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Burnside's lemma can compute the number of rotationally distinct colourings of the faces of a cube using three colours.. Let X be the set of 3 6 possible face color combinations that can be applied to a fixed cube, and let the rotation group G of the cube act on X by moving the colored faces: two colorings in X belong to the same orbit precisely when one is a rotation of the other.
The manipulations of the Rubik's Cube form the Rubik's Cube group. The Rubik's Cube group (,) represents the structure of the Rubik's Cube mechanical puzzle. Each element of the set corresponds to a cube move, which is the effect of any sequence of rotations of the cube's faces. With this representation, not only can any cube move be ...
In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold.The various charts on SO(3) set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation.
The rotation group is a Lie group of rotations about a fixed point. This (common) fixed point or center is called the center of rotation and is usually identified with the origin. The rotation group is a point stabilizer in a broader group of (orientation-preserving) motions. For a particular rotation: The axis of rotation is a line of its ...
The proper symmetry group is then a subgroup of the special orthogonal group SO(n), and is called the rotation group of the figure. In a discrete symmetry group, the points symmetric to a given point do not accumulate toward a limit point. That is, every orbit of the group (the images of a given point under all group elements) forms a discrete ...
In mathematics, a dihedral group is the group of symmetries of a regular polygon, [1] [2] 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 .
The rotation group is a group under function composition (or equivalently the product of linear transformations). It is a subgroup of the general linear group consisting of all invertible linear transformations of the real 3-space. [2] Furthermore, the rotation group is nonabelian. That is, the order in which rotations are composed makes a ...
These groups fall into two distinct families, according to whether they consist of rotations only, or include reflections. The cyclic groups, C n (abstract group type Z n), consist of rotations by 360°/n, and all integer multiples. For example, a four-legged stool has symmetry group C 4, consisting of