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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. Because the golden sections are ...
Construction from the vertices of a truncated octahedron, showing internal rectangles. The Cartesian coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedron with alternated vertices deleted.
The regular icosahedron can be constructed by intersecting three golden rectangles perpendicularly, arranged in two-by-two orthogonal, and connecting each of the golden rectangle's vertices with a segment line. There are 12 regular icosahedron vertices, considered as the center of 12 regular dodecahedron faces. [13]
The 92 vertices lie on the surfaces of three concentric spheres. The innermost group of 20 vertices form the vertices of a regular dodecahedron; the next layer of 12 form the vertices of a regular icosahedron; and the outer layer of 60 form the vertices of a nonuniform truncated icosahedron. The radii of these spheres are in the ratio [11]
Three golden rectangles touch all of the 12 vertices of a regular icosahedron. These geometric values can be calculated from their Cartesian coordinates, which also can be given using formulas involving . The coordinates of the dodecahedron are displayed on the figure to the right, while those of the icosahedron are:
In other words, it is constructed from the regular icosahedron by removing its polygonal faces without changing or creating new vertices. [1] Another way is to form a regular pentagon by each of the five vertices inside of a regular icosahedron, and twelve regular pentagons intersecting each other, making a pentagram as its vertex figure. [2] [3]
Let φ be the golden ratio.The 12 points given by (0, ±1, ±φ) and cyclic permutations of these coordinates are the vertices of a regular icosahedron.Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the 8 points (±1, ±1, ±1) together with the 12 points (0, ±φ, ± 1 / φ ) and cyclic permutations of these coordinates.
The coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform truncated octahedron, t{3,4} or , also called a snub octahedron, as s{3,4} or , and seen in the compound of two icosahedra. Eight of the vertices of the dodecahedron are shared with the cube.