Dimensions
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
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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.
Read more about this topic: Dodecahedron
Famous quotes containing the word dimensions:
“Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth. They break, chop, and impoverish it.”
—Ralph Waldo Emerson (1803–1882)
“It seems to me that we do not know nearly enough about ourselves; that we do not often enough wonder if our lives, or some events and times in our lives, may not be analogues or metaphors or echoes of evolvements and happenings going on in other people?—or animals?—even forests or oceans or rocks?—in this world of ours or, even, in worlds or dimensions elsewhere.”
—Doris Lessing (b. 1919)
“Why is it that many contemporary male thinkers, especially men of color, repudiate the imperialist legacy of Columbus but affirm dimensions of that legacy by their refusal to repudiate patriarchy?”
—bell hooks (b. c. 1955)