Re: [math-fun] rolling a ball along a fractal
Here's another way to clarify my question: Consider a level-zero snowflake curve (a path of length 1) and a level-one snowflake curve (a path consisting of four segments of length 1/3). If we roll a ball along the former, (with its south pole starting facing downward), the final orientation of the ball is NOT THE SAME as if we roll the ball along the latter (also with the south pole starting facing downward). (Actually, I suppose that if one chooses the radius cleverly, the orientation of the ball *could* be the same, but for generic radii, the ball will be in a different orientation.) If we replace the level-one curve by a level-two curve (consisting of sixteen segments of length 1/9), the final orientation of the ball could once again change. And so on. Is there any reason to think that this sequence of orientations of the ball should converge to some particular orientation in the configuration space of the ball (where we mod out by translation --- all we care about is rotation --- so we're really looking at the configuration space of a sphere)? And even if it does converge, is there any reason to think that a different sequence of polygonal approximations to the snowflake curve would give the same limit? Hopefully this is clear. So this I why I don't think the Pearse references are relevant. But thanks anyway to Mike for sending his message and "getting the ball rolling", as it were. :-) Jim Propp
On Mon, Aug 4, 2008 at 3:01 PM, James Propp <jpropp@cs.uml.edu> wrote:
Is there any reason to think that this sequence of orientations of the ball should converge to some particular orientation in the configuration space of the ball (where we mod out by translation --- all we care about is rotation --- so we're really looking at the configuration space of a sphere)?
This seems easy - since the limit of the length is infinite, you could only get a fixed point as you go from level n to n+1 if there's some very nice (rational) relationship between the size of the ball and the length of the level n snowflake. In other words, I think the answer to all your questions are "no, it doesn't make sense to talk about rolling a ball on a fractal". Am I missing something obvious? --Joshua Zucker
Ah, I see. Then I think that you will approach a limit; consider a fractal like the Koch snowflake with dimension between 1 and 2. As the features of the fractal get very small compared to the circle, rolling the ball over it starts to approximate the convex hull of the fractal. On Mon, Aug 4, 2008 at 3:01 PM, James Propp <jpropp@cs.uml.edu> wrote:
Here's another way to clarify my question:
Consider a level-zero snowflake curve (a path of length 1) and a level-one snowflake curve (a path consisting of four segments of length 1/3). If we roll a ball along the former, (with its south pole starting facing downward), the final orientation of the ball is NOT THE SAME as if we roll the ball along the latter (also with the south pole starting facing downward).
(Actually, I suppose that if one chooses the radius cleverly, the orientation of the ball *could* be the same, but for generic radii, the ball will be in a different orientation.)
If we replace the level-one curve by a level-two curve (consisting of sixteen segments of length 1/9), the final orientation of the ball could once again change.
And so on.
Is there any reason to think that this sequence of orientations of the ball should converge to some particular orientation in the configuration space of the ball (where we mod out by translation --- all we care about is rotation --- so we're really looking at the configuration space of a sphere)?
And even if it does converge, is there any reason to think that a different sequence of polygonal approximations to the snowflake curve would give the same limit?
Hopefully this is clear. So this I why I don't think the Pearse references are relevant. But thanks anyway to Mike for sending his message and "getting the ball rolling", as it were. :-)
Jim Propp
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
The difference between rolling the ball along one edge of the polygonal path, and the 4 smaller edges of the next iteration, should be O(edge^2). Even after multiplying by the number of edges, this is O(edge). This sum converges. So we'd expect the sphere-rolling-along-snowflake-path to converge. Rich -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Mike Stay Sent: Monday, August 04, 2008 4:07 PM To: math-fun Subject: Re: [math-fun] rolling a ball along a fractal Ah, I see. Then I think that you will approach a limit; consider a fractal like the Koch snowflake with dimension between 1 and 2. As the features of the fractal get very small compared to the circle, rolling the ball over it starts to approximate the convex hull of the fractal. On Mon, Aug 4, 2008 at 3:01 PM, James Propp <jpropp@cs.uml.edu> wrote:
Here's another way to clarify my question:
Consider a level-zero snowflake curve (a path of length 1) and a level-one snowflake curve (a path consisting of four segments of length 1/3). If we roll a ball along the former, (with its south pole starting facing downward), the final orientation of the ball is NOT THE SAME as if we roll the ball along the latter (also with the south pole starting facing downward).
(Actually, I suppose that if one chooses the radius cleverly, the orientation of the ball *could* be the same, but for generic radii, the ball will be in a different orientation.)
If we replace the level-one curve by a level-two curve (consisting of sixteen segments of length 1/9), the final orientation of the ball could once again change.
And so on.
Is there any reason to think that this sequence of orientations of the ball should converge to some particular orientation in the configuration space of the ball (where we mod out by translation --- all we care about is rotation --- so we're really looking at the configuration space of a sphere)?
And even if it does converge, is there any reason to think that a different sequence of polygonal approximations to the snowflake curve would give the same limit?
Hopefully this is clear. So this I why I don't think the Pearse references are relevant. But thanks anyway to Mike for sending his message and "getting the ball rolling", as it were. :-)
Jim Propp
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
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James Propp -
Joshua Zucker -
Mike Stay -
Schroeppel, Richard