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The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron. It is isomorphic to A 4. The conjugacy classes of T are: identity; 4 × rotation by 120°, order 3, cw; 4 × rotation by 120°, order 3, ccw; 3 × rotation by 180°, order 2; The octahedral group of order 24, rotational symmetry group of the cube and the ...
The order of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the full symmetry group, which includes reflections, and the proper symmetry group, which includes only rotations. The symmetry groups of the Platonic solids are a special class of three-dimensional point groups known as polyhedral ...
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation, [1] orbifold notation, [2] and order.
Polyhedra make a frequent appearance in modern computational geometry, computer graphics, and geometric design with topics including the reconstruction of polyhedral surfaces or surface meshes from scattered data points, [94] geodesics on polyhedral surfaces, [95] visibility and illumination in polyhedral scenes, [96] polycubes and other non ...
Another group of regular polyhedra comprise tilings of the real projective plane. These include the hemi-cube, hemi-octahedron, hemi-dodecahedron, and hemi-icosahedron. They are (globally) projective polyhedra, and are the projective counterparts of the Platonic solids. The tetrahedron does not have a projective counterpart as it does not have ...
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices.
Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters: [C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
In geometry, a point group is a mathematical group of symmetry operations ... and 7 additional polyhedral groups (also called Platonic). In Schönflies notation,