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
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