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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
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. Because the golden sections are ...
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'.
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 ...
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 ...
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)
Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest to that of a sphere of the same size (i.e ...
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 ...