[math-fun] Two topological visualization questions.
Since there's only one embedding of a circle in R^3 up to homotopy, there's an embedding of a mobius strip in R^3 where the edge is a geometric perfect circle. But I find myself unable to visualize such a thing. Has anyone seen a 3-d model of this surface? Second-best thing would be a graphic of such a thing, preferably one that you could rotate in 3 dimensions. I'd also like to better visualize Boy's surface, or any other immersion of RP^2 in R^3. It would also be interesting to have an insight into why immersing a Klein bottle in R^3 is easy, while immersing RP2 is "hard". I don't know of any formal sense in which this is true, but apparently Boy came up with this surface when challenged by Hilbert to prove that immersing RP^2 in R^3 was impossible. Also, are these two questions related? That is, can you immerse a mobius strip in R^3 in such a way that the boundary is a geometric circle, and that the union of this mobius strip and a disk with the same boundary is still an immersion (of RP^2 in R^3)? Andy Latto andy.latto@pobox.com -- Andy.Latto@pobox.com
I'm confused by there first sentence ("there's only one embedding of a circle in R^3 up to homotopy"), since a knot isn't homotopic to an unknot. But I think I understand and sympathize a lot of what follows; in particular, I'm pretty sure Klein bottles are easier to grok than Boy's surface for nearly everybody. I don't know whether this as a mathematical question or a psychological question or both, but I think it's an interesting one! Jim On Thu, Apr 2, 2020 at 3:07 PM Andy Latto <andy.latto@pobox.com> wrote:
Since there's only one embedding of a circle in R^3 up to homotopy, there's an embedding of a mobius strip in R^3 where the edge is a geometric perfect circle. But I find myself unable to visualize such a thing. Has anyone seen a 3-d model of this surface? Second-best thing would be a graphic of such a thing, preferably one that you could rotate in 3 dimensions.
I'd also like to better visualize Boy's surface, or any other immersion of RP^2 in R^3. It would also be interesting to have an insight into why immersing a Klein bottle in R^3 is easy, while immersing RP2 is "hard". I don't know of any formal sense in which this is true, but apparently Boy came up with this surface when challenged by Hilbert to prove that immersing RP^2 in R^3 was impossible.
Also, are these two questions related? That is, can you immerse a mobius strip in R^3 in such a way that the boundary is a geometric circle, and that the union of this mobius strip and a disk with the same boundary is still an immersion (of RP^2 in R^3)?
Andy Latto
andy.latto@pobox.com
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Thu, Apr 2, 2020 at 3:46 PM James Propp <jamespropp@gmail.com> wrote:
I'm confused by there first sentence ("there's only one embedding of a circle in R^3 up to homotopy"), since a knot isn't homotopic to an unknot.
Sorry; I should have said "the embedding of a circle in R^3 given by the edge of the most familiar embedding of the mobius strip in R3 is homotopic to the embedding of a geometric circle in R^3, so... So while my argument was completely wrong, the conclusion that you can embed a mobius strip in R^3 with a geometric circle as boundary is still true, as is the fact that my efforts to visualize this have proved completely unsuccessful.
But I think I understand and sympathize a lot of what follows; in particular, I'm pretty sure Klein bottles are easier to grok than Boy's surface for nearly everybody. I don't know whether this as a mathematical question or a psychological question or both, but I think it's an interesting one!
Jim
On Thu, Apr 2, 2020 at 3:07 PM Andy Latto <andy.latto@pobox.com> wrote:
Since there's only one embedding of a circle in R^3 up to homotopy, there's an embedding of a mobius strip in R^3 where the edge is a geometric perfect circle. But I find myself unable to visualize such a thing. Has anyone seen a 3-d model of this surface? Second-best thing would be a graphic of such a thing, preferably one that you could rotate in 3 dimensions.
I'd also like to better visualize Boy's surface, or any other immersion of RP^2 in R^3. It would also be interesting to have an insight into why immersing a Klein bottle in R^3 is easy, while immersing RP2 is "hard". I don't know of any formal sense in which this is true, but apparently Boy came up with this surface when challenged by Hilbert to prove that immersing RP^2 in R^3 was impossible.
Also, are these two questions related? That is, can you immerse a mobius strip in R^3 in such a way that the boundary is a geometric circle, and that the union of this mobius strip and a disk with the same boundary is still an immersion (of RP^2 in R^3)?
Andy Latto
andy.latto@pobox.com
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
<< embed a mobius strip in R^3 >> immerse, perhaps? WFL On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 3:46 PM James Propp <jamespropp@gmail.com> wrote:
I'm confused by there first sentence ("there's only one embedding of a circle in R^3 up to homotopy"), since a knot isn't homotopic to an unknot.
Sorry; I should have said "the embedding of a circle in R^3 given by the edge of the most familiar embedding of the mobius strip in R3 is homotopic to the embedding of a geometric circle in R^3, so...
So while my argument was completely wrong, the conclusion that you can embed a mobius strip in R^3 with a geometric circle as boundary is still true, as is the fact that my efforts to visualize this have proved completely unsuccessful.
But I think I understand and sympathize a lot of what follows; in particular, I'm pretty sure Klein bottles are easier to grok than Boy's surface for nearly everybody. I don't know whether this as a mathematical question or a psychological question or both, but I think it's an interesting one!
Jim
On Thu, Apr 2, 2020 at 3:07 PM Andy Latto <andy.latto@pobox.com> wrote:
Since there's only one embedding of a circle in R^3 up to homotopy, there's an embedding of a mobius strip in R^3 where the edge is a geometric perfect circle. But I find myself unable to visualize such a thing. Has anyone seen a 3-d model of this surface? Second-best thing would be a graphic of such a thing, preferably one that you could rotate in 3 dimensions.
I'd also like to better visualize Boy's surface, or any other immersion of RP^2 in R^3. It would also be interesting to have an insight into why immersing a Klein bottle in R^3 is easy, while immersing RP2 is "hard". I don't know of any formal sense in which this is true, but apparently Boy came up with this surface when challenged by Hilbert to prove that immersing RP^2 in R^3 was impossible.
Also, are these two questions related? That is, can you immerse a mobius strip in R^3 in such a way that the boundary is a geometric circle, and that the union of this mobius strip and a disk with the same boundary is still an immersion (of RP^2 in R^3)?
Andy Latto
andy.latto@pobox.com
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< embed a mobius strip in R^3 >> immerse, perhaps? WFL
No, embedded. You can embed a Mobius strip with edge being homoptopic to a geometric circle, so you can embed it with the edge actually being a geometric circle. There are illlustrations on the wikipedia page for mobius strip, but they aren't helping me visualize it. I want someone to 3-d print me one of these! Andy
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 3:46 PM James Propp <jamespropp@gmail.com> wrote:
I'm confused by there first sentence ("there's only one embedding of a circle in R^3 up to homotopy"), since a knot isn't homotopic to an unknot.
Sorry; I should have said "the embedding of a circle in R^3 given by the edge of the most familiar embedding of the mobius strip in R3 is homotopic to the embedding of a geometric circle in R^3, so...
So while my argument was completely wrong, the conclusion that you can embed a mobius strip in R^3 with a geometric circle as boundary is still true, as is the fact that my efforts to visualize this have proved completely unsuccessful.
But I think I understand and sympathize a lot of what follows; in particular, I'm pretty sure Klein bottles are easier to grok than Boy's surface for nearly everybody. I don't know whether this as a mathematical question or a psychological question or both, but I think it's an interesting one!
Jim
On Thu, Apr 2, 2020 at 3:07 PM Andy Latto <andy.latto@pobox.com> wrote:
Since there's only one embedding of a circle in R^3 up to homotopy, there's an embedding of a mobius strip in R^3 where the edge is a geometric perfect circle. But I find myself unable to visualize such a thing. Has anyone seen a 3-d model of this surface? Second-best thing would be a graphic of such a thing, preferably one that you could rotate in 3 dimensions.
I'd also like to better visualize Boy's surface, or any other immersion of RP^2 in R^3. It would also be interesting to have an insight into why immersing a Klein bottle in R^3 is easy, while immersing RP2 is "hard". I don't know of any formal sense in which this is true, but apparently Boy came up with this surface when challenged by Hilbert to prove that immersing RP^2 in R^3 was impossible.
Also, are these two questions related? That is, can you immerse a mobius strip in R^3 in such a way that the boundary is a geometric circle, and that the union of this mobius strip and a disk with the same boundary is still an immersion (of RP^2 in R^3)?
Andy Latto
andy.latto@pobox.com
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
Presumably the "Sudanese Möbius Band" (credited to Sue Goodman & Dan Asimov) at https://en.wikipedia.org/wiki/M%C3%B6bius_strip I found these easier to interpret than Gosper's old-tech renderings. Plainly apparent in the first frame is a caustic line where the surface intersects itself, as might be expected. WFL On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< embed a mobius strip in R^3 >> immerse, perhaps? WFL
No, embedded. You can embed a Mobius strip with edge being homoptopic to a geometric circle, so you can embed it with the edge actually being a geometric circle. There are illlustrations on the wikipedia page for mobius strip, but they aren't helping me visualize it.
I want someone to 3-d print me one of these!
Andy
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 3:46 PM James Propp <jamespropp@gmail.com> wrote:
I'm confused by there first sentence ("there's only one embedding of a circle in R^3 up to homotopy"), since a knot isn't homotopic to an unknot.
Sorry; I should have said "the embedding of a circle in R^3 given by the edge of the most familiar embedding of the mobius strip in R3 is homotopic to the embedding of a geometric circle in R^3, so...
So while my argument was completely wrong, the conclusion that you can embed a mobius strip in R^3 with a geometric circle as boundary is still true, as is the fact that my efforts to visualize this have proved completely unsuccessful.
But I think I understand and sympathize a lot of what follows; in particular, I'm pretty sure Klein bottles are easier to grok than Boy's surface for nearly everybody. I don't know whether this as a mathematical question or a psychological question or both, but I think it's an interesting one!
Jim
On Thu, Apr 2, 2020 at 3:07 PM Andy Latto <andy.latto@pobox.com> wrote:
Since there's only one embedding of a circle in R^3 up to homotopy, there's an embedding of a mobius strip in R^3 where the edge is a geometric perfect circle. But I find myself unable to visualize such a thing. Has anyone seen a 3-d model of this surface? Second-best thing would be a graphic of such a thing, preferably one that you could rotate in 3 dimensions.
I'd also like to better visualize Boy's surface, or any other immersion of RP^2 in R^3. It would also be interesting to have an insight into why immersing a Klein bottle in R^3 is easy, while immersing RP2 is "hard". I don't know of any formal sense in which this is true, but apparently Boy came up with this surface when challenged by Hilbert to prove that immersing RP^2 in R^3 was impossible.
Also, are these two questions related? That is, can you immerse a mobius strip in R^3 in such a way that the boundary is a geometric circle, and that the union of this mobius strip and a disk with the same boundary is still an immersion (of RP^2 in R^3)?
Andy Latto
andy.latto@pobox.com
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Presumably the "Sudanese Möbius Band" (credited to Sue Goodman & Dan Asimov) at
https://en.wikipedia.org/wiki/M%C3%B6bius_strip
I found these easier to interpret than Gosper's old-tech renderings. Plainly apparent in the first frame is a caustic line where the surface intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections. Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize. Andy
WFL
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< embed a mobius strip in R^3 >> immerse, perhaps? WFL
No, embedded. You can embed a Mobius strip with edge being homoptopic to a geometric circle, so you can embed it with the edge actually being a geometric circle. There are illlustrations on the wikipedia page for mobius strip, but they aren't helping me visualize it.
I want someone to 3-d print me one of these!
Andy
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 3:46 PM James Propp <jamespropp@gmail.com> wrote:
I'm confused by there first sentence ("there's only one embedding of a circle in R^3 up to homotopy"), since a knot isn't homotopic to an unknot.
Sorry; I should have said "the embedding of a circle in R^3 given by the edge of the most familiar embedding of the mobius strip in R3 is homotopic to the embedding of a geometric circle in R^3, so...
So while my argument was completely wrong, the conclusion that you can embed a mobius strip in R^3 with a geometric circle as boundary is still true, as is the fact that my efforts to visualize this have proved completely unsuccessful.
But I think I understand and sympathize a lot of what follows; in particular, I'm pretty sure Klein bottles are easier to grok than Boy's surface for nearly everybody. I don't know whether this as a mathematical question or a psychological question or both, but I think it's an interesting one!
