Russ, I don't get this at all. Two points: 1) There are an infinite number of ways that a line can be parallel to a plane; there is exactly one way it can be perpendicular to the plane. Is that the point?
2) The degree of orientation around the X and the Y axises don't have anything to do with each other. As far as the random distribution is concerned, you just pick a random number out of 360 degrees for the X axis orientation (the horizontal plane), then pick another random number of 360 for the Y axis orientation (the vertical plane). The random orientation out of the vertical plane determines to what degree the stick is parallel / perpendicular to the "ground," while the random orientation of the horizontal plane determines how short or long the stick appears, when viewed from the side. But there is absolutely no relationship between one orientation and the other. With both accounted for, you can have your stick pointing in any direction within a sphere. In other words, we won't* *see more oriented towards the horizontal. I suppose I could prove this to you with a few digital photos, but that just seems silly. (Maybe this image<http://cache2.asset-cache.net/xc/200265210-003.jpg?v=1&c=IWSAsset&k=2&d=F5B5107058D53DF5C79C86B68F684E92780EF085A319427220B5A96DDEC0522877997512DFCEB0B4>will help visualize it.) Also, to comment further on the source material, I'm pretty sure the only regularity in terms of which face is up (R or S) when I flip a tennis racket is determined by the bias of whatever spin I apply. There is no physical principle here - at least not relating the vertical and horizontal spin in a vacuum - and I can end up with either face pointing up. Or neither. It is harder to do flip without spin of course, due to air resistance, but it is still quite possible. I could also apply *more *spin to get one or the other. I can assure you that I have done this many times with a ping-pong racket. If Ashbaugh, Chicone, & Cushman 1991 are properly summarized here, then I would say they are full of it. Perhaps we should write a paper and call them silly buggers? (Perhaps it was intended for the April Fool's edition of the Journal of Dynamics and Differential Equations? I know math guys tend to have a pretty good sense of humor.) Cheers, Ted On Sat, Jul 17, 2010 at 8:10 PM, ERIC P. CHARLES <e...@psu.edu> wrote: > Russ, > This seems very weird to me (as, of course, it is intended to). First off, > I'm not sure it is an "explanation" any more then a "proof by definition". > Second, at least in the case of a 2D snapshot, there are just as many 3D > configurations that appear perfectly vertical as appear perfectly > horizontal. > > I'll have to meditate more on the more general case. > > Eric > > > On Sat, Jul 17, 2010 07:28 PM, *Russ Abbott <russ.abb...@gmail.com>*wrote: > > I just ran across > this<http://plato.stanford.edu/entries/mathematics-explanation/>. > (Call it the "horizontal force.") > > There appear to be physical explanations that are non-causal. Suppose that > a bunch of sticks are thrown into the air with a lot of spin so that they > twirl and tumble as they fall. We freeze the scene as the sticks are in free > fall and find that appreciably more of them are near the horizontal than > near the vertical orientation. Why is this? The reason is that there are > more ways for a stick to be the horizontal than near the vertical. To see > this, consider a single stick with a fixed midpoint position. There are many > ways this stick could be horizontal (spin it around in the horizontal > plane), but only two ways it could be vertical (up or down). This asymmetry > remains for positions near horizontal and vertical, as you can see if you > think about the full shell traced out by the stick as it takes all possible > orientations. This is a beautiful explanation for the physical distribution > of the sticks, but what is doing the explaining are broadly geometrical > facts that cannot be causes. > > > -- Russ Abbott > ______________________________________ > > Professor, Computer Science > California State University, Los Angeles > > cell: 310-621-3805 > Google vo! ice: 424-2Blue2 > > blog: http://russabbott.blogspot.com/ > vita: http://sites.google.com/site/russabbott/ > ______________________________________ > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org > > Eric Charles > > Professional Student and > Assistant Professor of Psychology > Penn State University > Altoona, PA 16601 > > > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org > -- Ted Carmichael, PhD Complex Systems Institute Department of Software and Information Systems College of Computing and Informatics 343-A Woodward Hall UNC Charlotte Charlotte, NC 28223 teds...@gmail.com tdcar...@uncc.edu Phone: 704-492-4902
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