Consider also the following lovely fact: choose a point on the unit sphere uniformly at random. It's z-coordinate is uniformly distributed in the interval [-1,+1]! To put it differently, you can generate a random point (x,y,z) like this: choose z uniformly in [-1,+1] choose theta uniformly in [0,2pi] set x = sqrt(1-z^2) cos theta, y = sqrt(1-z^2) sin theta This is only true for the 3-dimensional sphere, of course! - Cris On Jul 5, 2014, at 9:31 AM, "David Wilson" <davidwwilson@comcast.net> wrote:
At first I'm thinking, "why wouldn't that just be pi/2?" so I had to stare at the problem until I understood it correctly. It then became apparent that the answer must be greater than pi/2, and understanding the problem, I probably could have solved it after much torturous effort via standard analytic geometry and calculus, perhaps within my remaining lifetime. The answer is quite nice, though.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Friday, July 04, 2014 11:50 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Gary Antonick is edging away from the following bonus puzzle
because he thinks only Gene Salamin can solve it. Can any of you? (who didn't already know the answer)
For a ceremonial match, a soccer ball is colored with a map of the Globe instead of a traditional pattern. At the start of the match, the ball rests on its south pole, with its lat 0, long 0 point aimed due east. At the end of the match, the ball lies forgotten on the pitch, in a truly random orientation. For any orientation, there will always be an axis through the center about which a single rotation will restore the original north up, 0,0 east orientation. What is the (surprising) expected magnitude of this required rotation?
And yet last week Gary ran (with great satisfaction) Julian's tympanhedron, which, to my knowledge, has *never* been found when posed as a puzzle. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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