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:
“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
—Thomas Jefferson (1743–1826)
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)
“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)