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 popu ...

of order 8; that is, a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

in the

plane Plane most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface * Plane (mathematics), generalizations of a geometrical plane Plane or planes may also refer to: Biology * Plane ...

made of 8 equal-sized

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s connected edge to edge. When

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

s and

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, mirror-like reflection of waves from a surface *** Mirror image, a reflection in a mirror or in water ** Diffuse r ...

s are not considered to be distinct shapes, there are

369 __NOTOC__ Year 369 ( CCCLXIX) was a common year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Galates and Victor (or, less frequently, year 1122 ''Ab urbe condita''). The denomination ...

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 amb ...

s: * 316 octominoes (coloured grey) have no

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

. 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 (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...

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 (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...

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 In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...

of order 2, also known as the

Klein four-group In mathematics, the Klein four-group is an abelian 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 non-identi ...

. * 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 Necessity and sufficiency, sufficient rule for when a prototile will tile the plane. It consists of the following req ...

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 a ...

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

References

{{Polyforms Polyforms