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A decomino, or 10-omino, is a
The 4,655 free decominoes can be classified according to their
195 decominoes have holes. This makes it trivial to prove that the complete set of decominoes cannot be Iread.it: Maximal squares of polyominoes
/ref>
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 10, that is, a polygon in the plane made of 10 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 ...
s are not considered to be distinct shapes, there are 4,655 different ''free
Free may refer to:
Concept
* Freedom, having the ability to do something, without having to obey anyone/anything
* Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism
* Emancipate, to procur ...
'' decominoes (the free decominoes comprise 195 with holes and 4,460 without holes). When reflections are considered distinct, there are 9,189 ''one-sided'' decominoes. When rotations are also considered distinct, there are 36,446 ''fixed'' decominoes.
Symmetry
The 4,655 free decominoes can be classified 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:
* 4,461 decominoes 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, unch ...
.
* 90 decominoes 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.
* 22 decominoes have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection.
* 73 decominoes 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.
* 8 decominoes 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 decomino 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.
Unlike both octomino
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. When rotations and reflections are not considered to be distinct shapes, there are 369 different ''free'' ...
es and nonomino
A nonomino (or enneomino or 9-omino) 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 reflectio ...
es, no decomino has rotational symmetry of order 4.
Packing and tiling
195 decominoes have holes. This makes it trivial to prove that the complete set of decominoes 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 decominoes can be tiled.
The 4,460 decominos without holes comprise 44,600 unit squares. Thus, the largest square that can be tiled with distinct decominoes is at most 210 units on a side (210 squared is 44,100). Such a square containing 4,410 decominoes was constructed by Livio Zucca./ref>
References
{{Polyforms Polyforms