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A regular dodecahedron or pentagonal dodecahedron [notes 1] is a dodecahedron composed of regular pentagonal faces, three meeting at each vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron ...
In geometry, a dodecahedron (from Ancient Greek δωδεκάεδρον (dōdekáedron); from δώδεκα (dṓdeka) 'twelve' and ἕδρα (hédra) 'base, seat, face') or duodecahedron [1] is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid.
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.
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 ...
The truncated small stellated dodecahedron can be considered a degenerate uniform polyhedron since edges and vertices coincide, but it is included for completeness. Visually, it looks like a regular dodecahedron on the surface, but it has 24 faces in overlapping pairs. The spikes are truncated until they reach the plane of the pentagram beneath ...
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.
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.