Jim
On Thu, Apr 2, 2020 at 3:07 PM Andy Latto <andy.latto@pobox.com> wrote:
Since there's only one embedding of a circle in R^3 up to homotopy, there's an embedding of a mobius strip in R^3 where the edge is a geometric perfect circle. But I find myself unable to visualize such a thing. Has anyone seen a 3-d model of this surface? Second-best thing would be a graphic of such a thing, preferably one that you could rotate in 3 dimensions.
I'd also like to better visualize Boy's surface, or any other immersion of RP^2 in R^3. It would also be interesting to have an insight into why immersing a Klein bottle in R^3 is easy, while immersing RP2 is "hard". I don't know of any formal sense in which this is true, but apparently Boy came up with this surface when challenged by Hilbert to prove that immersing RP^2 in R^3 was impossible.
Also, are these two questions related? That is, can you immerse a mobius strip in R^3 in such a way that the boundary is a geometric circle, and that the union of this mobius strip and a disk with the same boundary is still an immersion (of RP^2 in R^3)?
Andy Latto
andy.latto@pobox.com
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
I now have a "visualization" that I can understand, although, like Andy, I would like to have a 3D model to hold in my hand. Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it. On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Presumably the "Sudanese Möbius Band" (credited to Sue Goodman & Dan Asimov) at
https://en.wikipedia.org/wiki/M%C3%B6bius_strip
I found these easier to interpret than Gosper's old-tech renderings. Plainly apparent in the first frame is a caustic line where the surface intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections.
Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize.
Andy
WFL
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon <fred.lunnon@gmail.com>
wrote:
<< embed a mobius strip in R^3 >> immerse, perhaps? WFL
No, embedded. You can embed a Mobius strip with edge being homoptopic to a geometric circle, so you can embed it with the edge actually being a geometric circle. There are illlustrations on the wikipedia page for mobius strip, but they aren't helping me visualize it.
I want someone to 3-d print me one of these!
Andy
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 3:46 PM James Propp <jamespropp@gmail.com> wrote:
I'm confused by there first sentence ("there's only one embedding
of a
circle in R^3 up to homotopy"), since a knot isn't homotopic to an unknot.
Sorry; I should have said "the embedding of a circle in R^3 given by the edge of the most familiar embedding of the mobius strip in R3 is homotopic to the embedding of a geometric circle in R^3, so...
So while my argument was completely wrong, the conclusion that you can embed a mobius strip in R^3 with a geometric circle as boundary is still true, as is the fact that my efforts to visualize this have proved completely unsuccessful.
But I think I understand and sympathize a lot of what follows; in particular, I'm pretty sure Klein bottles are easier to grok than Boy's surface for nearly everybody. I don't know whether this as a mathematical question or a psychological question or both, but I
think
it's an interesting one!
Jim
On Thu, Apr 2, 2020 at 3:07 PM Andy Latto <andy.latto@pobox.com> wrote:
> Since there's only one embedding of a circle in R^3 up to homotopy, > there's an embedding of a mobius strip in R^3 where the edge is a > geometric perfect circle. But I find myself unable to visualize such > a > thing. Has anyone seen a 3-d model of this surface? Second-best > thing > would be a graphic of such a thing, preferably one that you could > rotate in 3 dimensions. > > I'd also like to better visualize Boy's surface, or any other > immersion of RP^2 in R^3. It would also be interesting to have an > insight into why immersing a Klein bottle in R^3 is easy, while > immersing RP2 is "hard". I don't know of any formal sense in which > this is true, but apparently Boy came up with this surface when > challenged by Hilbert to prove that immersing RP^2 in R^3 was > impossible. > > Also, are these two questions related? That is, can you immerse a > mobius strip in R^3 in such a way that the boundary is a geometric > circle, and that the union of this mobius strip and a disk with the > same boundary is still an immersion (of RP^2 in R^3)? > > Andy Latto > > andy.latto@pobox.com > > -- > Andy.Latto@pobox.com > > _______________________________________________ > 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I think the surface you describe is a punctured Klein bottle, which is different from a mobius strip, which is a punctured projective plane. Andy On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com> wrote:
I now have a "visualization" that I can understand, although, like Andy, I would like to have a 3D model to hold in my hand.
Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it.
On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Presumably the "Sudanese Möbius Band" (credited to Sue Goodman & Dan Asimov) at
https://en.wikipedia.org/wiki/M%C3%B6bius_strip
I found these easier to interpret than Gosper's old-tech renderings. Plainly apparent in the first frame is a caustic line where the surface intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections.
Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize.
Andy
WFL
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon <fred.lunnon@gmail.com>
wrote:
<< embed a mobius strip in R^3 >> immerse, perhaps? WFL
No, embedded. You can embed a Mobius strip with edge being homoptopic to a geometric circle, so you can embed it with the edge actually being a geometric circle. There are illlustrations on the wikipedia page for mobius strip, but they aren't helping me visualize it.
I want someone to 3-d print me one of these!
Andy
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 3:46 PM James Propp <jamespropp@gmail.com> wrote: > > I'm confused by there first sentence ("there's only one embedding
of a
> circle in R^3 up to homotopy"), since a knot isn't homotopic to an > unknot.
Sorry; I should have said "the embedding of a circle in R^3 given by the edge of the most familiar embedding of the mobius strip in R3 is homotopic to the embedding of a geometric circle in R^3, so...
So while my argument was completely wrong, the conclusion that you can embed a mobius strip in R^3 with a geometric circle as boundary is still true, as is the fact that my efforts to visualize this have proved completely unsuccessful.
> > But I think I understand and sympathize a lot of what follows; in > particular, I'm pretty sure Klein bottles are easier to grok than > Boy's surface for nearly everybody. I don't know whether this as a > mathematical question or a psychological question or both, but I think > it's > an interesting one! > > Jim > > On Thu, Apr 2, 2020 at 3:07 PM Andy Latto <andy.latto@pobox.com> > wrote: > > > Since there's only one embedding of a circle in R^3 up to homotopy, > > there's an embedding of a mobius strip in R^3 where the edge is a > > geometric perfect circle. But I find myself unable to visualize such > > a > > thing. Has anyone seen a 3-d model of this surface? Second-best > > thing > > would be a graphic of such a thing, preferably one that you could > > rotate in 3 dimensions. > > > > I'd also like to better visualize Boy's surface, or any other > > immersion of RP^2 in R^3. It would also be interesting to have an > > insight into why immersing a Klein bottle in R^3 is easy, while > > immersing RP2 is "hard". I don't know of any formal sense in which > > this is true, but apparently Boy came up with this surface when > > challenged by Hilbert to prove that immersing RP^2 in R^3 was > > impossible. > > > > Also, are these two questions related? That is, can you immerse a > > mobius strip in R^3 in such a way that the boundary is a geometric > > circle, and that the union of this mobius strip and a disk with the > > same boundary is still an immersion (of RP^2 in R^3)? > > > > Andy Latto > > > > andy.latto@pobox.com > > > > -- > > Andy.Latto@pobox.com > > > > _______________________________________________ > > 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
Here is Henry Segerman's 3d-printable model: https://www.shapeways.com/product/YRBSFUEHB/round-mobius-strip On Thu, Apr 2, 2020 at 8:59 PM Andy Latto <andy.latto@pobox.com> wrote:
I think the surface you describe is a punctured Klein bottle, which is different from a mobius strip, which is a punctured projective plane.
Andy
On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com> wrote:
I now have a "visualization" that I can understand, although, like Andy,
I
would like to have a 3D model to hold in my hand.
Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it.
On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Presumably the "Sudanese Möbius Band" (credited to Sue Goodman & Dan Asimov) at
https://en.wikipedia.org/wiki/M%C3%B6bius_strip
I found these easier to interpret than Gosper's old-tech renderings. Plainly apparent in the first frame is a caustic line where the
surface
intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections.
Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize.
Andy
WFL
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon <fred.lunnon@gmail.com>
wrote:
<< embed a mobius strip in R^3 >> immerse, perhaps? WFL
No, embedded. You can embed a Mobius strip with edge being homoptopic to a geometric circle, so you can embed it with the edge actually being a geometric circle. There are illlustrations on the wikipedia page for mobius strip, but they aren't helping me visualize it.
I want someone to 3-d print me one of these!
Andy
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > On Thu, Apr 2, 2020 at 3:46 PM James Propp <
jamespropp@gmail.com>
> wrote: >> >> I'm confused by there first sentence ("there's only one embedding of a >> circle in R^3 up to homotopy"), since a knot isn't homotopic to an >> unknot. > > Sorry; I should have said "the embedding of a circle in R^3 given by > the edge of the most familiar embedding of the mobius strip in R3 is > homotopic to the embedding of a geometric circle in R^3, so... > > So while my argument was completely wrong, the conclusion that you can > embed a mobius strip in R^3 with a geometric circle as boundary is > still true, as is the fact that my efforts to visualize this have > proved completely unsuccessful. > >> >> But I think I understand and sympathize a lot of what follows; in >> particular, I'm pretty sure Klein bottles are easier to grok than >> Boy's surface for nearly everybody. I don't know whether this as a >> mathematical question or a psychological question or both, but I think >> it's >> an interesting one! >> >> Jim >> >> On Thu, Apr 2, 2020 at 3:07 PM Andy Latto < andy.latto@pobox.com> >> wrote: >> >> > Since there's only one embedding of a circle in R^3 up to homotopy, >> > there's an embedding of a mobius strip in R^3 where the edge is a >> > geometric perfect circle. But I find myself unable to visualize such >> > a >> > thing. Has anyone seen a 3-d model of this surface? Second-best >> > thing >> > would be a graphic of such a thing, preferably one that you could >> > rotate in 3 dimensions. >> > >> > I'd also like to better visualize Boy's surface, or any other >> > immersion of RP^2 in R^3. It would also be interesting to have an >> > insight into why immersing a Klein bottle in R^3 is easy, while >> > immersing RP2 is "hard". I don't know of any formal sense in which >> > this is true, but apparently Boy came up with this surface when >> > challenged by Hilbert to prove that immersing RP^2 in R^3 was >> > impossible. >> > >> > Also, are these two questions related? That is, can you immerse a >> > mobius strip in R^3 in such a way that the boundary is a geometric >> > circle, and that the union of this mobius strip and a disk with the >> > same boundary is still an immersion (of RP^2 in R^3)? >> > >> > Andy Latto >> > >> > andy.latto@pobox.com >> > >> > -- >> > Andy.Latto@pobox.com >> > >> > _______________________________________________ >> > 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 > > > > -- > Andy.Latto@pobox.com > > _______________________________________________ > 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.
Mmm ... looks pretty, but (1) Why is it a topological Moebius strip? (2) Is there a singularity lurking at infinity? (3) Also this argument << ... So you can gradually deform this figure, a[l]ways with no self-intersections >> would transform a closed knotted tube into a standard torus! WFL On 4/3/20, Michael Kleber <michael.kleber@gmail.com> wrote:
Here is Henry Segerman's 3d-printable model: https://www.shapeways.com/product/YRBSFUEHB/round-mobius-strip
On Thu, Apr 2, 2020 at 8:59 PM Andy Latto <andy.latto@pobox.com> wrote:
I think the surface you describe is a punctured Klein bottle, which is different from a mobius strip, which is a punctured projective plane.
Andy
On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com> wrote:
I now have a "visualization" that I can understand, although, like Andy,
I
would like to have a 3D model to hold in my hand.
Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it.
On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Presumably the "Sudanese Möbius Band" (credited to Sue Goodman & Dan Asimov) at
https://en.wikipedia.org/wiki/M%C3%B6bius_strip
I found these easier to interpret than Gosper's old-tech renderings. Plainly apparent in the first frame is a caustic line where the
surface
intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections.
Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize.
Andy
WFL
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon <fred.lunnon@gmail.com>
wrote:
> > << embed a mobius strip in R^3 >> immerse, perhaps? WFL
No, embedded. You can embed a Mobius strip with edge being homoptopic to a geometric circle, so you can embed it with the edge actually being a geometric circle. There are illlustrations on the wikipedia page for mobius strip, but they aren't helping me visualize it.
I want someone to 3-d print me one of these!
