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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 ...
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.
Area#Area formulas – Size of a two-dimensional surface; Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities
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.
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 ...
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 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 ...
That is, the 120 vertices of the polyhedron may be placed in one-to-one correspondence with the 5! permutations on five elements, in such a way that the three neighbors of each vertex are the three permutations formed from it by swapping the first two elements or circularly shifting (in either direction) the last four elements. [5]