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Icosahedral symmetry fundamental domains A soccer ball, a common example of a spherical truncated icosahedron, has full icosahedral symmetry. Rotations and reflections form the symmetry group of a great icosahedron. In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron.
A regular icosahedron is topologically identical to a cuboctahedron with its 6 square faces bisected on diagonals with pyritohedral symmetry. The icosahedra with pyritohedral symmetry constitute an infinite family of polyhedra which include the cuboctahedron, regular icosahedron, Jessen's icosahedron, and double cover octahedron. Cyclical ...
An object of C 3v symmetry under one of the 3-fold axes gives rise under the action of T d to an orbit consisting of four such objects, and T d corresponds to the set of permutations of these four objects. T d is a normal subgroup of O h. See also the isometries of the regular tetrahedron. T h, (3*2) [3 +,4] 2/m 3, m 3 order 24: pyritohedral ...
The icosahedral group of order 60, rotational symmetry group of the regular dodecahedron and the regular icosahedron. It is isomorphic to A 5. The conjugacy classes of I are: identity; 12 × rotation by ±72°, order 5; 12 × rotation by ±144°, order 5; 20 × rotation by ±120°, order 3; 15 × rotation by 180°, order 2
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.
All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron , a common crystal form in pyrite , has pyritohedral symmetry , while the tetartoid has ...
The pyritohedral group T h with fundamental domain The seams of a volleyball have pyritohedral symmetry. T h, 3*2, [4,3 +] or m 3, of order 24 – pyritohedral symmetry. [1] This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now S 6 (3) axes
The following table shows the solids in pairs of duals. In the top row they are shown with pyritohedral symmetry, in the bottom row with icosahedral symmetry (to which the mentioned colors refer). The table below shows orthographic projections from the 5-fold (red), 3-fold (yellow) and 2-fold (blue) symmetry axes.