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If reflections of a hexomino are considered distinct, as they are with one-sided hexominoes, then the first and fourth categories above would each double in size, resulting in an extra 25 hexominoes for a total of 60. If rotations are also considered distinct, then the hexominoes from the first
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However, there are other simple figures of 210 squares that can be packed with the hexominoes. For example, a 15 × 15 square with a 3 × 5 rectangle removed from the centre has 210 squares. With checkerboard colouring, it has 106 white and 104 black squares (or vice versa), so parity does not
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of black squares (3 white and 3 black). Overall, an even number of black squares will be covered in any arrangement. However, any rectangle of 210 squares will have 105 black squares and 105 white squares, and therefore cannot be covered by the 35 hexominoes.
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prevent a packing, and a packing is indeed possible. It is also possible for two sets of pieces to fit a rectangle of size 420, or for the set of 60 one-sided hexominoes (18 of which cover an even number of black squares) to fit a rectangle of size 360.
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category count eightfold, the ones from the next three categories count fourfold, and the ones from the last category count twice. This results in 20 × 8 + (6 + 2 + 5) × 4 + 2 × 2 = 216 fixed hexominoes.
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The two purple hexominoes have two axes of mirror symmetry, both parallel to the gridlines (thus one horizontal axis and one vertical axis). Their symmetry group has four elements. It is the
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167:, which can be packed into any of the rectangles 3 × 20, 4 × 15, 5 × 12 and 6 × 10.) A simple way to demonstrate that such a packing of hexominoes is not possible is via a
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pattern, then 11 of the hexominoes will cover an even number of black squares (either 2 white and 4 black or vice versa) and the other 24 hexominoes will cover an
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is necessarily a hexomino, with 11 hexominoes (shown at right) actually being nets. They appear on the right, again coloured according to their symmetry groups.
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The two green hexominoes have an axis of mirror symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection.
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parallel to the gridlines. Their symmetry group has two elements, the identity and a reflection in a line parallel to the sides of the squares.
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A polyhedral net for the cube cannot contain the O-tetromino, nor the I-pentomino, the U-pentomino, or the V-pentomino.
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Although a complete set of 35 hexominoes has a total of 210 squares, it is not possible to pack them into a
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of order 2. Their symmetry group has two elements, the identity and the 180° rotation.
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The figure above shows all 35 possible free hexominoes, coloured according to their
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Page by Jürgen Köller on hexominoes, including symmetry, packing and other aspects
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connected edge to edge. The name of this type of figure is formed with the prefix
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237:(2nd ed.). Princeton, New Jersey: Princeton University Press.
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hexominoes. When rotations are also considered distinct, there are
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hexominoes. When reflections are considered distinct, there are
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The five blue hexominoes have point symmetry, also known as
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Planar
Tilings and the Search for an Aperiodic Prototile
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are not considered to be distinct shapes, there are
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377:Eleven animations showing the patterns of the cube
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171:argument. If the hexominoes are placed on a
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282:"Counting polyominoes: yet another attack"
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262:. From MathWorld – A Wolfram Web Resource
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152:Each of the 35 hexominoes satisfies the
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317:. PhD dissertation, Rutgers University.
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114:The six red hexominoes have an axis of
16:Geometric shape formed from six squares
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328:Mathematische Basteleien: Hexominos
103:The twenty grey hexominoes have no
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135:of order 2, also known as the
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196:All 11 unfoldings of the cube
299:10.1016/0012-365X(81)90237-5
280:Redelmeier, D. Hugh (1981).
188:Polyhedral nets for the cube
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313:Rhoads, Glenn C. (2003).
39:of order 6; that is, a
342:Hexomino Constructions
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47:made of 6 equal-sized
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23:The 35 free hexominoes
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286:Discrete Mathematics
385:Polypolygon tilings
258:Weisstein, Eric W.
126:rotational symmetry
390:2007-10-18 at the
229:Golomb, Solomon W.
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148:Packing and tiling
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494:Higher dimensions
371:Geometry Junkyard
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351:External links
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264:. Retrieved
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173:checkerboard
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87:hexominoes.
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619:WikiProject
527:Polydrafter
501:Polyominoid
437:Polyominoes
380:(in French)
292:: 191–203.
234:Polyominoes
165:pentominoes
61:reflections
579:Snake cube
537:Polyiamond
266:2008-07-22
260:"Hexomino"
215:References
177:odd number
67:different
648:Polyforms
574:Soma cube
547:Polystick
522:Polyabolo
470:Heptomino
460:Pentomino
455:Tetromino
429:Polyforms
161:rectangle
78:one-sided
57:rotations
37:polyomino
642:Category
589:Hexastix
506:Polycube
485:Decomino
480:Nonomino
475:Octomino
465:Hexomino
388:Archived
231:(1994).
204:for the
105:symmetry
91:Symmetry
29:hexomino
595:Tantrix
584:Tangram
561:puzzles
532:Polyhex
450:Tromino
55:. When
53:hex(a)-
49:squares
43:in the
41:polygon
35:) is a
33:6-omino
629:Portal
602:Tetris
569:Blokus
515:Others
445:Domino
241:
169:parity
557:Games
85:fixed
45:plane
559:and
239:ISBN
206:cube
70:free
59:and
31:(or
368:'s
364:of
294:doi
82:216
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290:36
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200:A
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
