It seems to me that, in the limit, we have a behavior something like this: We have a unit line segment AB moving in the plane. Each of its endpoints is moving perpendicular to the line, toward the same side of the line, at speeds that add up to 1. Subject to that constraint, their speeds are an arbitrary function of time. Say the speed of point A is given by f(t); then point B is moving in the same direction at speed 1-f(t). Because the speeds of the endpoints can differ, the line can gradually change orientation; its angle (in radians) is changing at a speed 1/2 - f(t). It sweeps out area at a constant speed of 1/2. The curvatures of the curves traced out by A and B are related by the equation 1/a + 1/b = 1. The whole process continues until t = 2pi, so the total area swept out is pi. On Thu, Apr 19, 2018 at 9:26 AM, Michael Collins <mjcollins10@gmail.com> wrote:
I think (1) means that we have an infinite sequence of sets S_k where S_k is composed of k wedges (joined only along full edges), each with angle 2*pi/k; the limit is just the set of points p such that p is contained in all but finitely many S_k. You can definitely get an interesting collection of shapes this way.
On Wed, Apr 18, 2018 at 10:21 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I'm trying to guess what RWG meant without peeking at his drawings.
In order to make Jim Propp's statement exact, I would have to make precise
1) what "dissect and reassemble" mean
and
2) what "converges" to a 1-by-pi rectangle means.
A typical meaning for 1): For subsets A, B of R^2, to dissect A and reassemble it to B means that there is a partition
A = X_1 + ... + X_n
of A as a finite disjoint union, such that there exist isometries
f_1, ..., f_n of R^2
such that
B = f_1(X_1) + ... + f_n(X_n)
forms a partition of B as a finite disjoint union.
* * *
One meaning for 2) could be in the sense of Hausdorff distance between compact sets in the plane. The only problem I see here is that if strict partition are used in 1) as above, then the resulting rectangle B will not be compact, as it will not contain all of its boundary. I have complete faith that appropriate hand-waving will not incur the wrath of the math gods.
—Dan
----- Jim Propp wrote:
If you dissect a unit disk radially into a large number of equal wedges, it’s well known that you can reassemble them to form a shape that in the limit converges to a 1-by-pi rectangle.
RWG wrote: ----- gosper.org/picfzoom.gif gosper.org/semizoom.gif --rwg I don't see how to get anything other than allowing unequal wedges. ----- -----
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