Editing Tetromino

{{Short description|Four squares connected edge-to-edge}}
[[File:All 5 free tetrominoes.svg|thumb|200px|The 5 free tetrominoes]]
[[File:Typical Tetris Game.svg|thumb|A snapshot from a typical game of ''[[Tetris]]''.]]
A '''tetromino''' is a geometric shape composed of four [[square (geometry)|square]]s, connected [[orthogonality|orthogonally]] (i.e. at the edges and not the corners).<ref name=GolombPolyominoes>{{cite book |last=Golomb |first=Solomon W. |author-link=Solomon W. Golomb |title=Polyominoes |title-link= Polyominoes: Puzzles, Patterns, Problems, and Packings |year=1994 |publisher=Princeton University Press |location=Princeton, New Jersey |isbn=0-691-02444-8 |edition=2nd}}</ref><ref>{{cite journal |last=Redelmeier |first=D. Hugh |year=1981 |title=Counting polyominoes: yet another attack |journal=Discrete Mathematics |volume=36 |issue=2 |pages=191–203 |doi=10.1016/0012-365X(81)90237-5|doi-access=free}}</ref> Tetrominoes, like [[domino (mathematics)|dominoes]] and [[pentomino]]es, are a particular type of [[polyomino]]. The corresponding [[polycube]], called a '''tetracube''', is a geometric shape composed of four [[cube]]s connected orthogonally.

A popular use of tetrominoes is in the video game ''[[Tetris]]'' created by the [[Soviet]] game designer [[Alexey Pajitnov]], which refers to them as '''tetriminos'''.<ref>[http://tetris.com/about-tetris/ "About Tetris"], Tetris.com. Retrieved 2014-04-19.</ref> The tetrominoes used in the game are specifically the one-sided tetrominoes.

==Types of tetrominoes==
===Free tetrominoes===
Polyominos are formed by joining unit squares along their edges. A [[free polyomino]] is a polyomino considered up to [[congruence (geometry)|congruence]]. That is, two free polyominos are the same if there is a combination of [[translation (geometry)|translation]]s, [[rotation (mathematics)|rotation]]s, and [[reflection (mathematics)|reflection]]s that turns one into the other. A free tetromino is a free polyomino made from four squares. There are five free tetrominoes.

The free tetrominoes have the following symmetry:
* Straight: vertical and horizontal reflection symmetry, and two-fold rotational symmetry
* Square: vertical, horizontal, and diagonal reflection symmetry, and four-fold rotational symmetry
* T: vertical reflection symmetry only
* L: no symmetry
* S: two-fold rotational symmetry only

{{multiple image 
| align=left
| image_style = border:none;
| image_gap = 25
| image1 = Tetromino I Horizontal.svg
| width1=100
| caption1 = "straight tetromino"
| image2=Tetromino O Single.svg
| width2=50
| caption2= "square tetromino"
| image3=Tetromino T Up.svg
| width3=75
| caption3= "T-tetromino"
| image4=Tetromino L Up.svg
| width4=50
| caption4= "L-tetromino"
| image5=Tetromino S Horizontal.svg
| width5=75
| caption5= "skew tetromino"
}}
{{-}}

=== 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, ''[[Tetris]]''. There are seven distinct one-sided tetrominoes. These tetrominoes are named by the letter of the alphabet they most closely resemble. The "I", "O", and "T" tetrominoes have reflectional symmetry, so it does not matter whether they are considered as free tetrominoes or one-sided tetrominoes. The remaining four tetrominoes, "J", "L", "S", and "Z", exhibit a phenomenon called [[chirality (mathematics)|chirality]]. J and L are reflections of each other, and S and Z are reflections of each other.

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.

{{multiple image 
| align=left
| image_style = border:none;
| image_gap = 25
| image1 = Tetromino I Horizontal.svg
| width1=100
| caption1 = I
| image2=Tetromino O Single.svg
| width2=50
| caption2=O
| image3=Tetromino T Up.svg
| width3=75
| caption3=T
| image4=Tetromino J Up.svg
| width4=50
| caption4=J
| image5=Tetromino L Up.svg
| width5=50
| caption5=L
| image6=Tetromino S Horizontal.svg
| width6=75
| caption6=S
| image7=Tetromino Z Horizontal.svg
| width7=75
| caption7=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.

