"Shel Kaphan" <sjk@kaphan.org> asks:

A long time ago JHC gave (on math-fun) a nice way to see that the central angle of the icosahedron is atan(2) but I can't find or reproduce it :-( Does anyone still have that by any chance?

I don't know if this is it, but I think it's nice. Trim three index cards to the golden ratio (or let 3x5 be close enough) and cut a centered slit as long as the short edge parallel to the long edge of each. Then you can place one card through the slit of another, at right angles, so that their centers coincide. If you similarly place the third card through the slit of the second, and arrange the first to pierce the third ouroborromeanly, then the vertices of the cards will be the vertices of a regular icosahedron. /\ / \ ____________ \ \ \ \ \ \_________ \ /\ \ / / \ / \ \ / / \/ \_____\ / Istimirant stella! / / / / / / / / /_____\ /________/ \ \ / \ \ /________\ \ / \ / \/ Then show that the tangents of the central angles of a golden rectangle are 2 and -2. Dan Hoey@AIC.NRL.Navy.Mil