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. ----- -----