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The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent ...
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.
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.
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 ...
In addition to preserving length, proper rotations must also preserve orientation. A matrix will preserve or reverse orientation according to whether the determinant of the matrix is positive or negative. For an orthogonal matrix R, note that det R T = det R implies (det R) 2 = 1, so that det R = ±1.
The direct (orientation-preserving) symmetry operations, which form the group SO(3): The identity operation, denoted by E or the identity matrix I. Rotation about an axis through the origin by an angle θ. Rotation by θ = 360°/n for any positive integer n is denoted C n (from the Schoenflies notation for the group C n that it generates).
The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances. In 1D, all reflections are in the same class. In 2D, rotations by the same angle in either direction are in the same class. Glide reflections with translation by the same distance are in the same class. In 3D: