- published: 13 Feb 2015
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In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα, dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces, but usually a regular dodecahedron is meant: a Platonic solid. It is composed of 12 regular pentagonal faces, with three meeting at each vertex, and is represented by the Schläfli symbol {5,3}. It has 20 vertices and 30 edges. Its dual polyhedron is the icosahedron, with Schläfli symbol {3,5}.
A large number of other (nonregular) polyhedra also have 12 sides, but are given other names. Other dodecahedra include the hexagonal bipyramid and the rhombic dodecahedron.
If the edge length of a regular dodecahedron is a, the radius of a circumscribed sphere (one that touches the dodecahedron at all vertices) is
and the radius of an inscribed sphere (tangent to each of the dodecahedron's faces) is
while the midradius, which touches the middle of each edge, is
These quantities may also be expressed as
where φ is the golden ratio.
Note that, given a regular pentagonal dodecahedron of edge length one, ru is the radius of a circumscribing sphere about a cube of edge length φ, and ri is the apothem of a regular pentagon of edge length φ.
Out here at the timeline at lightspeed I travel although my thinking's barely moving
I'm recharging my senses
Awaiting tomorrow
One more place in time
Just a flash and I'll be back on earth again
Building dreams for free men
Just a flash and I'll be part of life again
Anywhen
Out here at the timeline
The dodecahedron retains the history we're living
A second home for the detached with room for reflection
Out of space and time
Just a flash and I'll be back on earth again
Building dreams for free men
Just a flash and I'll be part of life again
Anywhen
I penetrate the aerosphere
My destination's drawing near
Through halls of myth and dreamy light
The noise, the sand, the building site
Just a flash and I'll be back on earth again
Building dreams for free men
Just a flash and I'll be part of life again