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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 .
12 × rotation by ±144°, order 5, around the 6 axes through the face centers of the dodecahedron; 20 × rotation by ±120°, order 3, around the 10 axes through vertices of the dodecahedron; 15 × rotation by 180°, order 2, around the 15 axes through midpoints of edges of the dodecahedron; central inversion, order 2
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 ...
The dihedral angles for the edge-transitive polyhedra are: Picture ... Dodecahedron {5,3} (5.5.5) arccos (- √ 5 / 5 ) 116.565° Icosahedron
Rectified dodecahedron Rectified icosahedron Icosidodecahedron: t 1 {5,3}=r{5,3} 32 60 30 14 ... The number of edges is unchanged and are rotated 90 degrees. A ...
Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron; Regular spherical polyhedron. Dihedron, Hosohedron; Kepler–Poinsot polyhedron (Regular star polyhedra) Small stellated dodecahedron, Great stellated dodecahedron, Great icosahedron, Great dodecahedron; Abstract regular polyhedra (Projective polyhedron)
The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 12 congruent regular pentagons and 30 congruent flattened hexagons. It is constructed as a chamfer of a regular dodecahedron .