Andy
> > > > On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > > On Thu, Apr 2, 2020 at 3:46 PM James Propp < jamespropp@gmail.com> > > wrote: > >> > >> I'm confused by there first sentence ("there's only one embedding of a > >> circle in R^3 up to homotopy"), since a knot isn't homotopic to an > >> unknot. > > > > Sorry; I should have said "the embedding of a circle in R^3 given by > > the edge of the most familiar embedding of the mobius strip in R3 is > > homotopic to the embedding of a geometric circle in R^3, so... > > > > So while my argument was completely wrong, the conclusion that you can > > embed a mobius strip in R^3 with a geometric circle as boundary is > > still true, as is the fact that my efforts to visualize this have > > proved completely unsuccessful. > > > >> > >> But I think I understand and sympathize a lot of what follows; in > >> particular, I'm pretty sure Klein bottles are easier to grok than > >> Boy's surface for nearly everybody. I don't know whether this as a > >> mathematical question or a psychological question or both, but I think > >> it's > >> an interesting one! > >> > >> Jim > >> > >> On Thu, Apr 2, 2020 at 3:07 PM Andy Latto < andy.latto@pobox.com> > >> wrote: > >> > >> > Since there's only one embedding of a circle in R^3 up to homotopy, > >> > there's an embedding of a mobius strip in R^3 where the edge is a > >> > geometric perfect circle. But I find myself unable to visualize such > >> > a > >> > thing. Has anyone seen a 3-d model of this surface? Second-best > >> > thing > >> > would be a graphic of such a thing, preferably one that you could > >> > rotate in 3 dimensions. > >> > > >> > I'd also like to better visualize Boy's surface, or any > >> > other > >> > immersion of RP^2 in R^3. It would also be interesting to have an > >> > insight into why immersing a Klein bottle in R^3 is easy, while > >> > immersing RP2 is "hard". I don't know of any formal sense in which > >> > this is true, but apparently Boy came up with this surface when > >> > challenged by Hilbert to prove that immersing RP^2 in R^3 > >> > was > >> > impossible. > >> > > >> > Also, are these two questions related? That is, can you immerse a > >> > mobius strip in R^3 in such a way that the boundary is a geometric > >> > circle, and that the union of this mobius strip and a disk with the > >> > same boundary is still an immersion (of RP^2 in R^3)? > >> > > >> > Andy Latto > >> > > >> > andy.latto@pobox.com > >> > > >> > -- > >> > Andy.Latto@pobox.com > >> > > >> > _______________________________________________ > >> > 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 > > > > > > > > -- > > Andy.Latto@pobox.com > > > > _______________________________________________ > > 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
So, two things. First, the description I posted is NOT a Moebius strip; it is non-orientable, sure, and it has a single circular edge, but what it actually is is a punctured Klein bottle, and hence does not answer Andy Latto's request for an easily-visualizable discription of a circular-edge Moebius strip realization. Second, I concur with Fred Lunnon, that the Segerman model referenced by Michael Kleber isn't obviously the right object either. If we patch it into a sphere, is the resulting object obviously a Moebius strip? (Something in the back of my head is talking about winding numbers. Make it stop.) On Thu, Apr 2, 2020 at 10:14 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Mmm ... looks pretty, but
(1) Why is it a topological Moebius strip?
(2) Is there a singularity lurking at infinity?
(3) Also this argument << ... So you can gradually deform this figure, a[l]ways with no self-intersections >> would transform a closed knotted tube into a standard torus!
WFL
On 4/3/20, Michael Kleber <michael.kleber@gmail.com> wrote:
Here is Henry Segerman's 3d-printable model: https://www.shapeways.com/product/YRBSFUEHB/round-mobius-strip
On Thu, Apr 2, 2020 at 8:59 PM Andy Latto <andy.latto@pobox.com> wrote:
I think the surface you describe is a punctured Klein bottle, which is different from a mobius strip, which is a punctured projective plane.
Andy
On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com> wrote:
I now have a "visualization" that I can understand, although, like
Andy, I
would like to have a 3D model to hold in my hand.
Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it.
On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Presumably the "Sudanese Möbius Band" (credited to Sue Goodman & Dan Asimov) at
https://en.wikipedia.org/wiki/M%C3%B6bius_strip
I found these easier to interpret than Gosper's old-tech
renderings.
Plainly apparent in the first frame is a caustic line where the surface intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections.
Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize.
Andy
WFL
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon <
fred.lunnon@gmail.com> wrote:
>> >> << embed a mobius strip in R^3 >> immerse, perhaps? WFL > > No, embedded. You can embed a Mobius strip with edge being homoptopic > to a geometric circle, so you can embed it with the edge actually > being a geometric circle. There are illlustrations on the > wikipedia > page for mobius strip, but they aren't helping me visualize it. > > I want someone to 3-d print me one of these! > > Andy > > >> >> >> >> On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: >> > On Thu, Apr 2, 2020 at 3:46 PM James Propp < jamespropp@gmail.com> >> > wrote: >> >> >> >> I'm confused by there first sentence ("there's only one embedding of a >> >> circle in R^3 up to homotopy"), since a knot isn't homotopic to an >> >> unknot. >> > >> > Sorry; I should have said "the embedding of a circle in R^3 given by >> > the edge of the most familiar embedding of the mobius strip in R3 is >> > homotopic to the embedding of a geometric circle in R^3, so... >> > >> > So while my argument was completely wrong, the conclusion that you can >> > embed a mobius strip in R^3 with a geometric circle as boundary is >> > still true, as is the fact that my efforts to visualize this have >> > proved completely unsuccessful. >> > >> >> >> >> But I think I understand and sympathize a lot of what follows; in >> >> particular, I'm pretty sure Klein bottles are easier to grok than >> >> Boy's surface for nearly everybody. I don't know whether this as a >> >> mathematical question or a psychological question or both, but I think >> >> it's >> >> an interesting one! >> >> >> >> Jim >> >> >> >> On Thu, Apr 2, 2020 at 3:07 PM Andy Latto < andy.latto@pobox.com> >> >> wrote: >> >> >> >> > Since there's only one embedding of a circle in R^3 up to homotopy, >> >> > there's an embedding of a mobius strip in R^3 where the edge is a >> >> > geometric perfect circle. But I find myself unable to visualize such >> >> > a >> >> > thing. Has anyone seen a 3-d model of this surface? Second-best >> >> > thing >> >> > would be a graphic of such a thing, preferably one that you could >> >> > rotate in 3 dimensions. >> >> > >> >> > I'd also like to better visualize Boy's surface, or any >> >> > other >> >> > immersion of RP^2 in R^3. It would also be interesting to have an >> >> > insight into why immersing a Klein bottle in R^3 is easy, while >> >> > immersing RP2 is "hard". I don't know of any formal sense in which >> >> > this is true, but apparently Boy came up with this surface when >> >> > challenged by Hilbert to prove that immersing RP^2 in R^3 >> >> > was >> >> > impossible. >> >> > >> >> > Also, are these two questions related? That is, can you immerse a >> >> > mobius strip in R^3 in such a way that the boundary is a geometric >> >> > circle, and that the union of this mobius strip and a disk with the >> >> > same boundary is still an immersion (of RP^2 in R^3)? >> >> > >> >> > Andy Latto >> >> > >> >> > andy.latto@pobox.com >> >> > >> >> > -- >> >> > Andy.Latto@pobox.com >> >> > >> >> > _______________________________________________ >> >> > 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 >> > >> > >> > >> > -- >> > Andy.Latto@pobox.com >> > >> > _______________________________________________ >> > 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 > > > > -- > Andy.Latto@pobox.com > > _______________________________________________ > 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ 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
On Thu, Apr 2, 2020 at 10:14 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Mmm ... looks pretty, but
(1) Why is it a topological Moebius strip?
I think so, but even with a still picture of a 3d object, I'm still failing to visualize it completely. Now trying to decide whether I care enough about being able to visualize this to be willing to spend 40 bucks on it.
(2) Is there a singularity lurking at infinity?
This is a finite, compact, object, that does not extend to infinity. How can there be a singularity at infinity. Are you perhaps thinking of a Mobius strip as being a line bundle over the circle? That's not what I mean by a Mobius strip.I mean a compact manifold with boundary, a line-segment bundle over the circle. No part of it goes off to infinity.
(3) Also this argument << ... So you can gradually deform this figure, a[l]ways with no self-intersections >> would transform a closed knotted tube into a standard torus!
I don't see how my argument proves anything of the kind. My argument starts with a homotopy between two closed curves in R^3, and extends it to a homotopy of the mobius strip in R^3. What is the homotopy you are extending in your hypothetical proof that a closed knotted torus can be transformed into a standard torus? Look at it this way. Take the standard embedding of a Mobius strip in R^3, the one you get with a paper strip. Now forget the strip, and just look at the edge as a simple closed curve in R^3. It "loops around twice", but it is unknotted (if you give the paper strip 3 half twists instead of 1, you get a knotted edge, but that's not where I want to start)/ Since it's unknotted, the embedding in R^3 is homotopic to the circle; Loosely, you can "unfold" it into a figure-8 like curve (without a self-intersection; one side passes over the other) then "untwist" it to get an oval, then round the oval to a perfect circle. Are you saying there is a moment during this homotopy when an infinitesimal change in this curve suddenly changes it from one that can bound an embedded mobius strip to one that cannot bound an embedded mobius strip? Andy
WFL
On 4/3/20, Michael Kleber <michael.kleber@gmail.com> wrote:
Here is Henry Segerman's 3d-printable model: https://www.shapeways.com/product/YRBSFUEHB/round-mobius-strip
On Thu, Apr 2, 2020 at 8:59 PM Andy Latto <andy.latto@pobox.com> wrote:
I think the surface you describe is a punctured Klein bottle, which is different from a mobius strip, which is a punctured projective plane.
Andy
On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com> wrote:
I now have a "visualization" that I can understand, although, like Andy,
I
would like to have a 3D model to hold in my hand.
Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it.
On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Presumably the "Sudanese Möbius Band" (credited to Sue Goodman & Dan Asimov) at
https://en.wikipedia.org/wiki/M%C3%B6bius_strip
I found these easier to interpret than Gosper's old-tech renderings. Plainly apparent in the first frame is a caustic line where the
surface
intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections.
Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize.
Andy
WFL
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon <fred.lunnon@gmail.com>
wrote:
>> >> << embed a mobius strip in R^3 >> immerse, perhaps? WFL > > No, embedded. You can embed a Mobius strip with edge being homoptopic > to a geometric circle, so you can embed it with the edge actually > being a geometric circle. There are illlustrations on the > wikipedia > page for mobius strip, but they aren't helping me visualize it. > > I want someone to 3-d print me one of these! > > Andy > > >> >> >> >> On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: >> > On Thu, Apr 2, 2020 at 3:46 PM James Propp < jamespropp@gmail.com> >> > wrote: >> >> >> >> I'm confused by there first sentence ("there's only one embedding of a >> >> circle in R^3 up to homotopy"), since a knot isn't homotopic to an >> >> unknot. >> > >> > Sorry; I should have said "the embedding of a circle in R^3 given by >> > the edge of the most familiar embedding of the mobius strip in R3 is >> > homotopic to the embedding of a geometric circle in R^3, so... >> > >> > So while my argument was completely wrong, the conclusion that you can >> > embed a mobius strip in R^3 with a geometric circle as boundary is >> > still true, as is the fact that my efforts to visualize this have >> > proved completely unsuccessful. >> > >> >> >> >> But I think I understand and sympathize a lot of what follows; in >> >> particular, I'm pretty sure Klein bottles are easier to grok than >> >> Boy's surface for nearly everybody. I don't know whether this as a >> >> mathematical question or a psychological question or both, but I think >> >> it's >> >> an interesting one! >> >> >> >> Jim >> >> >> >> On Thu, Apr 2, 2020 at 3:07 PM Andy Latto < andy.latto@pobox.com> >> >> wrote: >> >> >> >> > Since there's only one embedding of a circle in R^3 up to homotopy, >> >> > there's an embedding of a mobius strip in R^3 where the edge is a >> >> > geometric perfect circle. But I find myself unable to visualize such >> >> > a >> >> > thing. Has anyone seen a 3-d model of this surface? Second-best >> >> > thing >> >> > would be a graphic of such a thing, preferably one that you could >> >> > rotate in 3 dimensions. >> >> > >> >> > I'd also like to better visualize Boy's surface, or any >> >> > other >> >> > immersion of RP^2 in R^3. It would also be interesting to have an >> >> > insight into why immersing a Klein bottle in R^3 is easy, while >> >> > immersing RP2 is "hard". I don't know of any formal sense in which >> >> > this is true, but apparently Boy came up with this surface when >> >> > challenged by Hilbert to prove that immersing RP^2 in R^3 >> >> > was >> >> > impossible. >> >> > >> >> > Also, are these two questions related? That is, can you immerse a >> >> > mobius strip in R^3 in such a way that the boundary is a geometric >> >> > circle, and that the union of this mobius strip and a disk with the >> >> > same boundary is still an immersion (of RP^2 in R^3)? >> >> > >> >> > Andy Latto >> >> > >> >> > andy.latto@pobox.com >> >> > >> >> > -- >> >> > Andy.Latto@pobox.com >> >> > >> >> > _______________________________________________ >> >> > 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 >> > >> > >> > >> > -- >> > Andy.Latto@pobox.com >> > >> > _______________________________________________ >> > 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 > > > > -- > Andy.Latto@pobox.com > > _______________________________________________ > 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ 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
-- Andy.Latto@pobox.com
On Thu, Apr 2, 2020 at 9:23 PM Michael Kleber <michael.kleber@gmail.com> wrote:
Here is Henry Segerman's 3d-printable model: https://www.shapeways.com/product/YRBSFUEHB/round-mobius-strip
And now that I look at it a bit more carefully, this isn't the object I'm looking for either! This object has a circular edge, but it also has a second edge, consisting of 4 circular arcs with right angles between them. Maybe this object is meant in some way to suggest the object I'm thinking of, but it isn't literally that object. Andy
On Thu, Apr 2, 2020 at 8:59 PM Andy Latto <andy.latto@pobox.com> wrote:
I think the surface you describe is a punctured Klein bottle, which is different from a mobius strip, which is a punctured projective plane.
