Nonomino
Encyclopedia
A nonomino is a polyomino
of order 9, that is, a polygon
in the plane
made of 9 equal-sized square
s connected edge-to-edge. The name of this type of figure is formed with the prefix non(a)-. When rotation
s and reflection
s 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.
s:
Unlike octominoes, there are no nonominoes with rotational symmetry of order 4 or with two axes of reflection symmetry aligned with the diagonals.
If reflections of a nonomino are considered distinct, as they are with one-sided nonominoes, then the first and fourth categories above double in size, resulting in an extra 1,215 nonominoes for a total of 2,500. If rotations are also considered distinct, then the nonominoes from the first category count eightfold, the ones from the next three categories count fourfold, the ones from the fifth category count twice, and the ones from the last category count only once. This results in 1,196 × 8 + (38+26+19) × 4 + 4 × 2 + 2 = 9,910 fixed nonominoes.
37 nonominoes have a hole. This makes it trivial to prove that the complete set of nonominoes cannot be packed
into a rectangle, and that not all nonominoes can be tiled
. However, it has been proven that 1,050 free nonominoes, or all but 235, do tile the plane.
One nonomino has a two-square hole (second rightmost in the top row). It is the smallest polyomino with a two-square hole.
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 with a connected interior....
of order 9, that is, a polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
in the plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
made of 9 equal-sized square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
s connected edge-to-edge. The name of this type of figure is formed with the prefix non(a)-. When rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s and reflection
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
s 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.
Symmetry
The 1,285 free nonominoes can be classified according to their symmetry groupSymmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
s:
- 1,196 nonominoes have no symmetrySymmetrySymmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
. Their symmetry group consists only of the identity mappingIdentity functionIn mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
. - 38 nonominoes have an axis of reflection symmetryReflection symmetryReflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...
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.
- 26 nonominoes have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection.
- 19 nonominoes have point symmetry, also known as rotational symmetryRotational symmetryGenerally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...
of order 2. Their symmetry group has two elements, the identity and the 180° rotation.
- 4 nonominoes 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 groupDihedral groupIn mathematics, a dihedral group is the group of symmetries of a regular polygon, including both 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.See also: Dihedral symmetry in three...
of order 2, also known as the Klein four-groupKlein four-groupIn mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...
.
- 2 nonominoes have four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4. Their symmetry group, the dihedral group of order 4, has eight elements.
Unlike octominoes, there are no nonominoes with rotational symmetry of order 4 or with two axes of reflection symmetry aligned with the diagonals.
If reflections of a nonomino are considered distinct, as they are with one-sided nonominoes, then the first and fourth categories above double in size, resulting in an extra 1,215 nonominoes for a total of 2,500. If rotations are also considered distinct, then the nonominoes from the first category count eightfold, the ones from the next three categories count fourfold, the ones from the fifth category count twice, and the ones from the last category count only once. This results in 1,196 × 8 + (38+26+19) × 4 + 4 × 2 + 2 = 9,910 fixed nonominoes.
Packing and tiling
Packing problem
Packing problems are a class of optimization problems in mathematics which involve attempting to pack objects together , as densely as possible. Many of these problems can be related to real life packaging, storage and transportation issues...
into a rectangle, and that not all nonominoes can be tiled
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
. However, it has been proven that 1,050 free nonominoes, or all but 235, do tile the plane.
One nonomino has a two-square hole (second rightmost in the top row). It is the smallest polyomino with a two-square hole.