Polyomino (EntityTopic, 5)

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Latest revision as of 23:30, 11 February 2014

A polyomino (also spelt polimino and polymino) is a collection of hypercubes joined so that one hypercell of one touches another hypercell of another. Polyominoes are named as follows:

Monomino - one hypercube.
Domino - two hypercubes.
Tromino - three hypercubes.
Tetromino - four hypercubes.

etc.

We can generalize sets of polyominoes as p(c,d) where c is the number of hypercubes and d is the number of dimensions.

In n dimensions, there are no new polyominoes that are not simply extrusions of n-1-dimensional polyominoes, if the number of hypercubes is equal to or less than the number of dimensions. In other words, p(c,d) = p(c,d-1) if d ≥ c.

Here is a table of the polyominoes available with various c and d:

Tetrominoes

Tetrominoes are arrangements of four orthogonally-connected cubes. There are seven tetrominoes:

O-block
Square block
Fat block

L-block
(J-block)
Leg block

T-block
Tee block

I-block
Line block

Z-block
(S-block)
Step block

Q-block
(G-block)
Screw block

X-block
Corner block

In Tetris, only the first five are available, but since you're only playing in 2D there, two of them are chiral, bringing the count back up to seven again.

The sixth is actually chiral in 3D, so that brings the count up to eight.

Octominoes

There are two special properties about octominoes:

The first polyomino with a hole is an octomino.
The first polyomino with no linear symmetry, but rotational symmetry of order 4 is an octomino.

@@ Line 1: / Line 1: @@
+<[#ontology [kind topic] [cats Shape Essays]]>
 A '''polyomino''' (also spelt ''polimino'' and ''polymino'') is a collection of [[hypercube]]s joined so that one [[hypercell]] of one touches another hypercell of another. Polyominoes are named as follows:
 * Monomino - one hypercube.
 * Domino - two hypercubes.
-* Triomino - three hypercubes.
+* Tromino - three hypercubes.
-* Quadromino - four hypercubes.
+* Tetromino - four hypercubes.
 etc.
@@ Line 12: / Line 13: @@
 Here is a table of the polyominoes available with various c and d:
-<blockquote>http://www.invhost.com/share/polyominoes.png</blockquote>
+<blockquote><[#embed [hash 9HYZJMFD9ECAMRA1KPA12F6FZT]]></blockquote>
+== Tetrominoes ==
+'''Tetrominoes''' are arrangements of four [[orthogonally]]-connected [[cube]]s. There are seven tetrominoes:
+<center>
+{|
+|valign="bottom" width="100" align="center"|<[#embed [hash TTWTY9RX2ZK66RWY7C3VTRAJXN]]><br />[[O-block|O-block<br />Square block<br />Fat block]]
+|valign="bottom" width="100" align="center"|<[#embed [hash J6GGEX0Z026WWCW0S97TW40D23]]><br />[[L-block|L-block<br />(J-block)<br />Leg block]]
+|valign="bottom" width="100" align="center"|<[#embed [hash FPSYWR20E0HZE3VGPKK4M8QYT6]]><br />[[T-block|T-block<br />Tee block<br />&nbsp;]]
+|valign="bottom" width="100" align="center"|<[#embed [hash DK63YXH89FDYRGXN8AKEAWM3X8]]><br />[[I-block|I-block<br />Line block<br />&nbsp;]]
+|valign="bottom" width="100" align="center"|<[#embed [hash 4DTEMEZTWSKB4W51PS21TRFY32]]><br />[[Z-block|Z-block<br />(S-block)<br />Step block]]
+|valign="bottom" width="100" align="center"|<[#embed [hash 42Z9WG799DZEJ34YGSV9C34Q25]]><br />[[Q-block|Q-block<br />(G-block)<br />Screw block]]
+|valign="bottom" width="100" align="center"|<[#embed [hash D0YTGGT3J3XRM1CS8GJBP21FQZ]]><br />[[X-block|X-block<br />Corner block<br />&nbsp;]]
+|}
+</center>
+In [[Tetris]], only the first five are available, but since you're only playing in [[2D]] there, two of them are [[chiral]], bringing the count back up to seven again.
+The sixth is actually chiral in [[3D]], so that brings the count up to ''eight''.
 == Octominoes ==
 There are two special properties about octominoes:
-* The first polyomino with a [[pocket]] is an octomino.
+* The first polyomino with a [[hole]] is an octomino.
 * The first polyomino with no linear symmetry, but rotational symmetry of order 4 is an octomino.
-{{Shapes}}