Addendum

I was all wet in my argument that the dodecahedron is more spherelike than the icosahedron. Physicist F. C. Frank was the first to inform me that although the dodecahedron is closer in both volume and surface area to a sphere in which both are inscribed, the icosahedron is closer in both volume and area to a sphere that the two Platonic solids circumscribe. If you stuff each solid until it expands to make a sphere, you need less stuffing (in proportion to volume) to make the dodecahedron spherical. But if you carve away portions of each solid until you have a sphere, you carve away a smaller proportion of the volume of the icosahedron. Thus with respect to the insphere and circumsphere there is a standoff concerning which is the most spherical.

However, as Frank, Gary Goodman, Tom McCormick, Robert Dewar, and others pointed out, the sphere is well known for its property of having the greatest volume per surface area of any other solid. If this property is taken as the essence of sphericity, the icosahedron comes out ahead. There is nothing deceptive about our intuition when we observe the five Platonic solids and conclude that the tetrahedron looks the least like a sphere and the icosahedron looks the most like a sphere.

A definitive paper on the question, "Platonic Sphericity," by Norman T. Gridgeman, of Ottawa, was published in the Journal of Recreational Mathematics in 1973. Gridgeman upholds the commonsense view that the icosahedron is the most spherical, then goes on to discuss less obvious ways to measure "sphericity." He thinks Plato could have been influenced by knowing that the dodecahedron is closer to the circumsphere, and that this may have been augmented by the fact that the dodecahedron's pentagonal faces are closer to circles than the triangular faces of the icosahedron. Perhaps Plato was also influenced, Gridgeman speculates, by the correspondence between the dodecahedron's 12 sides and the 12 signs of the zodiac.

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