Andy
On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com> wrote:
I now have a "visualization" that I can understand, although, like Andy,
I
would like to have a 3D model to hold in my hand.
Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it.
On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Presumably the "Sudanese Möbius Band" (credited to Sue Goodman & Dan Asimov) at
https://en.wikipedia.org/wiki/M%C3%B6bius_strip
I found these easier to interpret than Gosper's old-tech renderings. Plainly apparent in the first frame is a caustic line where the
surface
intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections.
Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize.
Andy
WFL
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon <fred.lunnon@gmail.com>
wrote:
> > << embed a mobius strip in R^3 >> immerse, perhaps? WFL
No, embedded. You can embed a Mobius strip with edge being homoptopic to a geometric circle, so you can embed it with the edge actually being a geometric circle. There are illlustrations on the wikipedia page for mobius strip, but they aren't helping me visualize it.
I want someone to 3-d print me one of these!
Andy
> > > > On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > > On Thu, Apr 2, 2020 at 3:46 PM James Propp < jamespropp@gmail.com> > > wrote: > >> > >> I'm confused by there first sentence ("there's only one embedding of a > >> circle in R^3 up to homotopy"), since a knot isn't homotopic to an > >> unknot. > > > > Sorry; I should have said "the embedding of a circle in R^3 given by > > the edge of the most familiar embedding of the mobius strip in R3 is > > homotopic to the embedding of a geometric circle in R^3, so... > > > > So while my argument was completely wrong, the conclusion that you can > > embed a mobius strip in R^3 with a geometric circle as boundary is > > still true, as is the fact that my efforts to visualize this have > > proved completely unsuccessful. > > > >> > >> But I think I understand and sympathize a lot of what follows; in > >> particular, I'm pretty sure Klein bottles are easier to grok than > >> Boy's surface for nearly everybody. I don't know whether this as a > >> mathematical question or a psychological question or both, but I think > >> it's > >> an interesting one! > >> > >> Jim > >> > >> On Thu, Apr 2, 2020 at 3:07 PM Andy Latto < andy.latto@pobox.com> > >> wrote: > >> > >> > Since there's only one embedding of a circle in R^3 up to homotopy, > >> > there's an embedding of a mobius strip in R^3 where the edge is a > >> > geometric perfect circle. But I find myself unable to visualize such > >> > a > >> > thing. Has anyone seen a 3-d model of this surface? Second-best > >> > thing > >> > would be a graphic of such a thing, preferably one that you could > >> > rotate in 3 dimensions. > >> > > >> > I'd also like to better visualize Boy's surface, or any other > >> > immersion of RP^2 in R^3. It would also be interesting to have an > >> > insight into why immersing a Klein bottle in R^3 is easy, while > >> > immersing RP2 is "hard". I don't know of any formal sense in which > >> > this is true, but apparently Boy came up with this surface when > >> > challenged by Hilbert to prove that immersing RP^2 in R^3 was > >> > impossible. > >> > > >> > Also, are these two questions related? That is, can you immerse a > >> > mobius strip in R^3 in such a way that the boundary is a geometric > >> > circle, and that the union of this mobius strip and a disk with the > >> > same boundary is still an immersion (of RP^2 in R^3)? > >> > > >> > Andy Latto > >> > > >> > andy.latto@pobox.com > >> > > >> > -- > >> > Andy.Latto@pobox.com > >> > > >> > _______________________________________________ > >> > 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 > > > > > > > > -- > > Andy.Latto@pobox.com > > > > _______________________________________________ > > 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
The outer "scalloped" edge is intended to suggest being extended to the point at infinity. (Notice how the scalloped edge is almost planar.) In principle you could deflate that embedding (at the cost of some pretty symmetry). Imagine the surface a sphere with a cloverleaf-shaped hole in it; fit Segerman's model into the hole. Mind, I am still not convinced that the resulting model is a Moebius strip, but I have to confess that it might be. On Fri, Apr 3, 2020 at 12:04 AM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 9:23 PM Michael Kleber <michael.kleber@gmail.com> wrote:
Here is Henry Segerman's 3d-printable model: https://www.shapeways.com/product/YRBSFUEHB/round-mobius-strip
And now that I look at it a bit more carefully, this isn't the object I'm looking for either! This object has a circular edge, but it also has a second edge, consisting of 4 circular arcs with right angles between them. Maybe this object is meant in some way to suggest the object I'm thinking of, but it isn't literally that object.
Andy
On Thu, Apr 2, 2020 at 8:59 PM Andy Latto <andy.latto@pobox.com> wrote:
I think the surface you describe is a punctured Klein bottle, which is different from a mobius strip, which is a punctured projective plane.
Andy
On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com>
wrote:
I now have a "visualization" that I can understand, although, like
Andy, I
would like to have a 3D model to hold in my hand.
Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it.
On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Presumably the "Sudanese Möbius Band" (credited to Sue Goodman & Dan Asimov) at
https://en.wikipedia.org/wiki/M%C3%B6bius_strip
I found these easier to interpret than Gosper's old-tech
renderings.
Plainly apparent in the first frame is a caustic line where the surface intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections.
Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize.
Andy
WFL
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon <
fred.lunnon@gmail.com> wrote:
>> >> << embed a mobius strip in R^3 >> immerse, perhaps? WFL > > No, embedded. You can embed a Mobius strip with edge being homoptopic > to a geometric circle, so you can embed it with the edge actually > being a geometric circle. There are illlustrations on the wikipedia > page for mobius strip, but they aren't helping me visualize it. > > I want someone to 3-d print me one of these! > > Andy > > >> >> >> >> On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: >> > On Thu, Apr 2, 2020 at 3:46 PM James Propp < jamespropp@gmail.com> >> > wrote: >> >> >> >> I'm confused by there first sentence ("there's only one embedding of a >> >> circle in R^3 up to homotopy"), since a knot isn't homotopic to an >> >> unknot. >> > >> > Sorry; I should have said "the embedding of a circle in R^3 given by >> > the edge of the most familiar embedding of the mobius strip in R3 is >> > homotopic to the embedding of a geometric circle in R^3, so... >> > >> > So while my argument was completely wrong, the conclusion that you can >> > embed a mobius strip in R^3 with a geometric circle as boundary is >> > still true, as is the fact that my efforts to visualize this have >> > proved completely unsuccessful. >> > >> >> >> >> But I think I understand and sympathize a lot of what follows; in >> >> particular, I'm pretty sure Klein bottles are easier to grok than >> >> Boy's surface for nearly everybody. I don't know whether this as a >> >> mathematical question or a psychological question or both, but I think >> >> it's >> >> an interesting one! >> >> >> >> Jim >> >> >> >> On Thu, Apr 2, 2020 at 3:07 PM Andy Latto < andy.latto@pobox.com> >> >> wrote: >> >> >> >> > Since there's only one embedding of a circle in R^3 up to homotopy, >> >> > there's an embedding of a mobius strip in R^3 where the edge is a >> >> > geometric perfect circle. But I find myself unable to visualize such >> >> > a >> >> > thing. Has anyone seen a 3-d model of this surface? Second-best >> >> > thing >> >> > would be a graphic of such a thing, preferably one that you could >> >> > rotate in 3 dimensions. >> >> > >> >> > I'd also like to better visualize Boy's surface, or any other >> >> > immersion of RP^2 in R^3. It would also be interesting to have an >> >> > insight into why immersing a Klein bottle in R^3 is easy, while >> >> > immersing RP2 is "hard". I don't know of any formal sense in which >> >> > this is true, but apparently Boy came up with this surface when >> >> > challenged by Hilbert to prove that immersing RP^2 in R^3 was >> >> > impossible. >> >> > >> >> > Also, are these two questions related? That is, can you immerse a >> >> > mobius strip in R^3 in such a way that the boundary is a geometric >> >> > circle, and that the union of this mobius strip and a disk with the >> >> > same boundary is still an immersion (of RP^2 in R^3)? >> >> > >> >> > Andy Latto >> >> > >> >> > andy.latto@pobox.com >> >> > >> >> > -- >> >> > Andy.Latto@pobox.com >> >> > >> >> > _______________________________________________ >> >> > 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 >> > >> > >> > >> > -- >> > Andy.Latto@pobox.com >> > >> > _______________________________________________ >> > 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 > > > > -- > Andy.Latto@pobox.com > > _______________________________________________ > 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Here's a movie of a rotating Sudanese surface. Quite informative. No singularities or going to infinity or self-intersecting...just a remarkably hard to grok surface in ordinary 3-space. https://vimeo.com/2037835 The natural question is if you go in the hole on one side of the circle do you come out the other side? Yes you do...this is a deformation of a loop, after all, so you can link with it. I once made a polygonal version of this surface, which I find slightly easier to understand. It's based on the observation that the edges of an octahedron are three equatorial squares. To make it, start with a cube that is missing one face. Attach four equilateral triangles to the four open edges of the cube. Fold these triangular flaps alternately up and down, so opposite triangles share a vertex. One of these two vertices will be inside the box, and one will be outside. The edges of these four triangles form the 12 edges of a regular octahedron. Finally fill in one of the two equatorial squares of the octahedron that is NOT edges of the cube with a square panel, and voila, a Möbius strip, with the remaining equatorial square being the sole edge. Well it still hurts my brain, but I do find it a little easier to grasp than the slippery smooth skin of the Sudanese surface. On Thu, Apr 2, 2020 at 9:22 PM Allan Wechsler <acwacw@gmail.com> wrote:
The outer "scalloped" edge is intended to suggest being extended to the point at infinity. (Notice how the scalloped edge is almost planar.) In principle you could deflate that embedding (at the cost of some pretty symmetry). Imagine the surface a sphere with a cloverleaf-shaped hole in it; fit Segerman's model into the hole. Mind, I am still not convinced that the resulting model is a Moebius strip, but I have to confess that it might be.
On Fri, Apr 3, 2020 at 12:04 AM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 9:23 PM Michael Kleber <michael.kleber@gmail.com> wrote:
Here is Henry Segerman's 3d-printable model: https://www.shapeways.com/product/YRBSFUEHB/round-mobius-strip
And now that I look at it a bit more carefully, this isn't the object I'm looking for either! This object has a circular edge, but it also has a second edge, consisting of 4 circular arcs with right angles between them. Maybe this object is meant in some way to suggest the object I'm thinking of, but it isn't literally that object.
Andy
On Thu, Apr 2, 2020 at 8:59 PM Andy Latto <andy.latto@pobox.com>
wrote:
I think the surface you describe is a punctured Klein bottle, which
is
different from a mobius strip, which is a punctured projective plane.
