At 01:41 PM 12/6/02 -0800, R. William Gosper wrote:
Sucker bet: Which is greater, the edge of a regular icosahedron or its circumradius? Answer: e/r = sqrt(2-2/sqrt(5)) . I had to stare at a picture to believe this. --rwg
The near equality of these two numbers has tripped up many an amateur geometer (and some professionals too). If the two numbers were equal, five non-interpenetrating regular tetrahedra could just be arranged around a common edge. In fact, the five barely fail to go all the way around the edge. So if you try to make an icosahedron with edges equal to its outradius, the edges will be too tight: they have to be a little longer than the outradius in order for everything to meet. Bonnie M. Stewart recounts how he was tricked by the same false identity at the beginning of _Adventures Among the Toroids_. If the identity were true, space could be tiled with tetrahedra. But if you try, you find yourself having to fold the tetrahedra toward each other in 4-space in order to force the last two faces to touch. The result is a regular 4-dimensional polytope. Because the identity is so close to true, very little folding is necessary, so the "curvature" of the resulting polyhedron is gentle, and it takes an astonishing 600 tetrahedra to complete the figure. -A