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A regular tetrahedron, an example of a solid with full tetrahedral symmetry. A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.
Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The full symmetry group is the Coxeter group of type H 3. It may be represented by Coxeter notation [5,3] and Coxeter diagram.
A cube has the same set of symmetries, since it is the polyhedron that is dual to an octahedron. The group of orientation-preserving symmetries is S 4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the cube.
The subgroup of orientation-preserving symmetries (translations, rotations, and compositions of these) is called its proper symmetry group. An object is chiral when it has no orientation-reversing symmetries, so that its proper symmetry group is equal to its full symmetry group.
Achiral and chiral operators are also called local symmetry-preserving operations (LSP) and local operations that preserve orientation-preserving symmetries (LOPSP), respectively. [7] [8] [9] LSPs should be understood as local operations that preserve symmetry, not operations that preserve local symmetry. Again, these are symmetries in a ...
By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation.
The direct isometries (i.e., isometries preserving the handedness of chiral subsets) comprise a subgroup of E(n), called the special Euclidean group and usually denoted by E + (n) or SE(n). They include the translations and rotations, and combinations thereof; including the identity transformation, but excluding any reflections.
In Euclidean or pseudo-Euclidean spaces, a point reflection is an isometry (preserves distance). [1] In the Euclidean plane, a point reflection is the same as a half-turn rotation (180° or π radians), while in three-dimensional Euclidean space a point reflection is an improper rotation which preserves distances but reverses orientation.