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If the edge length of a regular dodecahedron is , the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices), the radius of an inscribed sphere (tangent to each of the regular dodecahedron's faces), and the midradius (one that touches the middle of each edge) are: [21] =, =, =. Given a regular dodecahedron ...
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 ...
In 4-dimensional geometry, the dodecahedral bipyramid is the direct sum of a dodecahedron and a segment, {5,3} + { }. Each face of a central dodecahedron is attached with two pentagonal pyramids, creating 24 pentagonal pyramidal cells, 72 isosceles triangular faces, 70 edges, and 22 vertices.
It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. [1] It is given a Schläfli symbol sr{ 5 ⁄ 2 ,5}, as a snub great dodecahedron . Cartesian coordinates
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron. As a parallelohedron, the rhombic dodecahedron can be used to tesselate its copies in space creating a rhombic dodecahedral honeycomb.
Dodecahedrane is a chemical compound, a hydrocarbon with formula C 20 H 20, whose carbon atoms are arranged as the vertices (corners) of a regular dodecahedron. Each carbon is bound to three neighbouring carbon atoms and to a hydrogen atom. This compound is one of the three possible Platonic hydrocarbons, the other two being cubane and ...
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.
Let be the golden ratio.The 12 points given by (,,) 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 points (,,) together with the points (, /,) and cyclic permutations of these coordinates.