Branch: unit cube placed on bend. Not chiral in 3D. (Letter B in the diagrams below)

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. ^  
  2. ^ Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics 36: 191–203.  
  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

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
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.