Fractal Bagatelle
in [Math]
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From: Eric Baird on 15 May 2010 22:56 Hi People! There's a 3D fractal I've been wondering about for a while now� I haven't actually plotted it myself, and was wondering if anybody's seen it elsewhere. Here's the idea: You release a ball-bearing so that it's allowed to roll down an inclined plane, and plot the position of the ball, against time, against a range of starting positions along the plane that all have the same height. For each position, the ball rolls down the plane in a straight line, in exactly the same way. That should give a 3D plot that's a simple �sheet� � it'll have a curve because of the ball's acceleration under gravity, but there'll be no //intrinsic// curvature. Now hammer a nail into the inclined plane. Things get complicated � if the ball misses the nail altogether, then the nail has no effect. If the ball's path is close enough to the nail to strike it a glancing blow, the ball bounces off to one side or the other, producing a spray of vectors around the nail (like the effect you get when you aim a jet of water at a lamp-post). So the resulting surface has a sort of bulgy growth protruding from it. If the ball-bearing hits the nail closer to the centre, it might bounce back a little way up the plane, before descending again with a slightly greater sideways deflection that results in it hitting the nail a second time before being thrown off to one side. If we increase the accuracy of the hit, the bearing should theoretically bounce three, four, five, six, or an infinite number of times before leaving the nail. So the plotted sheet should have a sort of protruding �spine� that corresponds to the nail's spatial position, that generates an infinite succession of new lobes. If you now hammer in an //array// of nails in a �bagatelle board�-type configuration, it gets really complex. The lobes now have side-lobes, and the ball has multiple ways of reaching the other nails, either by being dropped directly onto them, or by getting there by bouncing off another higher nail. The different paths to a strikepoint have different results. And there are nails directly underneath other nails, that can only be reached indirectly. So I was thinking, it sounds like quite a fun surface. If you wanted to get more complex, I guess you could add additional dimensions for the ratio between nail spacing and ball radius, ball drop-height, and maybe even an vector for the ball. But I'd personally be happy just to see a simple "3D surface" plot of one representative example. =Erk= (Eric Baird) http://www.scribd.com/Eric_Baird |