Andy
On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com> wrote:
I now have a "visualization" that I can understand, although, like
Andy, I
would like to have a 3D model to hold in my hand.
Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it.
On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon < fred.lunnon@gmail.com> wrote: > > Presumably the "Sudanese Möbius Band" (credited to Sue Goodman > & Dan Asimov) at > > https://en.wikipedia.org/wiki/M%C3%B6bius_strip > > I found these easier to interpret than Gosper's old-tech renderings. > Plainly apparent in the first frame is a caustic line where the surface > intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections.
Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize.
Andy
> > WFL > > > On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > > On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon < fred.lunnon@gmail.com> wrote: > >> > >> << embed a mobius strip in R^3 >> immerse, perhaps? WFL > > > > No, embedded. You can embed a Mobius strip with edge being homoptopic > > to a geometric circle, so you can embed it with the edge actually > > being a geometric circle. There are illlustrations on the wikipedia > > page for mobius strip, but they aren't helping me visualize it. > > > > I want someone to 3-d print me one of these! > > > > Andy > > > > > >> > >> > >> > >> On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > >> > On Thu, Apr 2, 2020 at 3:46 PM James Propp < jamespropp@gmail.com> > >> > wrote: > >> >> > >> >> I'm confused by there first sentence ("there's only one embedding of a > >> >> circle in R^3 up to homotopy"), since a knot isn't homotopic to an > >> >> unknot. > >> > > >> > Sorry; I should have said "the embedding of a circle in R^3 given by > >> > the edge of the most familiar embedding of the mobius strip in R3 is > >> > homotopic to the embedding of a geometric circle in R^3, so... > >> > > >> > So while my argument was completely wrong, the conclusion that you can > >> > embed a mobius strip in R^3 with a geometric circle as boundary is > >> > still true, as is the fact that my efforts to visualize this have > >> > proved completely unsuccessful. > >> > > >> >> > >> >> But I think I understand and sympathize a lot of what follows; in > >> >> particular, I'm pretty sure Klein bottles are easier to grok than > >> >> Boy's surface for nearly everybody. I don't know whether this as a > >> >> mathematical question or a psychological question or both, but I think > >> >> it's > >> >> an interesting one! > >> >> > >> >> Jim > >> >> > >> >> On Thu, Apr 2, 2020 at 3:07 PM Andy Latto < andy.latto@pobox.com> > >> >> wrote: > >> >> > >> >> > Since there's only one embedding of a circle in R^3 up to homotopy, > >> >> > there's an embedding of a mobius strip in R^3 where the edge is a > >> >> > geometric perfect circle. But I find myself unable to visualize such > >> >> > a > >> >> > thing. Has anyone seen a 3-d model of this surface? Second-best > >> >> > thing > >> >> > would be a graphic of such a thing, preferably one that you could > >> >> > rotate in 3 dimensions. > >> >> > > >> >> > I'd also like to better visualize Boy's surface, or any other > >> >> > immersion of RP^2 in R^3. It would also be interesting to have an > >> >> > insight into why immersing a Klein bottle in R^3 is easy, while > >> >> > immersing RP2 is "hard". I don't know of any formal sense in which > >> >> > this is true, but apparently Boy came up with this surface when > >> >> > challenged by Hilbert to prove that immersing RP^2 in R^3 was > >> >> > impossible. > >> >> > > >> >> > Also, are these two questions related? That is, can you immerse a > >> >> > mobius strip in R^3 in such a way that the boundary is a geometric > >> >> > circle, and that the union of this mobius strip and a disk with the > >> >> > same boundary is still an immersion (of RP^2 in R^3)? > >> >> > > >> >> > Andy Latto > >> >> > > >> >> > andy.latto@pobox.com > >> >> > > >> >> > -- > >> >> > Andy.Latto@pobox.com > >> >> > > >> >> > _______________________________________________ > >> >> > 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 > >> > > >> > > >> > > >> > -- > >> > Andy.Latto@pobox.com > >> > > >> > _______________________________________________ > >> > 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 > > > > > > > > -- > > Andy.Latto@pobox.com > > > > _______________________________________________ > > 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ 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
Even better, the first comment on that movie says: Daniel Piker <https://vimeo.com/user798992>PLUS <https://vimeo.com/plus>11 years ago You can now order an actual physical 3D print of this from shapeways, here: shapeways.com/model/12988/sudanese_m__bius.html <http://www.shapeways.com/model/12988/sudanese_m__bius.html> It costs only $26.01. And even better than THAT, the image on the Shapeways site is a “digital preview not a photo” that you can rotate as you wish, to inspect it from different angles and move however slowly you need! (Click the 3D button in the upper right corner of the image.) I am humbled that even some great mathematicians on this list find visualizing this surface challenging. I’m ordering one from Shapeways, hoping it will cure my headache. — Mike
On Apr 3, 2020, at 1:43 AM, Scott Kim <scott@scottkim.com> wrote:
Here's a movie of a rotating Sudanese surface. Quite informative. No singularities or going to infinity or self-intersecting...just a remarkably hard to grok surface in ordinary 3-space. https://vimeo.com/2037835
The natural question is if you go in the hole on one side of the circle do you come out the other side? Yes you do...this is a deformation of a loop, after all, so you can link with it.
I once made a polygonal version of this surface, which I find slightly easier to understand. It's based on the observation that the edges of an octahedron are three equatorial squares. To make it, start with a cube that is missing one face. Attach four equilateral triangles to the four open edges of the cube. Fold these triangular flaps alternately up and down, so opposite triangles share a vertex. One of these two vertices will be inside the box, and one will be outside. The edges of these four triangles form the 12 edges of a regular octahedron. Finally fill in one of the two equatorial squares of the octahedron that is NOT edges of the cube with a square panel, and voila, a Möbius strip, with the remaining equatorial square being the sole edge. Well it still hurts my brain, but I do find it a little easier to grasp than the slippery smooth skin of the Sudanese surface.
I think I know how the Segerman surface is produced. 1. Consider the horizontal plane, P, in R^3, passing through the origin. 2. Take the unit sphere centred at the origin, and let it intersect P to form a unit circle C. This will form the boundary of our surface. 3. For each angle theta in [0, pi), take the vertical plane Q_theta through the origin with an azimuth angle of theta. Within this plane Q_theta, we draw a (possibly degenerate) circular arc defined by the following properties: -- the centre of the arc lies on the vertical line through the origin; -- the endpoints of the arc are the two intersections of Q_theta with C; -- the angle of approach where the arc meets P is 2*theta; Two of these arcs are degenerate circles: one is the intersection of the x-axis with the disc bounded by C, and the other is the intersection of the y-axis with the *complement* of the disc bounded by C. The latter is the arc which passes through the point at infinity in the one-point compactification of R^3. 4. Take the union of all of these arcs. Now, I claim that this surface (considered as a subset of the one-point compactification of R^3) is homeomorphic to a Moebius strip. Specifically, if you take the usual construction of a Moebius strip from taking a long skinny rectangle and identifying the two short sides in opposite orientations, the short line segments parallel to those two short sides correspond exactly to the arcs in the above construction. The only annoyance is the point at infinity, but that's easily amendable: stereographically project from (R^3 union {infinity}) to S^3, then rotate the 3-sphere slighty so there's no longer a point of the surface on the north pole, and then stereographically project back to R^3. (This is topologically equivalent to Allan's suggestion of inserting Segerman's model into a hole in a sphere.) Best wishes, Adam P. Goucher
Sent: Friday, April 03, 2020 at 5:17 AM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Two topological visualization questions.
The outer "scalloped" edge is intended to suggest being extended to the point at infinity. (Notice how the scalloped edge is almost planar.) In principle you could deflate that embedding (at the cost of some pretty symmetry). Imagine the surface a sphere with a cloverleaf-shaped hole in it; fit Segerman's model into the hole. Mind, I am still not convinced that the resulting model is a Moebius strip, but I have to confess that it might be.
On Fri, Apr 3, 2020 at 12:04 AM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 9:23 PM Michael Kleber <michael.kleber@gmail.com> wrote:
Here is Henry Segerman's 3d-printable model: https://www.shapeways.com/product/YRBSFUEHB/round-mobius-strip
And now that I look at it a bit more carefully, this isn't the object I'm looking for either! This object has a circular edge, but it also has a second edge, consisting of 4 circular arcs with right angles between them. Maybe this object is meant in some way to suggest the object I'm thinking of, but it isn't literally that object.
Andy
On Thu, Apr 2, 2020 at 8:59 PM Andy Latto <andy.latto@pobox.com> wrote:
I think the surface you describe is a punctured Klein bottle, which is different from a mobius strip, which is a punctured projective plane.
Andy
On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com>
wrote:
I now have a "visualization" that I can understand, although, like
Andy, I
would like to have a 3D model to hold in my hand.
Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it.
On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote: > > Presumably the "Sudanese Möbius Band" (credited to Sue Goodman > & Dan Asimov) at > > https://en.wikipedia.org/wiki/M%C3%B6bius_strip > > I found these easier to interpret than Gosper's old-tech renderings. > Plainly apparent in the first frame is a caustic line where the surface > intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections.
Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize.
Andy
> > WFL > > > On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > > On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon < fred.lunnon@gmail.com> wrote: > >> > >> << embed a mobius strip in R^3 >> immerse, perhaps? WFL > > > > No, embedded. You can embed a Mobius strip with edge being homoptopic > > to a geometric circle, so you can embed it with the edge actually > > being a geometric circle. There are illlustrations on the wikipedia > > page for mobius strip, but they aren't helping me visualize it. > > > > I want someone to 3-d print me one of these! > > > > Andy > > > > > >> > >> > >> > >> On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > >> > On Thu, Apr 2, 2020 at 3:46 PM James Propp < jamespropp@gmail.com> > >> > wrote: > >> >> > >> >> I'm confused by there first sentence ("there's only one embedding of a > >> >> circle in R^3 up to homotopy"), since a knot isn't homotopic to an > >> >> unknot. > >> > > >> > Sorry; I should have said "the embedding of a circle in R^3 given by > >> > the edge of the most familiar embedding of the mobius strip in R3 is > >> > homotopic to the embedding of a geometric circle in R^3, so... > >> > > >> > So while my argument was completely wrong, the conclusion that you can > >> > embed a mobius strip in R^3 with a geometric circle as boundary is > >> > still true, as is the fact that my efforts to visualize this have > >> > proved completely unsuccessful. > >> > > >> >> > >> >> But I think I understand and sympathize a lot of what follows; in > >> >> particular, I'm pretty sure Klein bottles are easier to grok than > >> >> Boy's surface for nearly everybody. I don't know whether this as a > >> >> mathematical question or a psychological question or both, but I think > >> >> it's > >> >> an interesting one! > >> >> > >> >> Jim > >> >> > >> >> On Thu, Apr 2, 2020 at 3:07 PM Andy Latto < andy.latto@pobox.com> > >> >> wrote: > >> >> > >> >> > Since there's only one embedding of a circle in R^3 up to homotopy, > >> >> > there's an embedding of a mobius strip in R^3 where the edge is a > >> >> > geometric perfect circle. But I find myself unable to visualize such > >> >> > a > >> >> > thing. Has anyone seen a 3-d model of this surface? Second-best > >> >> > thing > >> >> > would be a graphic of such a thing, preferably one that you could > >> >> > rotate in 3 dimensions. > >> >> > > >> >> > I'd also like to better visualize Boy's surface, or any other > >> >> > immersion of RP^2 in R^3. It would also be interesting to have an > >> >> > insight into why immersing a Klein bottle in R^3 is easy, while > >> >> > immersing RP2 is "hard". I don't know of any formal sense in which > >> >> > this is true, but apparently Boy came up with this surface when > >> >> > challenged by Hilbert to prove that immersing RP^2 in R^3 was > >> >> > impossible. > >> >> > > >> >> > Also, are these two questions related? That is, can you immerse a > >> >> > mobius strip in R^3 in such a way that the boundary is a geometric > >> >> > circle, and that the union of this mobius strip and a disk with the > >> >> > same boundary is still an immersion (of RP^2 in R^3)? > >> >> > > >> >> > Andy Latto > >> >> > > >> >> > andy.latto@pobox.com > >> >> > > >> >> > -- > >> >> > Andy.Latto@pobox.com > >> >> > > >> >> > _______________________________________________ > >> >> > 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 > >> > > >> > > >> > > >> > -- > >> > Andy.Latto@pobox.com > >> > > >> > _______________________________________________ > >> > 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 > > > > > > > > -- > > Andy.Latto@pobox.com > > > > _______________________________________________ > > 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ 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
The circles (other than the central one) do not form a "second edge" they are just a grid on the surface of the object, which extends to infinity, but could be bent up to form a sphere, like the Riemann sphere, if you want a finite object. Adam P. Goucher explains in detail. And Mike Beeler indicates where you can see this object as a rotatable shapeways preview, but without the orthogonal grid of circles. On Thu, Apr 2, 2020 at 10:04 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 9:23 PM Michael Kleber <michael.kleber@gmail.com> wrote:
Here is Henry Segerman's 3d-printable model: https://www.shapeways.com/product/YRBSFUEHB/round-mobius-strip
And now that I look at it a bit more carefully, this isn't the object I'm looking for either! This object has a circular edge, but it also has a second edge, consisting of 4 circular arcs with right angles between them. Maybe this object is meant in some way to suggest the object I'm thinking of, but it isn't literally that object.