{{multiple image 
| align=left
| perrow=10
| image_style = border:none;
| image_gap = 10
| image1 = Tetromino J Up.svg
| width1=50
| image2= Tetromino J Left.svg
| width2=75
| image3= Tetromino J Down.svg
| width3=50
| image4= Tetromino J Right.svg
| width4=75
| image5= Tetromino S Horizontal.svg
| width5=75
| image6= Tetromino S Vertical.svg
| width6=50
| image7= Tetromino T Up.svg
| width7=75
| image8= Tetromino T Left.svg
| width8=50
| image9= Tetromino I Horizontal.svg
| width9=100
| image10= Tetromino O Single.svg
| width10=50
| image11 = Tetromino L Down.svg
| width11=50
| image12= Tetromino L Left.svg
| width12=75
| image13= Tetromino L Up.svg
| width13=50
| image14= Tetromino L Right.svg
| width14=75
| image15= Tetromino Z Horizontal.svg
| width15=75
| image16= Tetromino Z Vertical.svg
| width16=50
| image17= Tetromino T Down.svg
| width17=75
| image18= Tetromino T Right.svg
| width18=50
| image19= Tetromino I Vertical.svg
| width19=25
}}
{{-}}

==Tiling a rectangle==
===Filling a rectangle with one set of tetrominoes===
A single set of free tetrominoes or one-sided tetrominoes cannot fit in a rectangle. This can be shown with a proof similar to the [[Mutilated chessboard problem|mutilated chessboard argument]]. A 5×4 rectangle with a checkerboard pattern has 20 squares, containing 10 light squares and 10 dark squares, but a complete set of free tetrominoes has either 11 dark squares and 9 light squares, or 11 light squares and 9 dark squares. This is due to the T tetromino having either 3 dark squares and one light square, or 3 light squares and one dark square, while all other tetrominoes each have 2 dark squares and 2 light squares. Similarly, a 7×4 rectangle has 28 squares, containing 14 squares of each shade, but the set of one-sided tetrominoes has either 15 dark squares and 13 light squares, or 15 light squares and 13 dark squares. By extension, any odd number of sets for either type cannot fit in a rectangle. Additionally, the 19 fixed tetrominoes cannot fit in a 4×19 rectangle. This was discovered by exhausting all possibilities in a computer search.

{{multiple image 
| align = left
| image_style = border:none;
| image_gap = 25
| image1 = Tetrominoes with Checkerboard Squares.svg
| width1=690
| caption1 = The free tetrominoes (left side of line) have 11 dark squares and 9 light squares.<br>The one-sided tetrominoes (all 7 shown above) have 15 dark squares and 13 light squares.
| image2 = Checkerboard Rectangle 7x4.svg
| width2=240
| caption2 = A 5×4 board has 10 squares each color.<br>A 7×4 board has 14 squares each color.
}}
{{-}}

=== Parity ===
A further consequence of the T tetromino having 3 squares of one colour and 1 square of the other is that any rectangle containing an even number of squares must contain an even number of T tetrominoes. Conversely, any rectangles containing an odd number of squares must contain an odd number of T tetrominoes.

===Filling a modified rectangle with one set of tetrominoes===
All three sets of tetrominoes can fit rectangles with holes: 
<ul>
<li>All 5 free tetrominoes fit a 7×3 rectangle with a hole.</li>
<li>All 7 one-sided tetrominoes fit a 6×5 rectangle with two holes of the same "checkerboard color".</li>
<li>All 19 fixed tetrominoes fit a 11×7 rectangle with a hole.</li>
</ul>

{{multiple image 
| align = left
| image_style = border:none;
| image_gap = 25
| image1 = Tetromino Tiling - 7x3.svg
| width1 = 116
| caption1 = Free tetrominoes in a rectangle with one hole
| image2 = Tetromino Tiling - 6x5.svg
| width2 = 100
| caption2 = One-sided tetrominoes in a rectangle with two holes
| image3 = Tetromino Tiling 11x7.svg
| width3 = 175
| caption3 = Fixed tetrominoes in rectangle with one hole
}}
{{-}}

===Filling a rectangle with two sets of tetrominoes===
Two sets of free or one-sided tetrominoes can fit into a rectangle in different ways, as shown below:

