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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 .
Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection ), for a total ...
A dodecahedron and its dual icosahedron The intersection of both solids is the icosidodecahedron , and their convex hull is the rhombic triacontahedron . Seen from 2-fold, 3-fold and 5-fold symmetry axes
Icosahedron {3,5} (3.3.3.3.3) arccos (- √ 5 / 3 ) 138.190° Kepler–Poinsot solids (regular nonconvex) Small stellated dodecahedron { 5 / 2 ,5} ( 5 / 2 . 5 / 2 . 5 / 2 . 5 / 2 . 5 / 2 ) arccos (- √ 5 / 5 ) 116.565° Great dodecahedron {5, 5 / 2 } (5.5.5.5.5) / 2 arccos ...
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 ...
When a regular dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.55%). [10] The resulting of both spheres' volumes initially began from the problem by ancient Greeks, determining which of two shapes has a larger volume: an icosahedron inscribed in a ...
The 5 Platonic solids are called a tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron with 4, 6, 8, 12, and 20 sides respectively. The regular hexahedron is a cube . Table of polyhedra