An octomino (or 8-omino) is a polyomino of order 8; that is, a polygon in the plane made of 8 equal-sized squares connected edge to edge.[1] When rotations and reflections are not considered to be distinct shapes, there are 369 different free octominoes. When reflections are considered distinct, there are 704 one-sided octominoes. When rotations are also considered distinct, there are 2,725 fixed octominoes.[2][3]
The 369 free octominoes
Symmetryedit
The figure shows all possible free octominoes, coloured according to their symmetry groups:
316 octominoes (coloured grey) have no symmetry. Their symmetry group consists only of the identity mapping.
23 octominoes (coloured red) have an axis of reflection symmetry 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.
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.
18 octominoes (coloured blue) have point symmetry, also known as rotational symmetry of order 2. Their symmetry group has two elements, the identity and the 180° rotation.
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.
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 dihedral group of order 2, also known as the Klein four-group.
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.
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 elements.
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.[3]
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 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.
Packing and tilingedit
Of the 369 free octominoes, 320 satisfy the Conway criterion and 23 more can form a patch satisfying the criterion.[4] The other 26 octominoes (including the 6 with holes) are unable to tessellate the plane.[5]
Since 6 of the free octominoes have a hole, it is trivial to prove that the complete set of octominoes cannot be packed into a rectangle, and that not all octominoes can be tiled.
^Weisstein, Eric W. "Octomino". From MathWorld – A Wolfram Web Resource. Retrieved 2008-07-22.
^ abRedelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics. 36 (2): 191–203. doi:10.1016/0012-365X(81)90237-5.
^Rhoads, Glenn C. (2005). "Planar tilings by polyominoes, polyhexes, and polyiamonds". Journal of Computational and Applied Mathematics. 174 (2): 329–353. doi:10.1016/j.cam.2004.05.002.
^Gardner, Martin (August 1975). "More about tiling the plane: the possibilities of polyominoes, polyiamonds and polyhexes". Scientific American. 233 (2): 112–115.
May 07, 2024
octomino, confused, with, otomino, octomino, omino, polyomino, order, that, polygon, plane, made, equal, sized, squares, connected, edge, edge, when, rotations, reflections, considered, distinct, shapes, there, different, free, octominoes, when, reflections, c. Not to be confused with Otomino An octomino or 8 omino is a polyomino of order 8 that is a polygon in the plane made of 8 equal sized squares connected edge to edge 1 When rotations and reflections are not considered to be distinct shapes there are 369 different free octominoes When reflections are considered distinct there are 704 one sided octominoes When rotations are also considered distinct there are 2 725 fixed octominoes 2 3 The 369 free octominoesSymmetry editThe figure shows all possible free octominoes coloured according to their symmetry groups 316 octominoes coloured grey have no symmetry Their symmetry group consists only of the identity mapping 23 octominoes coloured red have an axis of reflection symmetry 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 nbsp dd 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 nbsp dd 18 octominoes coloured blue have point symmetry also known as rotational symmetry of order 2 Their symmetry group has two elements the identity and the 180 rotation nbsp dd 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 nbsp dd 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 dihedral group of order 2 also known as the Klein four group 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 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 elements nbsp dd 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 3 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 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 Packing and tiling editOf the 369 free octominoes 320 satisfy the Conway criterion and 23 more can form a patch satisfying the criterion 4 The other 26 octominoes including the 6 with holes are unable to tessellate the plane 5 Since 6 of the free octominoes have a hole it is trivial to prove that the complete set of octominoes cannot be packed into a rectangle and that not all octominoes can be tiled nbsp dd References edit Golomb Solomon W 1994 Polyominoes 2nd ed Princeton New Jersey Princeton University Press ISBN 0 691 02444 8 Weisstein Eric W Octomino From MathWorld A Wolfram Web Resource Retrieved 2008 07 22 a b Redelmeier D Hugh 1981 Counting polyominoes yet another attack Discrete Mathematics 36 2 191 203 doi 10 1016 0012 365X 81 90237 5 Rhoads Glenn C 2005 Planar tilings by polyominoes polyhexes and polyiamonds Journal of Computational and Applied Mathematics 174 2 329 353 doi 10 1016 j cam 2004 05 002 Gardner Martin August 1975 More about tiling the plane the possibilities of polyominoes polyiamonds and polyhexes Scientific American 233 2 112 115 Retrieved from https en wikipedia org w index php title Octomino amp oldid 1196854908, wikipedia, wiki, book, books, library,
