An octomino (or 8-omino) is a

polyomino 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. Polyominoes have been used in pop ...

of order 8, that is, a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

in the

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

made of 8 equal-sized

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

s connected edge-to-edge. When

rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

s and

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

s 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.

Symmetry

The figure shows all possible free octominoes, coloured according to their

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...

s: * 316 octominoes (coloured grey) have no

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

. Their symmetry group consists only of the

identity mapping Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

. * 23 octominoes (coloured red) have an axis of

reflection symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...

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. :: Reflection symmetrical octominoes 90 deg

Reflection symmetrical octominoes 90 deg

* 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. :: Reflection symmetrical octominoes 45 deg

Reflection symmetrical octominoes 45 deg

* 18 octominoes (coloured blue) have point symmetry, also known as

rotational symmetry 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 i ...

of order 2. Their symmetry group has two elements, the identity and the 180° rotation. :: Rotation symmetrical octominoes (C2)

* 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. :: Rotation symmetrical octomino (C4)

* 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 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, ge ...

of order 2, also known as the

Klein four-group In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three ...

. * 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. 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

Of the 369 free octominoes, 320 satisfy the

Conway criterion In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, is a sufficient rule for when a prototile will tile the plane. It consists of the following requirements:Will It Tile? Try ...

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. Since 6 of the free octominoes have a hole, it is trivial to prove that the complete set of octominoes cannot be

packed Data structure alignment is the way data is arranged and accessed in computer memory. It consists of three separate but related issues: data alignment, data structure padding, and packing. The CPU in modern computer hardware performs reads and ...

into a rectangle, and that not all octominoes can be tiled. ::

References

{{Polyforms Polyforms