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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.
The truncated icosahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex. [5] It has the same symmetry as the regular icosahedron, the icosahedral symmetry, and it also has the property of vertex-transitivity.
Convex regular icosahedron A tensegrity icosahedron. In geometry, an icosahedron (/ ˌ aɪ k ɒ s ə ˈ h iː d r ən,-k ə-,-k oʊ-/ or / aɪ ˌ k ɒ s ə ˈ h iː d r ən / [1]) is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and ἕδρα (hédra) 'seat'.
The icosahedron has a large number of stellations. Coxeter et al. (1938) in their work stated fifty-nine stellations were identified for the regular icosahedron. [24] Regular icosahedron itself is the first stellation of an icosahedron, and the subsequent stellation obtained by radiating spikes from the faces of a regular icosahedron.
In geometry, the rectified truncated icosahedron is a convex polyhedron. It has 92 faces: 60 isosceles triangles , 12 regular pentagons , and 20 regular hexagons . It is constructed as a rectified , truncated icosahedron , rectification truncating vertices down to mid-edges.
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, [1] great rhombicosidodecahedron, [2] [3] omnitruncated dodecahedron or omnitruncated icosahedron [4] is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
3D model of a truncated great icosahedron. In geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U 55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices. [1] It is given a Schläfli symbol t{3, 5 ⁄ 2} or t 0,1 {3, 5 ⁄ 2} as a truncated great ...
It can be seen as the compound of an icosahedron and dodecahedron. It is one of four compounds constructed from a Platonic solid or Kepler-Poinsot solid, and its dual. It has icosahedral symmetry (I h) and the same vertex arrangement as a rhombic triacontahedron.
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