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If reflections of an octomino are considered distinct, as they are with one-sided octominoes, then the first, fourth and fifth categories above double in size, resulting in an extra 335 octominoes for a total of 704. If rotations are also considered distinct, then the octominoes from the first
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category count eightfold, the ones from the next three categories count fourfold, the ones from categories five to seven count twice, and the last octomino counts only once. This results in 316 × 8 + (23+5+18) × 4 + (1+4+1) × 2 + 1 = 2,725 fixed octominoes.
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1 octomino (coloured cyan) has four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4. Its symmetry group, the dihedral group of order 4, has eight
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4 octominoes (coloured purple) have two axes of reflection symmetry, both aligned with the gridlines. Their symmetry group has four elements, the identity, two reflections and the 180° rotation. It is the
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The set of octominoes is the lowest polyomino set in which all eight possible symmetries are realized. The next higher set with this property is the dodecomino (12-omino) set.
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1 octomino (coloured orange) has two axes of reflection symmetry, both aligned with the diagonals. Its symmetry group is also the dihedral group of order 2 with four elements.
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5 octominoes (coloured green) have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection.
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1 octomino (coloured yellow) has rotational symmetry of order 4. Its symmetry group has four elements, the identity and the 90°, 180° and 270° rotations.
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aligned with the gridlines. Their symmetry group has two elements, the identity and the reflection in a line parallel to the sides of the squares.
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and 23 more can form a patch satisfying the criterion. The other 26 octominoes (including the 6 with holes) are unable to tessellate the plane.
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Gardner, Martin (August 1975). "More about tiling the plane: the possibilities of polyominoes, polyiamonds and polyhexes".
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Since 6 of the free octominoes have a hole, it is trivial to prove that the complete set of octominoes cannot be
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of order 2. Their symmetry group has two elements, the identity and the 180° rotation.
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Rhoads, Glenn C. (2005). "Planar tilings by polyominoes, polyhexes, and polyiamonds".
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The figure shows all possible free octominoes, coloured according to their
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octominoes. When rotations are also considered distinct, there are 2,725
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octominoes. When reflections are considered distinct, there are 704
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290:(2nd ed.). Princeton, New Jersey: Princeton University Press.
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18 octominoes (coloured blue) have point symmetry, also known as
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into a rectangle, and that not all octominoes can be
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are not considered to be distinct shapes, there are
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371:Journal of Computational and Applied Mathematics
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111:23 octominoes (coloured red) have an axis of
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236:Of the 369 free octominoes, 320 satisfy the
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338:"Counting polyominoes: yet another attack"
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315:. From MathWorld – A Wolfram Web Resource
16:Geometric shape formed from eight squares
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58:connected edge to edge. When
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336:Redelmeier, D. Hugh (1981).
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54:made of 8 equal-sized
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342:Discrete Mathematics
399:Scientific American
311:Weisstein, Eric W.
153:rotational symmetry
113:reflection symmetry
282:Golomb, Solomon W.
232:Packing and tiling
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317:. Retrieved
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84:octominoes.
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635:WikiProject
543:Polydrafter
517:Polyominoid
453:Polyominoes
287:Polyominoes
64:reflections
595:Snake cube
553:Polyiamond
319:2008-07-22
313:"Octomino"
268:References
70:different
664:Polyforms
590:Soma cube
563:Polystick
538:Polyabolo
486:Heptomino
476:Pentomino
471:Tetromino
445:Polyforms
207:elements.
78:one-sided
60:rotations
44:polyomino
658:Category
605:Hexastix
522:Polycube
501:Decomino
496:Nonomino
491:Octomino
481:Hexomino
284:(1994).
102:symmetry
88:Symmetry
36:octomino
611:Tantrix
600:Tangram
577:puzzles
548:Polyhex
466:Tromino
56:squares
50:in the
48:polygon
42:) is a
40:8-omino
21:Otomino
645:Portal
618:Tetris
585:Blokus
531:Others
461:Domino
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245:packed
573:Games
249:tiled
82:fixed
52:plane
575:and
292:ISBN
73:free
62:and
38:(or
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379:doi
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
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