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/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
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