{{multiple image 
| align=left
| image_style = border:none;
| image_gap = 25
| image1 = Tetromino Tiling 8x5.svg
| width1=114
| caption1 = Two sets of free tetrominoes in a 5×8 rectangle
| image2 = Tetromino Tiling 10x4.svg
| width2=150
| caption2 = Two sets of free tetrominoes in a 4×10 rectangle
| image3 = Tetromino Tiling 8x7.svg
| width3=120
| caption3 = Two sets of one-sided tetrominoes in a 8×7 rectangle
| image4 = Tetromino Tiling 14x4.svg
| width4=200
| caption4 = Two sets of one-sided tetrominoes in a 14×4 rectangle
}}
{{-}}

==Etymology==
The name "tetromino" is a combination of the [[prefix]] ''tetra-'' 'four' (from [[Ancient Greek]] {{lang|grc|τετρα-}}), and "[[domino]]". The name was introduced by [[Solomon W. Golomb]] in 1953 along with other nomenclature related to polyominos.<ref>{{cite web |last1=Darling |first1=David |title=Polyomino |url=https://www.daviddarling.info/encyclopedia/P/polyomino.html |website=daviddarling.info |access-date=May 23, 2020}}</ref><ref name=GolombPolyominoes/>

==Filling a box with tetracubes==
Each of the five free tetrominoes has a corresponding tetracube, which is the tetromino [[extrusion|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 [[Tromino|tricube]]:

{{multiple image 
| align=left
| image_style = border:none;
| image_gap = 25
| image1 = Tetromino I.svg
| width1=71
| caption1 = I <br>"straight tetracube"
| image2=Tetromino O.svg
| width2=39
| caption2=O <br>"square tetracube"
| image3=Tetromino T.svg
| width3=54
| caption3=T <br>"T-tetracube"
| image4=Tetromino L.svg
| width4=56
| caption4=L <br>"L-tetracube"
| image5=Tetromino J.svg
| width5=56
| caption5=J is the same as L in 3D
| image6=Tetromino S.svg
| width6=56
| caption6=S <br>"skew tetracube"
| image7=Tetromino Z.svg
| width7=56
| caption7=Z is the same as S in 3D
| image8=Tetracube branch.svg
| width8=41
| caption8=B <br>"Branch"
| image9=Tetracube r-screw.svg
| width9=42
| caption9=D <br>"Right Screw"
| image10=Tetracube l-screw.svg
| width10=39
| caption10=F <br>"Left Screw"
}}

The tetracubes can be packed into two-layer 3D boxes in several different ways, based on the dimensions of the box and criteria for inclusion. They are shown in both a pictorial diagram and a text diagram. For boxes using two sets of the same pieces, the pictorial diagram depicts each set as a lighter or darker shade of the same color. The text diagram depicts each set as having a capital or lower-case letter. In the text diagram, the top layer is on the left, and the bottom layer is on the right.
 
[[File:Tetromino_tetracube_packing.svg|right|325px]]
<pre style="margin-left:2em;">
1.) 2×4×5 box filled with two sets of free tetrominoes: 

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.) 2×2×10 box filled with two sets of free tetrominoes:

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





3.) 2×4×4 box filled with one set of all tetrominoes:

F T T T        F Z Z B
F F T B        Z Z B B
O O L D        L L L D
O O D D        I I I I





4.) 2×2×8 box filled with one set of all tetrominoes: 

D Z Z L O T T T        D L L L O B F F
D D Z Z O B T F        I I I I O B B F





5.) 2×2×7 box filled with tetrominoes, with mirror-image pieces removed:

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
</pre>

==See also==
* [[Soma cube]]

===Previous and next orders===
* [[Tromino]]
* [[Pentomino]]

==References==
{{reflist}}

==External links==
* [[Vadim Gerasimov]]: [http://vadim.oversigma.com/Tetris.htm The story of Tetris]
* [http://www.tetris-today.com/story/original-tetris0.shtml The Father of Tetris] ([https://web.archive.org/web/20061202094148/http://www.tetris-today.com/story/original-tetris0.shtml web archive copy of the page here])

{{Tetris}}

{{Polyforms}}

[[Category:Polyforms]]
[[Category:Tetris]]
[[Category:Mathematical games]]