Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a compound of five octahedra, which comes under our definition of stellated icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the " golden section". By reciprocation, this leads to an octahedron circumscribed about an icosahedron. Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. Five opposing tetrahedra (not shown) can also be inscribed.Īs two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair).
1 / 4(5 + √ 5) V = 1 / 4(14 ϕ + 8) a 3 Relation to the regular tetrahedron įive tetrahedra inscribed in a dodecahedron. Thus, the difference in volume between the encompassing regular dodecahedron and the enclosed cube is always one half the volume of the cube times ϕ.įrom these ratios are derived simple formulas for the volume of a regular dodecahedron with edge length a in terms of the golden mean: The ratio of a regular dodecahedron's volume to the volume of a cube embedded inside such a regular dodecahedron is 1 : 2 / 2 + ϕ, or 1 + ϕ / 2 : 1, or (5 + √ 5) : 4.įor example, an embedded cube with a volume of 64 (and edge length of 4), will nest within a regular dodecahedron of volume 64 + 32 ϕ (and edge length of 4 ϕ − 4). The ratio of the edge of a regular dodecahedron to the edge of a cube embedded inside such a regular dodecahedron is 1 : ϕ, or ( ϕ − 1) : 1. In fact, five cubes may overlap and interlock inside the regular dodecahedron to result in the compound of five cubes. compared with 2.181.), which ratio is approximately 3.512 461 179 75, or in exact terms: 3 / 5(3 ϕ + 1) or (1.8 ϕ + 0.6).Ī cube can be embedded within a regular dodecahedron, affixed to eight of its equidistant vertices, in five different positions. 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%).Ī regular dodecahedron with edge length 1 has more than three and a half times the volume of an icosahedron with the same length edges (7.663. A regular dodecahedron has 12 faces and 20 vertices, whereas a regular icosahedron has 20 faces and 12 vertices. The dodecahedron and icosahedron are dual polyhedra. The regular dodecahedron has icosahedral symmetry I h, Coxeter group, order 120, with an abstract group structure of A 5 × Z 2. The stellations of the regular dodecahedron make up three of the four Kepler–Poinsot polyhedra.Ī rectified regular dodecahedron forms an icosidodecahedron. 
The regular dodecahedron is the third in an infinite set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron.
#CUBE 20TH PROJECTOR CENTRAL FULL#
The diagonal element counts are the ratio of the full Coxeter group H 3, order 120, divided by the order of the subgroup with mirror removal. Here is the configuration expanded with k-face elements and k-figures. It is represented by the Schläfli symbol It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals). 2023, DOI:10.1038/s44166-9 Ī regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. Crystal structure of Co20L12 dodecahedron reported by Kai Wu, Jonathan Nitschke and co-workers at University of Cambridge in Nat.
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