Andy
On Thu, Apr 2, 2020 at 8:59 PM Andy Latto <andy.latto@pobox.com> wrote:
I think the surface you describe is a punctured Klein bottle, which is different from a mobius strip, which is a punctured projective plane.
Andy
On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com>
wrote:
I now have a "visualization" that I can understand, although, like
Andy, I
would like to have a 3D model to hold in my hand.
Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it.
On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Presumably the "Sudanese Möbius Band" (credited to Sue Goodman & Dan Asimov) at
https://en.wikipedia.org/wiki/M%C3%B6bius_strip
I found these easier to interpret than Gosper's old-tech
renderings.
Plainly apparent in the first frame is a caustic line where the surface intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections.
Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize.
Andy
WFL
On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon <
fred.lunnon@gmail.com> wrote:
>> >> << embed a mobius strip in R^3 >> immerse, perhaps? WFL > > No, embedded. You can embed a Mobius strip with edge being homoptopic > to a geometric circle, so you can embed it with the edge actually > being a geometric circle. There are illlustrations on the wikipedia > page for mobius strip, but they aren't helping me visualize it. > > I want someone to 3-d print me one of these! > > Andy > > >> >> >> >> On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: >> > On Thu, Apr 2, 2020 at 3:46 PM James Propp < jamespropp@gmail.com> >> > wrote: >> >> >> >> I'm confused by there first sentence ("there's only one embedding of a >> >> circle in R^3 up to homotopy"), since a knot isn't homotopic to an >> >> unknot. >> > >> > Sorry; I should have said "the embedding of a circle in R^3 given by >> > the edge of the most familiar embedding of the mobius strip in R3 is >> > homotopic to the embedding of a geometric circle in R^3, so... >> > >> > So while my argument was completely wrong, the conclusion that you can >> > embed a mobius strip in R^3 with a geometric circle as boundary is >> > still true, as is the fact that my efforts to visualize this have >> > proved completely unsuccessful. >> > >> >> >> >> But I think I understand and sympathize a lot of what follows; in >> >> particular, I'm pretty sure Klein bottles are easier to grok than >> >> Boy's surface for nearly everybody. I don't know whether this as a >> >> mathematical question or a psychological question or both, but I think >> >> it's >> >> an interesting one! >> >> >> >> Jim >> >> >> >> On Thu, Apr 2, 2020 at 3:07 PM Andy Latto < andy.latto@pobox.com> >> >> wrote: >> >> >> >> > Since there's only one embedding of a circle in R^3 up to homotopy, >> >> > there's an embedding of a mobius strip in R^3 where the edge is a >> >> > geometric perfect circle. But I find myself unable to visualize such >> >> > a >> >> > thing. Has anyone seen a 3-d model of this surface? Second-best >> >> > thing >> >> > would be a graphic of such a thing, preferably one that you could >> >> > rotate in 3 dimensions. >> >> > >> >> > I'd also like to better visualize Boy's surface, or any other >> >> > immersion of RP^2 in R^3. It would also be interesting to have an >> >> > insight into why immersing a Klein bottle in R^3 is easy, while >> >> > immersing RP2 is "hard". I don't know of any formal sense in which >> >> > this is true, but apparently Boy came up with this surface when >> >> > challenged by Hilbert to prove that immersing RP^2 in R^3 was >> >> > impossible. >> >> > >> >> > Also, are these two questions related? That is, can you immerse a >> >> > mobius strip in R^3 in such a way that the boundary is a geometric >> >> > circle, and that the union of this mobius strip and a disk with the >> >> > same boundary is still an immersion (of RP^2 in R^3)? >> >> > >> >> > Andy Latto >> >> > >> >> > andy.latto@pobox.com >> >> > >> >> > -- >> >> > Andy.Latto@pobox.com >> >> > >> >> > _______________________________________________ >> >> > 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 >> > >> > >> > >> > -- >> > Andy.Latto@pobox.com >> > >> > _______________________________________________ >> > 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 > > > > -- > Andy.Latto@pobox.com > > _______________________________________________ > 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Apr 3, 2020, at 1:43 AM, Scott Kim <scott@scottkim.com> wrote:
Here's a movie of a rotating Sudanese surface. Quite informative. No singularities or going to infinity or self-intersecting...just a remarkably hard to grok surface in ordinary 3-space. https://vimeo.com/2037835
Seeing is believing --- can't argue with that! Andy Latto<andy.latto@pobox.com> --- << I don't see how my argument proves anything of the kind. My argument starts with a homotopy between two closed curves in R^3, and extends it to a homotopy of the mobius strip in R^3. What is the homotopy you are extending in your hypothetical proof that a closed knotted torus can be transformed into a standard torus? >> The identity homotopy from the empty set to the empty set. What theorem were you employing to deduce that such an extension exists? WFL On 4/3/20, James Buddenhagen <jbuddenh@gmail.com> wrote:
The circles (other than the central one) do not form a "second edge" they are just a grid on the surface of the object, which extends to infinity, but could be bent up to form a sphere, like the Riemann sphere, if you want a finite object. Adam P. Goucher explains in detail. And Mike Beeler indicates where you can see this object as a rotatable shapeways preview, but without the orthogonal grid of circles.
On Thu, Apr 2, 2020 at 10:04 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 9:23 PM Michael Kleber <michael.kleber@gmail.com> wrote:
Here is Henry Segerman's 3d-printable model: https://www.shapeways.com/product/YRBSFUEHB/round-mobius-strip
And now that I look at it a bit more carefully, this isn't the object I'm looking for either! This object has a circular edge, but it also has a second edge, consisting of 4 circular arcs with right angles between them. Maybe this object is meant in some way to suggest the object I'm thinking of, but it isn't literally that object.
Andy
On Thu, Apr 2, 2020 at 8:59 PM Andy Latto <andy.latto@pobox.com> wrote:
I think the surface you describe is a punctured Klein bottle, which is different from a mobius strip, which is a punctured projective plane.
Andy
On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com>
wrote:
I now have a "visualization" that I can understand, although, like
Andy, I
would like to have a 3D model to hold in my hand.
Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it.
On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote: > > Presumably the "Sudanese Möbius Band" (credited to Sue Goodman > & Dan Asimov) at > > https://en.wikipedia.org/wiki/M%C3%B6bius_strip > > I found these easier to interpret than Gosper's old-tech renderings. > Plainly apparent in the first frame is a caustic line where the surface > intersects itself, as might be expected.
If you're right that this figure has a self-intersecting line, it's not the figure I'm looking for. I want a Mobius strip that is *embedded* not just *immersed*, which means no self-intersections.
Why the "as can be expected"? The standard embedding of a mobius strip in R^3, the one you get by giving a strip of paper a half-twist and joining it into a band, has no self-intersections, and the embedding of the boundary into R^3 is homotopic to the embedding of the geometric circle. So you can gradually deform this figure, aways with no self-intersections, into a figure where the edge is a geometric circle. The fact that it feels like there must be a self-intersection shows how difficult the resulting surface (which has no self-intersections) is to visualize.
Andy
> > WFL > > > On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > > On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon < fred.lunnon@gmail.com> wrote: > >> > >> << embed a mobius strip in R^3 >> immerse, perhaps? WFL > > > > No, embedded. You can embed a Mobius strip with edge being homoptopic > > to a geometric circle, so you can embed it with the edge actually > > being a geometric circle. There are illlustrations on the wikipedia > > page for mobius strip, but they aren't helping me visualize > > it. > > > > I want someone to 3-d print me one of these! > > > > Andy > > > > > >> > >> > >> > >> On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > >> > On Thu, Apr 2, 2020 at 3:46 PM James Propp < jamespropp@gmail.com> > >> > wrote: > >> >> > >> >> I'm confused by there first sentence ("there's only one embedding of a > >> >> circle in R^3 up to homotopy"), since a knot isn't homotopic to an > >> >> unknot. > >> > > >> > Sorry; I should have said "the embedding of a circle in R^3 given by > >> > the edge of the most familiar embedding of the mobius strip in R3 is > >> > homotopic to the embedding of a geometric circle in R^3, so... > >> > > >> > So while my argument was completely wrong, the conclusion that you can > >> > embed a mobius strip in R^3 with a geometric circle as boundary is > >> > still true, as is the fact that my efforts to visualize > >> > this have > >> > proved completely unsuccessful. > >> > > >> >> > >> >> But I think I understand and sympathize a lot of what follows; in > >> >> particular, I'm pretty sure Klein bottles are easier to grok than > >> >> Boy's surface for nearly everybody. I don't know whether this as a > >> >> mathematical question or a psychological question or both, but I think > >> >> it's > >> >> an interesting one! > >> >> > >> >> Jim > >> >> > >> >> On Thu, Apr 2, 2020 at 3:07 PM Andy Latto < andy.latto@pobox.com> > >> >> wrote: > >> >> > >> >> > Since there's only one embedding of a circle in R^3 up > >> >> > to homotopy, > >> >> > there's an embedding of a mobius strip in R^3 where the edge is a > >> >> > geometric perfect circle. But I find myself unable to visualize such > >> >> > a > >> >> > thing. Has anyone seen a 3-d model of this surface? Second-best > >> >> > thing > >> >> > would be a graphic of such a thing, preferably one that you could > >> >> > rotate in 3 dimensions. > >> >> > > >> >> > I'd also like to better visualize Boy's surface, or any other > >> >> > immersion of RP^2 in R^3. It would also be interesting > >> >> > to have an > >> >> > insight into why immersing a Klein bottle in R^3 is > >> >> > easy, while > >> >> > immersing RP2 is "hard". I don't know of any formal sense in which > >> >> > this is true, but apparently Boy came up with this surface when > >> >> > challenged by Hilbert to prove that immersing RP^2 in R^3 was > >> >> > impossible. > >> >> > > >> >> > Also, are these two questions related? That is, can you immerse a > >> >> > mobius strip in R^3 in such a way that the boundary is a geometric > >> >> > circle, and that the union of this mobius strip and a disk with the > >> >> > same boundary is still an immersion (of RP^2 in R^3)? > >> >> > > >> >> > Andy Latto > >> >> > > >> >> > andy.latto@pobox.com > >> >> > > >> >> > -- > >> >> > Andy.Latto@pobox.com > >> >> > > >> >> > _______________________________________________ > >> >> > 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 > >> > > >> > > >> > > >> > -- > >> > Andy.Latto@pobox.com > >> > > >> > _______________________________________________ > >> > 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 > > > > > > > > -- > > Andy.Latto@pobox.com > > > > _______________________________________________ > > 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ 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
On Fri, Apr 3, 2020 at 12:38 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
On Apr 3, 2020, at 1:43 AM, Scott Kim <scott@scottkim.com> wrote:
Here's a movie of a rotating Sudanese surface. Quite informative. No singularities or going to infinity or self-intersecting...just a remarkably hard to grok surface in ordinary 3-space. https://vimeo.com/2037835
Seeing is believing --- can't argue with that!
Andy Latto<andy.latto@pobox.com> --- << I don't see how my argument proves anything of the kind. My argument starts with a homotopy between two closed curves in R^3, and extends it to a homotopy of the mobius strip in R^3. What is the homotopy you are extending in your hypothetical proof that a closed knotted torus can be transformed into a standard torus? >>
The identity homotopy from the empty set to the empty set.
