[math-fun] Locally isometric polyhedral embedding of flat torus
Where can I learn about, and see pictures of, polyhedral surfaces in R^3 that are locally flat (the angles at each vertex add up to 360 degrees) and have the global topology of R^2/Z^2? More specifically and concretely, how can I crease and fold a square sheet of paper [0,1]x[0,1] so that I can actually glue (t,0) to (t,1) and (0,t) to (1,t) for every t in [0,1]? Jim Propp
This isn't exactly what you asked for, but it does have some lovely pictures of what is said to be a C^1 embedding of a flat torus in R^3: < http://math.univ-lyon1.fr/~borrelli/Hevea/PNAS_version_soumise.pdf > --Dan On May 27, 2014, at 8:50 PM, James Propp <jamespropp@gmail.com> wrote:
Where can I learn about, and see pictures of, polyhedral surfaces in R^3 that are locally flat (the angles at each vertex add up to 360 degrees) and have the global topology of R^2/Z^2?
More specifically and concretely, how can I crease and fold a square sheet of paper [0,1]x[0,1] so that I can actually glue (t,0) to (t,1) and (0,t) to (1,t) for every t in [0,1]?
Fred Lunnon, "Origami torus" math-fun thread, 2009. --Michael On May 27, 2014 11:51 PM, "James Propp" <jamespropp@gmail.com> wrote:
Where can I learn about, and see pictures of, polyhedral surfaces in R^3 that are locally flat (the angles at each vertex add up to 360 degrees) and have the global topology of R^2/Z^2?
More specifically and concretely, how can I crease and fold a square sheet of paper [0,1]x[0,1] so that I can actually glue (t,0) to (t,1) and (0,t) to (1,t) for every t in [0,1]?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
See picture at https://www.dropbox.com/s/t8iqaeoe5e86ld1/solitore3.pdf and flat net at https://www.dropbox.com/s/42grmh6o3re4ulf/flattore3.pdf WFL On 5/28/14, Michael Kleber <michael.kleber@gmail.com> wrote:
Fred Lunnon, "Origami torus" math-fun thread, 2009.
--Michael On May 27, 2014 11:51 PM, "James Propp" <jamespropp@gmail.com> wrote:
Where can I learn about, and see pictures of, polyhedral surfaces in R^3 that are locally flat (the angles at each vertex add up to 360 degrees) and have the global topology of R^2/Z^2?
More specifically and concretely, how can I crease and fold a square sheet of paper [0,1]x[0,1] so that I can actually glue (t,0) to (t,1) and (0,t) to (1,t) for every t in [0,1]?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Fred's pictures are nice (and the solution they depict could well be minimal and therefore of intrinsic mathematical interest), but I'm more interested in seeing an origami torus with lots of sides that, viewed from afar, looks doughnut-like. I'm 90% sure I've seen one; in fact, I think it was made of equilateral triangles, with six meeting at each vertex. And (this part may be delusional) I think it had a degree of freedom that permitted it to swim through itself. Can anyone provide a link? Jim On Wed, May 28, 2014 at 5:40 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
See picture at https://www.dropbox.com/s/t8iqaeoe5e86ld1/solitore3.pdf and flat net at https://www.dropbox.com/s/42grmh6o3re4ulf/flattore3.pdf
WFL
On 5/28/14, Michael Kleber <michael.kleber@gmail.com> wrote:
Fred Lunnon, "Origami torus" math-fun thread, 2009.
--Michael On May 27, 2014 11:51 PM, "James Propp" <jamespropp@gmail.com> wrote:
Where can I learn about, and see pictures of, polyhedral surfaces in R^3 that are locally flat (the angles at each vertex add up to 360 degrees) and have the global topology of R^2/Z^2?
More specifically and concretely, how can I crease and fold a square sheet of paper [0,1]x[0,1] so that I can actually glue (t,0) to (t,1) and (0,t) to (1,t) for every t in [0,1]?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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A google trawl turned up some promising links. Two relevant names are Will Webber and Ron Resch. http://www.flickriver.com/photos/tactom/8471902275/ http://11011110.livejournal.com/163853.html << Will Webber here. My dissertation did discuss tori that are made from congruent triangles connected 6 at each vertex. There were two types described. The first type was a generalization a construction of Alaoglu and Giese published in the 50's. These were stacks of isohedral octahedra that were calculated to have the proper angle between the top and the bottom of the stacks so that an integer number of stacks would fit together to make a torus. The second type was folded from a plane. The arch described above is an example of this type. Much of my inspiration was the work of Ron Resch who is mentioned in a post below. When I started this research we were looking to use equilateral triangles. When I built the first model my adviser (Branko Grunbaum) bet me his pension that it did not exist. As it turned out he was right (and got to keep his pension). That particular model did not exist with equilateral triangular faces. It did however exist with isosceles faces that were almost equilateral. Once we allowed for non equilateral faces then we found many (infinite) fold patterns that yielded polyhedral tori with 6 congruent faces at each vertex. A simple continuity argument proves their existence. I used a modified bisection algorithm and a multivarible newton's method algorithm to calculate a bunch of examples. As for using equilateral triangles, I have not found one of my folded examples to work. One has isosceles triangles of side lengths 1, .99995 and .99995. Close but not quite. Bonnie Stewart describes several examples that do exist with equilateral triangles in his book "Adventures Among the Toroids." These do not fit 6 at a vertex, but are equilateral. The smallest example has 48 faces and is attributed to Kurt Schmucker in 1972.
