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An icosahedron can be inscribed in a dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa. [ 17 ] An icosahedron can be inscribed in an octahedron by placing its 12 vertices on the 12 edges of the octahedron such that they divide each edge into its two golden sections .
A regular icosahedron can be distorted or marked up as a lower pyritohedral symmetry, [2] [3] and is called a snub octahedron, snub tetratetrahedron, snub tetrahedron, and pseudo-icosahedron. [4] This can be seen as an alternated truncated octahedron .
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.
t 1 {p,q} The edges are fully truncated into single points. The polyhedron now has the combined faces of the parent and dual. Polyhedra are named by the number of sides of the two regular forms: {p,q} and {q,p}, like cuboctahedron for r{4,3} between a cube and octahedron. Birectified (2r) (also dual) 2r{p,q} t 2 {p,q}
A dodecahedron is GP(1,0), and a truncated icosahedron is GP(1,1). A similar technique can be applied to construct polyhedra with tetrahedral symmetry and octahedral symmetry. These polyhedra will have triangles or squares rather than pentagons.
The icosidodecahedron is a rectified dodecahedron and also a rectified icosahedron, existing as the full-edge truncation between these regular solids. The icosidodecahedron contains 12 pentagons of the dodecahedron and 20 triangles of the icosahedron: