22 Apr
2018
22 Apr
'18
10:37 a.m.
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 > > > >> > > > > > > > > > > > _______________________________________________ > > > 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 >