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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