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It shares the dual polyhedron of a regular icosahedron, the regular dodecahedron: a regular icosahedron can be inscribed in a regular dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa. [16] The rotational symmetry group of the regular icosahedron is isomorphic to the alternating group on five letters.
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 quantity h (called the Coxeter number) is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively. The angular deficiency at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2 π .
The concave equilateral dodecahedron, called an endo-dodecahedron. [clarification needed] A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions. A regular dodecahedron is an intermediate case with equal edge lengths. A rhombic dodecahedron is a degenerate case with the 6 crossedges reduced to ...
arccos (- 1 / 2 ) = 2 π / 3 120° Great rhombic triacontahedron (Dual of great icosidodecahedron) — V(3. 5 / 2 .3. 5 / 2 ) arccos ( √ 5-1 / 4 ) = 2 π / 5 72° Duals of the ditrigonal polyhedra Small triambic icosahedron (Dual of small ditrigonal icosidodecahedron) — V(3. 5 / 2 .3. 5 / ...
1 icosahedron 1 dodecahedron: Faces: 20 triangles 12 pentagons: Edges: 60 Vertices: 32 Symmetry group: icosahedral (I h)
The regular dodecahedron is a polyhedron with twelve pentagonal faces, thirty edges, and twenty vertices. [1] It is one of the Platonic solids, a set of polyhedrons in which the faces are regular polygons that are congruent and the same number of faces meet at a vertex. [2] This set of polyhedrons is named after Plato.
For example, the icosahedron is {3,5+} 1,0, and pentakis dodecahedron, {3,5+} 1,1 is seen as a regular dodecahedron with pentagonal faces divided into 5 triangles. The primary face of the subdivision is called a principal polyhedral triangle (PPT) or the breakdown structure. Calculating a single PPT allows the entire figure to be created.