WFL On 5/28/14, James Propp <jamespropp@gmail.com> wrote:
Fred's pictures are nice (and the solution they depict could well be minimal and therefore of intrinsic mathematical interest), but I'm more interested in seeing an origami torus with lots of sides that, viewed from afar, looks doughnut-like.
I'm 90% sure I've seen one; in fact, I think it was made of equilateral triangles, with six meeting at each vertex. And (this part may be delusional) I think it had a degree of freedom that permitted it to swim through itself.
Can anyone provide a link?
Jim
On Wed, May 28, 2014 at 5:40 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
See picture at https://www.dropbox.com/s/t8iqaeoe5e86ld1/solitore3.pdf and flat net at https://www.dropbox.com/s/42grmh6o3re4ulf/flattore3.pdf
WFL
On 5/28/14, Michael Kleber <michael.kleber@gmail.com> wrote:
Fred Lunnon, "Origami torus" math-fun thread, 2009.
--Michael On May 27, 2014 11:51 PM, "James Propp" <jamespropp@gmail.com> wrote:
Where can I learn about, and see pictures of, polyhedral surfaces in R^3 that are locally flat (the angles at each vertex add up to 360 degrees) and have the global topology of R^2/Z^2?
More specifically and concretely, how can I crease and fold a square sheet of paper [0,1]x[0,1] so that I can actually glue (t,0) to (t,1) and (0,t) to (1,t) for every t in [0,1]?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Other flat tori would be interesting, too, but especially the other maximally symmetric one, the hexagonal torus. Which has a triangulation by N equilateral triangles, 6/vertex, at least for any N = 2 x 7^k or N = 6 x 7^k, k >= 0. --Dan On May 27, 2014, at 8:50 PM, James Propp <jamespropp@gmail.com> wrote:
Where can I learn about, and see pictures of, polyhedral surfaces in R^3 that are locally flat (the angles at each vertex add up to 360 degrees) and have the global topology of R^2/Z^2?
More specifically and concretely, how can I crease and fold a square sheet of paper [0,1]x[0,1] so that I can actually glue (t,0) to (t,1) and (0,t) to (1,t) for every t in [0,1]?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
There are many tube-like accordion folds. If any of them permit any bending, they could be bent into a ring that would almost certainly permit the toroidal rotation that Jim remembers from that cubical toy. (I think we may have one.) On Wed, May 28, 2014 at 2:17 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Other flat tori would be interesting, too, but especially the other maximally symmetric one, the hexagonal torus. Which has a triangulation by N equilateral triangles, 6/vertex, at least for any N = 2 x 7^k or N = 6 x 7^k, k >= 0.
--Dan
On May 27, 2014, at 8:50 PM, James Propp <jamespropp@gmail.com> wrote:
Where can I learn about, and see pictures of, polyhedral surfaces in R^3 that are locally flat (the angles at each vertex add up to 360 degrees) and have the global topology of R^2/Z^2?
More specifically and concretely, how can I crease and fold a square sheet of paper [0,1]x[0,1] so that I can actually glue (t,0) to (t,1) and (0,t) to (1,t) for every t in [0,1]?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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The Bellows Theorem* (formerly Conjecture) states that for any closed, triangulated polyhedron in 3-space that flexes, its volume remains constant. This would impose severe constraints on a circular concertina (its polyhedron could always be triangulated), but maybe it's possible. --Dan _________________________________________________________________ * http://www.emis.ams.org/journals/BAG/vol.38/no.1/b38h1csw.ps.gz On May 28, 2014, at 11:37 AM, Allan Wechsler <acwacw@gmail.com> wrote:
There are many tube-like accordion folds. If any of them permit any bending, they could be bent into a ring that would almost certainly permit the toroidal rotation that Jim remembers from that cubical toy. (I think we may have one.)
I have only just worked out (I think) what was meant by "swim through itself" here, reminding me of this animation of a 7-link rotating smoke-ring robot with just one nontrivial degree of freedom, which I don't recall having mentioned on math-fun before: https://www.dropbox.com/s/aeo6rxtc5j4p291/sevenring.gif Open in a browser for the animation to execute: it should do so in situ, with no need to downoad first. Fred Lunnon On 5/28/14, Dan Asimov <dasimov@earthlink.net> wrote:
The Bellows Theorem* (formerly Conjecture) states that for any closed, triangulated polyhedron in 3-space that flexes, its volume remains constant. This would impose severe constraints on a circular concertina (its polyhedron could always be triangulated), but maybe it's possible.
--Dan
_________________________________________________________________ * http://www.emis.ams.org/journals/BAG/vol.38/no.1/b38h1csw.ps.gz
On May 28, 2014, at 11:37 AM, Allan Wechsler <acwacw@gmail.com> wrote:
There are many tube-like accordion folds. If any of them permit any bending, they could be bent into a ring that would almost certainly permit the toroidal rotation that Jim remembers from that cubical toy. (I think we may have one.)
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Very nice! (I think it takes 8 tetrahedra to form a flexible ring like that if they are all *regular*, according to unpublished work of J. Gerver.) --Dan On May 28, 2014, at 12:43 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I have only just worked out (I think) what was meant by "swim through itself" here, reminding me of this animation of a 7-link rotating smoke-ring robot with just one nontrivial degree of freedom, which I don't recall having mentioned on math-fun before:
https://www.dropbox.com/s/aeo6rxtc5j4p291/sevenring.gif
Open in a browser for the animation to execute: it should do so in situ, with no need to downoad first.
participants (5)
-
Allan Wechsler -
Dan Asimov -
Fred Lunnon -
James Propp -
Michael Kleber