I'm not sure I understand the question about the critical angle. Certainly you can get some overlap even if there are no contiguous sectors (hard right or hard left turn). But perhaps you are talking about total "swerve", essentially the difference in arc length between the two sides. My intuition is that unless the swerve exceeds pi, you can't get any overlap. On Sun, Apr 22, 2018 at 12:39 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
Replacing 3 Pi/2 in my argument with Pi+epsilon still gives overlap, so the bound is at most Pi+epsilon for arbitrarily small epsilon.
I don't think Pi by itself is sufficient and I think this can be shown formally.
On Sun, Apr 22, 2018 at 4:36 PM, James Propp <jamespropp@gmail.com> wrote:
There are two fun problems here. One is Allan’s question about the maximum overlap when a full disk is dissected. Another is, what is the critical angle theta such that, for all sectors exceeding theta, positive overlap can be achieved by some dissection/recomposition? Is it 3 Pi / 2?
Jim Propp
On Sunday, April 22, 2018, Tomas Rokicki <rokicki@gmail.com> wrote:
Render three-quarters of a circle, using 3Pi/2 edges. Extend each flat edge straight out by Pi/8; this gives an overlap of Pi/8 x Pi/8.
This can almost certainly be improved by extending by one eighth of a circle using the other point as a radius, rather than extending straight out, but I am too lazy to do the math to determine if this is actually an improvement.
This is almost certainly not the best one can do and I am interested in Allan's question about what the maximum possible overlap might be.
-tom
On Sat, Apr 21, 2018 at 3:03 AM, James Propp <jamespropp@gmail.com> wrote:
A picture, please?
Jim
On Friday, April 20, 2018, Allan Wechsler <acwacw@gmail.com> wrote:
Oh, wait. I see it. Tomas is right, of course. I wonder what the maximum self-overlap is. I can see how to get pi^2 / 64.
On Fri, Apr 20, 2018 at 10:18 PM, Allan Wechsler <acwacw@gmail.com
wrote:
Really? You only have 2pi time units to do it in. Also ... I
want a clean
intersection: the overlap has to have area.
Pics or it didn't happen.
On Fri, Apr 20, 2018 at 10:12 PM, Tomas Rokicki < rokicki@gmail.com
wrote:
> The answer is yes. > > On Fri, Apr 20, 2018 at 8:21 PM Allan Wechsler <
acwacw@gmail.com> wrote:
> > > Guys! Guys! Here is a question! You'll see, it will end with a question > > mark and everything. > > > > Can such a ribbon self-intersect? > > > > I'm guessing the answer is no, but I can't see a proof path. > > > > On Fri, Apr 20, 2018 at 1:28 PM, James Propp < jamespropp@gmail.com> > wrote: > > > > > Perfect! Thanks. > > > > > > I guess the theorem here is that the area of such a ribbon is equal to > > half > > > the sum of the lengths of the two non-straight sides. > > > > > > Come to think of it, this is just a consequence of what’s-his-name’s > > > theorem about the area swept out by a line segment that moves > > perpendicular > > > to itself. The only nonobvious step is relating the distance traveled > by > > > the midpoint of the segment to the distances traveled by the > endpoints of > > > the segment. > > > > > > Jim > > > > > > On Friday, April 20, 2018, Allan Wechsler <acwacw@gmail.com
wrote:
> > > > > > > Maybe what you are looking for is this. The "ribbon" has two > > curvilinear > > > > edges. From any point A on one edge, draw a perpendicular line; it > will > > > > turn out to be perpendicular to the other edge as well. (By > > > "perpendicular" > > > > I mean "perpendicular to the tangent at that point".) > > > > > > > > On Fri, Apr 20, 2018 at 12:52 PM, James Propp < jamespropp@gmail.com > > > > > > wrote: > > > > > > > > > Thanks, Allan! The relation 1/a(t) + 1/b(t) is close to what I > > wanted. > > > > But > > > > > it requires a time-parametrization. Is there a way to characterize > > such > > > > > shapes directly? > > > > > > > > > > Jim > > > > > > > > > > On Thursday, April 19, 2018, Allan Wechsler < acwacw@gmail.com
> > wrote: > > > > > > > > > > > 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. > > > > > > > > ----- > > > > > > > > ----- > > > > > > > > > > > > > > > > _______________________________________________ > > > > > > > > math-fun mailing list > > > > > > > > math-fun@mailman.xmission.com > > > > > > > > https://mailman.xmission.com/c > gi-bin/mailman/listinfo/math-fun > > > > > > > > > > > > > > > _______________________________________________ > > > > > > > math-fun mailing list > > > > > > > math-fun@mailman.xmission.com > > > > > > > https://mailman.xmission.com/ cgi-bin/mailman/listinfo/math- f > un > > > > > > > > > > > > > _______________________________________________ > > > > > > math-fun mailing list > > > > > > math-fun@mailman.xmission.com > > > > > > https://mailman.xmission.com/ cgi-bin/mailman/listinfo/math- fun > > > > > > > > > > > _______________________________________________ > > > > > math-fun mailing list > > > > > math-fun@mailman.xmission.com > > > > > https://mailman.xmission.com/ cgi-bin/mailman/listinfo/math- fun > > > > > > > > > _______________________________________________ > > > > math-fun mailing list > > > > math-fun@mailman.xmission.com > > > > https://mailman.xmission.com/ cgi-bin/mailman/listinfo/math- fun > > > > > > > _______________________________________________ > > > math-fun mailing list > > > math-fun@mailman.xmission.com > > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math- fun > > > > > _______________________________________________ > > math-fun mailing list > > math-fun@mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math- fun > > > -- > -- http://cube20.org/ -- http://golly.sf.net/ -- > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
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