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 figure shows all possible free octominoes, coloured according to their symmetry groups:
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.
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.
Derived from the Greek word for '5', and "domino", a pentomino is a polyomino of order 5; that is, a polygon in the plane made of 5 equal-sized squares connected edge to edge. When rotations and reflections are not considered to be distinct shapes, there are 12 different free pentominoes. When reflections are considered distinct, there are 18 one-sided pentominoes. When rotations are also considered distinct, there are 63 fixed pentominoes.
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling.
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.
A polyiamond is a polyform whose base form is an equilateral triangle. The word polyiamond is a back-formation from diamond, because this word is often used to describe the shape of a pair of equilateral triangles placed base to base, and the initial 'di-' looks like a Greek prefix meaning 'two-'. The name was suggested by recreational mathematics writer Thomas H. O'Beirne in New Scientist 1961 number 1, page 164.
In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetries. The set of symmetries of a frieze pattern is called a frieze group.
A hexomino is a polyomino of order 6; that is, a polygon in the plane made of 6 equal-sized squares connected edge to edge. The name of this type of figure is formed with the prefix hex(a)-. When rotations and reflections are not considered to be distinct shapes, there are 35 different free hexominoes. When reflections are considered distinct, there are 60 one-sided hexominoes. When rotations are also considered distinct, there are 216 fixed hexominoes.
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. For each wallpaper there corresponds a group of congruent transformations, with function composition as the group operation. Thus, a wallpaper group is a mathematical classification of a two‑dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tessellations, tiles and physical wallpaper.
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.
In recreational mathematics, a polyhex is a polyform with a regular hexagon as the base form, constructed by joining together 1 or more hexagons. Specific forms are named by their number of hexagons: monohex, dihex, trihex, tetrahex, etc. They were named by David Klarner who investigated them.
A polycube is a solid figure formed by joining one or more equal cubes face to face. Polycubes are the three-dimensional analogues of the planar polyominoes. The Soma cube, the Bedlam cube, the Diabolical cube, the Slothouber–Graatsma puzzle, and the Conway puzzle are examples of packing problems based on polycubes.
The Schoenfliesnotation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the symmetry of a molecule, the notation is often sufficient and commonly used for spectroscopy. However, in crystallography, there is additional translational symmetry, and point groups are not enough to describe the full symmetry of crystals, so the full space group is usually used instead. The naming of full space groups usually follows another common convention, the Hermann–Mauguin notation, also known as the international notation.
A tromino or triomino is a polyomino of size 3, that is, a polygon in the plane made of three equal-sized squares connected edge-to-edge.
A heptomino is a polyomino of order 7; that is, a polygon in the plane made of 7 equal-sized squares connected edge to edge. The name of this type of figure is formed with the prefix hept(a)-. When rotations and reflections are not considered to be distinct shapes, there are 108 different free heptominoes. When reflections are considered distinct, there are 196 one-sided heptominoes. When rotations are also considered distinct, there are 760 fixed heptominoes.
A nonomino is a polyomino of order 9; that is, a polygon in the plane made of 9 equal-sized squares connected edge to edge. The name of this type of figure is formed with the prefix non(a)-. When rotations and reflections are not considered to be distinct shapes, there are 1,285 different free nonominoes. When reflections are considered distinct, there are 2,500 one-sided nonominoes. When rotations are also considered distinct, there are 9,910 fixed nonominoes.
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.
A regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dihn.
In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a 1⁄3 turn rotation of a regular triangle about its center, a reflection of a square across its diagonal, a translation of the Euclidean plane, or a point reflection of a sphere through its center are all symmetry operations. Each symmetry operation is performed with respect to some symmetry element.
In mathematics, a domino is a polyomino of order 2, that is, a polygon in the plane made of two equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there is only one free domino.
A decomino, or 10-omino, is a polyomino of order 10; that is, a polygon in the plane made of 10 equal-sized squares connected edge to edge. When rotations and reflections are not considered to be distinct shapes, there are 4,655 different free decominoes. When reflections are considered distinct, there are 9,189 one-sided decominoes. When rotations are also considered distinct, there are 36,446 fixed decominoes.
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