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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 ...
A tetartoid (also tetragonal pentagonal dodecahedron, pentagon-tritetrahedron, and tetrahedric pentagon dodecahedron) is a dodecahedron with chiral tetrahedral symmetry (T). Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the ...
The shape of the "Crystal Dome" used in the popular TV game show The Crystal Maze was based on a pentakis dodecahedron. In Doctor Atomic, the shape of the first atomic bomb detonated in New Mexico was a pentakis dodecahedron. In De Blob 2 in the Prison Zoo, domes are made up of parts of a Pentakis Dodecahedron. These Domes also appear whenever ...
Great stellated dodecahedron { 5 / 2 ,3} ( 5 / 2 . 5 / 2 . 5 / 2 ) arccos ( √ 5 / 5 ) 63.435° Great icosahedron {3, 5 / 2 } (3.3.3.3.3) / 2 arccos ( √ 5 / 3 ) 41.810° Quasiregular polyhedra (Rectified regular) Tetratetrahedron: r{3,3} (3.3.3.3) arccos (- 1 / 3 ) 109.471 ...
There are many relations among the uniform polyhedra. [1] [2] [3] Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron.Others share the same vertices and edges as other polyhedron.
A vertex configuration can also be represented as a polygonal vertex figure showing the faces around the vertex. This vertex figure has a 3-dimensional structure since the faces are not in the same plane for polyhedra, but for vertex-uniform polyhedra all the neighboring vertices are in the same plane and so this plane projection can be used to visually represent the vertex configuration.
The blue vertices lie at (± 1 / ϕ , 0, ±ϕ) and form a rectangle on the xz-plane. (The red, green and blue coordinate triples are circular permutations of each other.) The distance between adjacent vertices is 2 / ϕ , and the distance from the origin to any vertex is √ 3. ϕ = 1 + √ 5 / 2 is the golden ratio.
The compound of small stellated dodecahedron and great dodecahedron is a polyhedron compound where the great dodecahedron is internal to its dual, the small stellated dodecahedron. This can be seen as one of the two three-dimensional equivalents of the compound of two pentagrams ({10/4} " decagram "); this series continues into the fourth ...