What theorem were you employing to deduce that such an extension exists?
Hmmm. I was going to say the Homotopy Extension Property or the Homotopy Lifting property. But neither of those guarantees that the lifting/extension of an injective homotopy (by which I mean a homotopy injective on fibers) is injective. I don't have a formal proof, but it seems "intuitively obvious" to me that the injective homotopy from the "twice-around" circle to the geometric circle (or any injective homotopy of the circle to R^3) extends to an injective homotopy that starts from the identity map from R^3 to R^3. Restricting the "other end" of this ambient space homotopy to the mobius strip gives the desired object. Andy
WFL
On 4/3/20, James Buddenhagen <jbuddenh@gmail.com> wrote:
The circles (other than the central one) do not form a "second edge" they are just a grid on the surface of the object, which extends to infinity, but could be bent up to form a sphere, like the Riemann sphere, if you want a finite object. Adam P. Goucher explains in detail. And Mike Beeler indicates where you can see this object as a rotatable shapeways preview, but without the orthogonal grid of circles.
On Thu, Apr 2, 2020 at 10:04 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 9:23 PM Michael Kleber <michael.kleber@gmail.com> wrote:
Here is Henry Segerman's 3d-printable model: https://www.shapeways.com/product/YRBSFUEHB/round-mobius-strip
And now that I look at it a bit more carefully, this isn't the object I'm looking for either! This object has a circular edge, but it also has a second edge, consisting of 4 circular arcs with right angles between them. Maybe this object is meant in some way to suggest the object I'm thinking of, but it isn't literally that object.
Andy
On Thu, Apr 2, 2020 at 8:59 PM Andy Latto <andy.latto@pobox.com> wrote:
I think the surface you describe is a punctured Klein bottle, which is different from a mobius strip, which is a punctured projective plane.
Andy
On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com>
wrote:
I now have a "visualization" that I can understand, although, like
Andy, I
would like to have a 3D model to hold in my hand.
Imagine an ordinary circular disk in a horizontal plane. Punch two circular holes in the disk, and starting with the first hole, build a tube with circular cross section. The tube should run straight downward for a short distance, then bend to run horizontally until it is well clear of the underside of the original disk. Once clear, it can bend upward to cross the plane of the disk without interference from the disk itself, until it is comfortably above the plane of the disk. Now it can bend horizontal again and head for a point above the second hole, and on arrival above the second hole, bend downward to seal with it.
On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote:
> On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon <fred.lunnon@gmail.com> wrote: > > > > Presumably the "Sudanese Möbius Band" (credited to Sue Goodman > > & Dan Asimov) at > > > > https://en.wikipedia.org/wiki/M%C3%B6bius_strip > > > > I found these easier to interpret than Gosper's old-tech renderings. > > Plainly apparent in the first frame is a caustic line where the surface > > intersects itself, as might be expected. > > If you're right that this figure has a self-intersecting line, > it's > not the figure I'm looking for. I want a Mobius strip that is > *embedded* not just *immersed*, which means no self-intersections. > > Why the "as can be expected"? The standard embedding of a mobius strip > in R^3, the one you get by giving a strip of paper a half-twist > and > joining it into a band, has no self-intersections, and the embedding > of the boundary into R^3 is homotopic to the embedding of the > geometric circle. So you can gradually deform this figure, aways with > no self-intersections, into a figure where the edge is a geometric > circle. The fact that it feels like there must be a self-intersection > shows how difficult the resulting surface (which has no > self-intersections) is to visualize. > > Andy > > > > > WFL > > > > > > On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > > > On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon < fred.lunnon@gmail.com> > wrote: > > >> > > >> << embed a mobius strip in R^3 >> immerse, perhaps? WFL > > > > > > No, embedded. You can embed a Mobius strip with edge being homoptopic > > > to a geometric circle, so you can embed it with the edge actually > > > being a geometric circle. There are illlustrations on the wikipedia > > > page for mobius strip, but they aren't helping me visualize > > > it. > > > > > > I want someone to 3-d print me one of these! > > > > > > Andy > > > > > > > > >> > > >> > > >> > > >> On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > > >> > On Thu, Apr 2, 2020 at 3:46 PM James Propp < jamespropp@gmail.com> > > >> > wrote: > > >> >> > > >> >> I'm confused by there first sentence ("there's only one embedding > of a > > >> >> circle in R^3 up to homotopy"), since a knot isn't homotopic to an > > >> >> unknot. > > >> > > > >> > Sorry; I should have said "the embedding of a circle in R^3 given by > > >> > the edge of the most familiar embedding of the mobius strip in R3 is > > >> > homotopic to the embedding of a geometric circle in R^3, so... > > >> > > > >> > So while my argument was completely wrong, the conclusion that you > can > > >> > embed a mobius strip in R^3 with a geometric circle as boundary is > > >> > still true, as is the fact that my efforts to visualize > > >> > this have > > >> > proved completely unsuccessful. > > >> > > > >> >> > > >> >> But I think I understand and sympathize a lot of what follows; in > > >> >> particular, I'm pretty sure Klein bottles are easier to grok than > > >> >> Boy's surface for nearly everybody. I don't know whether this as a > > >> >> mathematical question or a psychological question or both, but I > think > > >> >> it's > > >> >> an interesting one! > > >> >> > > >> >> Jim > > >> >> > > >> >> On Thu, Apr 2, 2020 at 3:07 PM Andy Latto < andy.latto@pobox.com> > > >> >> wrote: > > >> >> > > >> >> > Since there's only one embedding of a circle in R^3 up > > >> >> > to > homotopy, > > >> >> > there's an embedding of a mobius strip in R^3 where the edge is a > > >> >> > geometric perfect circle. But I find myself unable to visualize > such > > >> >> > a > > >> >> > thing. Has anyone seen a 3-d model of this surface? Second-best > > >> >> > thing > > >> >> > would be a graphic of such a thing, preferably one that you could > > >> >> > rotate in 3 dimensions. > > >> >> > > > >> >> > I'd also like to better visualize Boy's surface, or any other > > >> >> > immersion of RP^2 in R^3. It would also be interesting > > >> >> > to have an > > >> >> > insight into why immersing a Klein bottle in R^3 is > > >> >> > easy, while > > >> >> > immersing RP2 is "hard". I don't know of any formal sense in > which > > >> >> > this is true, but apparently Boy came up with this surface when > > >> >> > challenged by Hilbert to prove that immersing RP^2 in R^3 was > > >> >> > impossible. > > >> >> > > > >> >> > Also, are these two questions related? That is, can you immerse a > > >> >> > mobius strip in R^3 in such a way that the boundary is a > geometric > > >> >> > circle, and that the union of this mobius strip and a disk with > the > > >> >> > same boundary is still an immersion (of RP^2 in R^3)? > > >> >> > > > >> >> > Andy Latto > > >> >> > > > >> >> > andy.latto@pobox.com > > >> >> > > > >> >> > -- > > >> >> > Andy.Latto@pobox.com > > >> >> > > > >> >> > _______________________________________________ > > >> >> > 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 > > >> > > > >> > > > >> > > > >> > -- > > >> > Andy.Latto@pobox.com > > >> > > > >> > _______________________________________________ > > >> > 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 > > > > > > > > > > > > -- > > > Andy.Latto@pobox.com > > > > > > _______________________________________________ > > > 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 > > > > -- > Andy.Latto@pobox.com > > _______________________________________________ > 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
I should confess straight away that I am very rusty about all this stuff. However surely no such restricted argument can possibly produce your desired conclusion --- despite the fact that it got lucky here, as Piker's visually low-key but pedagogically bang on-target animation amply demonstrates. The underlying obstruction is that embedded deformation involves the entire space: for example, a prerequisite is that not only the initial and final "Moebius strips" concerned are homotopic, but also their complements in |R^3 . And even that constraint is insufficient, as the example of mirror-image trefoil knots illustrates. WFL On 4/3/20, Andy Latto <andy.latto@pobox.com> wrote:
On Fri, Apr 3, 2020 at 12:38 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
On Apr 3, 2020, at 1:43 AM, Scott Kim <scott@scottkim.com> wrote:
Here's a movie of a rotating Sudanese surface. Quite informative. No singularities or going to infinity or self-intersecting...just a remarkably hard to grok surface in ordinary 3-space. https://vimeo.com/2037835
Seeing is believing --- can't argue with that!
Andy Latto<andy.latto@pobox.com> --- << I don't see how my argument proves anything of the kind. My argument starts with a homotopy between two closed curves in R^3, and extends it to a homotopy of the mobius strip in R^3. What is the homotopy you are extending in your hypothetical proof that a closed knotted torus can be transformed into a standard torus? >>
The identity homotopy from the empty set to the empty set.
What theorem were you employing to deduce that such an extension exists?
Hmmm. I was going to say the Homotopy Extension Property or the Homotopy Lifting property. But neither of those guarantees that the lifting/extension of an injective homotopy (by which I mean a homotopy injective on fibers) is injective.
I don't have a formal proof, but it seems "intuitively obvious" to me that the injective homotopy from the "twice-around" circle to the geometric circle (or any injective homotopy of the circle to R^3) extends to an injective homotopy that starts from the identity map from R^3 to R^3. Restricting the "other end" of this ambient space homotopy to the mobius strip gives the desired object.
Andy
WFL
On 4/3/20, James Buddenhagen <jbuddenh@gmail.com> wrote:
The circles (other than the central one) do not form a "second edge" they are just a grid on the surface of the object, which extends to infinity, but could be bent up to form a sphere, like the Riemann sphere, if you want a finite object. Adam P. Goucher explains in detail. And Mike Beeler indicates where you can see this object as a rotatable shapeways preview, but without the orthogonal grid of circles.
On Thu, Apr 2, 2020 at 10:04 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 9:23 PM Michael Kleber <michael.kleber@gmail.com> wrote:
Here is Henry Segerman's 3d-printable model: https://www.shapeways.com/product/YRBSFUEHB/round-mobius-strip
And now that I look at it a bit more carefully, this isn't the object I'm looking for either! This object has a circular edge, but it also has a second edge, consisting of 4 circular arcs with right angles between them. Maybe this object is meant in some way to suggest the object I'm thinking of, but it isn't literally that object.
Andy
On Thu, Apr 2, 2020 at 8:59 PM Andy Latto <andy.latto@pobox.com> wrote:
I think the surface you describe is a punctured Klein bottle, which is different from a mobius strip, which is a punctured projective plane.
Andy
On Thu, Apr 2, 2020 at 7:38 PM Allan Wechsler <acwacw@gmail.com>
wrote:
> > I now have a "visualization" that I can understand, although, > like Andy, I > would like to have a 3D model to hold in my hand. > > Imagine an ordinary circular disk in a horizontal plane. Punch > two circular > holes in the disk, and starting with the first hole, build a tube with > circular cross section. The tube should run straight downward for > a short > distance, then bend to run horizontally until it is well clear of > the > underside of the original disk. Once clear, it can bend upward to cross the > plane of the disk without interference from the disk itself, > until it is > comfortably above the plane of the disk. Now it can bend > horizontal again > and head for a point above the second hole, and on arrival above > the second > hole, bend downward to seal with it. > > On Thu, Apr 2, 2020 at 6:54 PM Andy Latto <andy.latto@pobox.com> wrote: > > > On Thu, Apr 2, 2020 at 6:35 PM Fred Lunnon > > <fred.lunnon@gmail.com> wrote: > > > > > > Presumably the "Sudanese Möbius Band" (credited to Sue > > > Goodman > > > & Dan Asimov) at > > > > > > https://en.wikipedia.org/wiki/M%C3%B6bius_strip > > > > > > I found these easier to interpret than Gosper's old-tech renderings. > > > Plainly apparent in the first frame is a caustic line where > > > the surface > > > intersects itself, as might be expected. > > > > If you're right that this figure has a self-intersecting line, > > it's > > not the figure I'm looking for. I want a Mobius strip that is > > *embedded* not just *immersed*, which means no > > self-intersections. > > > > Why the "as can be expected"? The standard embedding of a > > mobius strip > > in R^3, the one you get by giving a strip of paper a half-twist > > and > > joining it into a band, has no self-intersections, and the embedding > > of the boundary into R^3 is homotopic to the embedding of the > > geometric circle. So you can gradually deform this figure, > > aways with > > no self-intersections, into a figure where the edge is a > > geometric > > circle. The fact that it feels like there must be a self-intersection > > shows how difficult the resulting surface (which has no > > self-intersections) is to visualize. > > > > Andy > > > > > > > > WFL > > > > > > > > > On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > > > > On Thu, Apr 2, 2020 at 4:55 PM Fred Lunnon < fred.lunnon@gmail.com> > > wrote: > > > >> > > > >> << embed a mobius strip in R^3 >> immerse, perhaps? > > > >> WFL > > > > > > > > No, embedded. You can embed a Mobius strip with edge being homoptopic > > > > to a geometric circle, so you can embed it with the edge actually > > > > being a geometric circle. There are illlustrations on the wikipedia > > > > page for mobius strip, but they aren't helping me visualize > > > > it. > > > > > > > > I want someone to 3-d print me one of these! > > > > > > > > Andy > > > > > > > > > > > >> > > > >> > > > >> > > > >> On 4/2/20, Andy Latto <andy.latto@pobox.com> wrote: > > > >> > On Thu, Apr 2, 2020 at 3:46 PM James Propp < jamespropp@gmail.com> > > > >> > wrote: > > > >> >> > > > >> >> I'm confused by there first sentence ("there's only one embedding > > of a > > > >> >> circle in R^3 up to homotopy"), since a knot isn't homotopic to an > > > >> >> unknot. > > > >> > > > > >> > Sorry; I should have said "the embedding of a circle in > > > >> > R^3 given by > > > >> > the edge of the most familiar embedding of the mobius > > > >> > strip in R3 is > > > >> > homotopic to the embedding of a geometric circle in R^3, so... > > > >> > > > > >> > So while my argument was completely wrong, the > > > >> > conclusion that you > > can > > > >> > embed a mobius strip in R^3 with a geometric circle as boundary is > > > >> > still true, as is the fact that my efforts to visualize > > > >> > this have > > > >> > proved completely unsuccessful. > > > >> > > > > >> >> > > > >> >> But I think I understand and sympathize a lot of what follows; in > > > >> >> particular, I'm pretty sure Klein bottles are easier to grok than > > > >> >> Boy's surface for nearly everybody. I don't know > > > >> >> whether this as a > > > >> >> mathematical question or a psychological question or > > > >> >> both, but I > > think > > > >> >> it's > > > >> >> an interesting one! > > > >> >> > > > >> >> Jim > > > >> >> > > > >> >> On Thu, Apr 2, 2020 at 3:07 PM Andy Latto < andy.latto@pobox.com> > > > >> >> wrote: > > > >> >> > > > >> >> > Since there's only one embedding of a circle in R^3 > > > >> >> > up > > > >> >> > to > > homotopy, > > > >> >> > there's an embedding of a mobius strip in R^3 where > > > >> >> > the edge is a > > > >> >> > geometric perfect circle. But I find myself unable to visualize > > such > > > >> >> > a > > > >> >> > thing. Has anyone seen a 3-d model of this surface? Second-best > > > >> >> > thing > > > >> >> > would be a graphic of such a thing, preferably one > > > >> >> > that you could > > > >> >> > rotate in 3 dimensions. > > > >> >> > > > > >> >> > I'd also like to better visualize Boy's surface, or > > > >> >> > any other > > > >> >> > immersion of RP^2 in R^3. It would also be > > > >> >> > interesting > > > >> >> > to have an > > > >> >> > insight into why immersing a Klein bottle in R^3 is > > > >> >> > easy, while > > > >> >> > immersing RP2 is "hard". I don't know of any formal sense in > > which > > > >> >> > this is true, but apparently Boy came up with this surface when > > > >> >> > challenged by Hilbert to prove that immersing RP^2 in R^3 was > > > >> >> > impossible. > > > >> >> > > > > >> >> > Also, are these two questions related? That is, can > > > >> >> > you immerse a > > > >> >> > mobius strip in R^3 in such a way that the boundary > > > >> >> > is a > > geometric > > > >> >> > circle, and that the union of this mobius strip and a disk with > > the > > > >> >> > same boundary is still an immersion (of RP^2 in R^3)? > > > >> >> > > > > >> >> > Andy Latto > > > >> >> > > > > >> >> > andy.latto@pobox.com > > > >> >> > > > > >> >> > -- > > > >> >> > Andy.Latto@pobox.com > > > >> >> > > > > >> >> > _______________________________________________ > > > >> >> > 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 > > > >> > > > > >> > > > > >> > > > > >> > -- > > > >> > Andy.Latto@pobox.com > > > >> > > > > >> > _______________________________________________ > > > >> > 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 > > > > > > > > > > > > > > > > -- > > > > Andy.Latto@pobox.com > > > > > > > > _______________________________________________ > > > > 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 > > > > > > > > -- > > Andy.Latto@pobox.com > > > > _______________________________________________ > > 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Fri, Apr 3, 2020 at 12:38 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
What theorem were you employing to deduce that such an extension exists?
I found the theorem that confirms my intuitions; its the Isotopy Extension Theorem, which only works for manifolds in the smooth category (Alexander's Horned Sphere is a counterexample in the topological category). http://math.ucr.edu/~res/math260s10/isotopyextension.pdf Isotopy is what I was calling "homotopy injective on fibers"; a homotopy I X N -> M is an isotopy if the restriction {t} X N -> M is an embedding. The theorem says that if N is a smooth compact manifold M is a smooth manifold, and f: N X I -> M is a smooth isotopy, that it extends to an ambient isotopy, that is an isotopy f* : M x I -> M. Applying this to the isotopy that "unfolds" the boundary of the standard embedding of the mobius strip to the geometric circle, and then restricting the endpoint of this map to the original mobius strip, we have the desired embedding of the mobius strip with a geometric circle as an edge. The isotopic extension property (in the case where N is the circle and M is R^3, which is the same case we used here) is what you need to show that the fundamental group of the knot complement is a knot invariant. Andy
So what this is saying is that ensuring sufficient (smooth) structure obviates any need explicitly to involve the rest of the exterior space. Which I could sort of see from the start, but failed to appreciate the significance of involving the fibre/fiber. Overall, excellently instructive problem & discussion! WFL On 4/6/20, Andy Latto <andy.latto@pobox.com> wrote:
On Fri, Apr 3, 2020 at 12:38 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
What theorem were you employing to deduce that such an extension exists?
I found the theorem that confirms my intuitions; its the Isotopy Extension Theorem, which only works for manifolds in the smooth category (Alexander's Horned Sphere is a counterexample in the topological category). http://math.ucr.edu/~res/math260s10/isotopyextension.pdf
Isotopy is what I was calling "homotopy injective on fibers"; a homotopy I X N -> M is an isotopy if the restriction {t} X N -> M is an embedding.
The theorem says that if N is a smooth compact manifold M is a smooth manifold, and f: N X I -> M is a smooth isotopy, that it extends to an ambient isotopy, that is an isotopy f* : M x I -> M. Applying this to the isotopy that "unfolds" the boundary of the standard embedding of the mobius strip to the geometric circle, and then restricting the endpoint of this map to the original mobius strip, we have the desired embedding of the mobius strip with a geometric circle as an edge.
The isotopic extension property (in the case where N is the circle and M is R^3, which is the same case we used here) is what you need to show that the fundamental group of the knot complement is a knot invariant.
Andy
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
When I search Google for moebius + circular + boundary, I see several images that are clearly the embedding Andy Latto is asking about. They look vaguely conch-like, and from these still images, I can't really grok what's going on. An intermediate stage would be one whose edge traced a figure-8 path, displaced a little at the crossing so the edge didn't self-intersect. On Thu, Apr 2, 2020 at 4:14 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 3:46 PM James Propp <jamespropp@gmail.com> wrote:
I'm confused by there first sentence ("there's only one embedding of a circle in R^3 up to homotopy"), since a knot isn't homotopic to an
unknot.
Sorry; I should have said "the embedding of a circle in R^3 given by the edge of the most familiar embedding of the mobius strip in R3 is homotopic to the embedding of a geometric circle in R^3, so...
So while my argument was completely wrong, the conclusion that you can embed a mobius strip in R^3 with a geometric circle as boundary is still true, as is the fact that my efforts to visualize this have proved completely unsuccessful.
But I think I understand and sympathize a lot of what follows; in particular, I'm pretty sure Klein bottles are easier to grok than Boy's surface for nearly everybody. I don't know whether this as a mathematical question or a psychological question or both, but I think
it's
an interesting one!
Jim
On Thu, Apr 2, 2020 at 3:07 PM Andy Latto <andy.latto@pobox.com> wrote:
Since there's only one embedding of a circle in R^3 up to homotopy, there's an embedding of a mobius strip in R^3 where the edge is a geometric perfect circle. But I find myself unable to visualize such a thing. Has anyone seen a 3-d model of this surface? Second-best thing would be a graphic of such a thing, preferably one that you could rotate in 3 dimensions.
I'd also like to better visualize Boy's surface, or any other immersion of RP^2 in R^3. It would also be interesting to have an insight into why immersing a Klein bottle in R^3 is easy, while immersing RP2 is "hard". I don't know of any formal sense in which this is true, but apparently Boy came up with this surface when challenged by Hilbert to prove that immersing RP^2 in R^3 was impossible.
Also, are these two questions related? That is, can you immerse a mobius strip in R^3 in such a way that the boundary is a geometric circle, and that the union of this mobius strip and a disk with the same boundary is still an immersion (of RP^2 in R^3)?
Andy Latto
andy.latto@pobox.com
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Presumably these are Gosper's (Macsyma generated?) plots http://www.tweedledum.com/rwg/Moebius.htm The trouble is that the distance cues just don't work ... we need a complete animation at this juncture! WFL On 4/2/20, Allan Wechsler <acwacw@gmail.com> wrote:
When I search Google for moebius + circular + boundary, I see several images that are clearly the embedding Andy Latto is asking about. They look vaguely conch-like, and from these still images, I can't really grok what's going on.
An intermediate stage would be one whose edge traced a figure-8 path, displaced a little at the crossing so the edge didn't self-intersect.
On Thu, Apr 2, 2020 at 4:14 PM Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 2, 2020 at 3:46 PM James Propp <jamespropp@gmail.com> wrote:
I'm confused by there first sentence ("there's only one embedding of a circle in R^3 up to homotopy"), since a knot isn't homotopic to an
unknot.
Sorry; I should have said "the embedding of a circle in R^3 given by the edge of the most familiar embedding of the mobius strip in R3 is homotopic to the embedding of a geometric circle in R^3, so...
So while my argument was completely wrong, the conclusion that you can embed a mobius strip in R^3 with a geometric circle as boundary is still true, as is the fact that my efforts to visualize this have proved completely unsuccessful.
But I think I understand and sympathize a lot of what follows; in particular, I'm pretty sure Klein bottles are easier to grok than Boy's surface for nearly everybody. I don't know whether this as a mathematical question or a psychological question or both, but I think
it's
an interesting one!
Jim
On Thu, Apr 2, 2020 at 3:07 PM Andy Latto <andy.latto@pobox.com> wrote:
Since there's only one embedding of a circle in R^3 up to homotopy, there's an embedding of a mobius strip in R^3 where the edge is a geometric perfect circle. But I find myself unable to visualize such a thing. Has anyone seen a 3-d model of this surface? Second-best thing would be a graphic of such a thing, preferably one that you could rotate in 3 dimensions.
I'd also like to better visualize Boy's surface, or any other immersion of RP^2 in R^3. It would also be interesting to have an insight into why immersing a Klein bottle in R^3 is easy, while immersing RP2 is "hard". I don't know of any formal sense in which this is true, but apparently Boy came up with this surface when challenged by Hilbert to prove that immersing RP^2 in R^3 was impossible.
Also, are these two questions related? That is, can you immerse a mobius strip in R^3 in such a way that the boundary is a geometric circle, and that the union of this mobius strip and a disk with the same boundary is still an immersion (of RP^2 in R^3)?
Andy Latto
andy.latto@pobox.com
-- Andy.Latto@pobox.com
_______________________________________________ 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
-- Andy.Latto@pobox.com
_______________________________________________ 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
participants (9)
-
Adam P. Goucher -
Allan Wechsler -
Andy Latto -
Fred Lunnon -
James Buddenhagen -
James Propp -
Michael Kleber -
Mike Beeler -
Scott Kim