Re: [math-fun] Everting a torus
Sure it's honest. IF by that you mean: ----- Is it an isotopy* of the torus T^2 minus a disk D into R^3 that starts with a standard torus with a small round disk removed, and ends with the torus-with-a-hole occupying the same image in R^3 but with the generators** interchanged? ----- You can easily see that it's honest by letting it progress slowly (I have to start and stop it to do this; YMMV.) It's easy to follow either of the two types of holes** and see that it ends up as the other type. (OK, the teensy cheat is only that the removed disk ends up elsewhere from where it started, but a rotation about the z-axis will easily fix that.) —Dan ——————————————————————— * that is, a continuous family of homeomorphisms, or in technical terms, a map H: H: (T^2 - D)x[0,1] —> R^3 ** the conventional types of basis "holes" in a torus: a longitude (the long way around) and a meridian (the short way). -----Original Message-----
From: James Propp <jamespropp@gmail.com> Sent: Jul 14, 2016 1:27 PM To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Everting a torus
Compare the easy-to-follow choreography at https://www.youtube.com/watch?v=S4ddRPvwcZI (due to Greg McShane, I'm guessing) with the much subtler choreography at https://www.youtube.com/watch?v=jA86M6fdm_Q (due to Arnaud Cheritat, I'm guessing).
Is the former eversion honest? Note that a circular patch has been cut away from the torus to make the action easier to follow; this permits unintended cheating. After all, you could make a video of a sphere-with-a-hole-in-it being turned inside out, and it would look convincing, unless you think hard about what happens to the boundary of the patch that bounds the hole.
The same goes for a recent gif of Simon Pampena's (which I can view on Twitter but can't find on the web).
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Actually, by "honest" I mean "free of pinch singularities". And now that I'm looking more carefully and thinking harder, I'm pretty sure there IS a pinch at 0:04. That is, I don't think the video shows what Smale would call an eversion of the torus. Jim Propp On Thursday, July 14, 2016, Dan Asimov <dasimov@earthlink.net> wrote:
Sure it's honest. IF by that you mean:
----- Is it an isotopy* of the torus T^2 minus a disk D into R^3 that starts with a standard torus with a small round disk removed, and ends with the torus-with-a-hole occupying the same image in R^3 but with the generators** interchanged? -----
You can easily see that it's honest by letting it progress slowly (I have to start and stop it to do this; YMMV.) It's easy to follow either of the two types of holes** and see that it ends up as the other type.
(OK, the teensy cheat is only that the removed disk ends up elsewhere from where it started, but a rotation about the z-axis will easily fix that.)
—Dan
——————————————————————— * that is, a continuous family of homeomorphisms, or in technical terms, a map H:
H: (T^2 - D)x[0,1] —> R^3
** the conventional types of basis "holes" in a torus: a longitude (the long way around) and a meridian (the short way).
-----Original Message-----
From: James Propp <jamespropp@gmail.com <javascript:;>> Sent: Jul 14, 2016 1:27 PM To: math-fun <math-fun@mailman.xmission.com <javascript:;>> Subject: [math-fun] Everting a torus
Compare the easy-to-follow choreography at https://www.youtube.com/watch?v=S4ddRPvwcZI (due to Greg McShane, I'm guessing) with the much subtler choreography at https://www.youtube.com/watch?v=jA86M6fdm_Q (due to Arnaud Cheritat, I'm guessing).
Is the former eversion honest? Note that a circular patch has been cut away from the torus to make the action easier to follow; this permits unintended cheating. After all, you could make a video of a sphere-with-a-hole-in-it being turned inside out, and it would look convincing, unless you think hard about what happens to the boundary of the patch that bounds the hole.
The same goes for a recent gif of Simon Pampena's (which I can view on Twitter but can't find on the web).
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Let me be more precise: I don't think the video in any way corresponds to an eversion of an UNMUTILATED torus. (It does show an interesting way to deform a mutilated torus with self-intersections allowed, but that's not the same.) Am I right? Jim On Thursday, July 14, 2016, James Propp <jamespropp@gmail.com> wrote:
Actually, by "honest" I mean "free of pinch singularities".
And now that I'm looking more carefully and thinking harder, I'm pretty sure there IS a pinch at 0:04.
That is, I don't think the video shows what Smale would call an eversion of the torus.
Jim Propp
On Thursday, July 14, 2016, Dan Asimov <dasimov@earthlink.net <javascript:_e(%7B%7D,'cvml','dasimov@earthlink.net');>> wrote:
Sure it's honest. IF by that you mean:
----- Is it an isotopy* of the torus T^2 minus a disk D into R^3 that starts with a standard torus with a small round disk removed, and ends with the torus-with-a-hole occupying the same image in R^3 but with the generators** interchanged? -----
You can easily see that it's honest by letting it progress slowly (I have to start and stop it to do this; YMMV.) It's easy to follow either of the two types of holes** and see that it ends up as the other type.
(OK, the teensy cheat is only that the removed disk ends up elsewhere from where it started, but a rotation about the z-axis will easily fix that.)
—Dan
——————————————————————— * that is, a continuous family of homeomorphisms, or in technical terms, a map H:
H: (T^2 - D)x[0,1] —> R^3
** the conventional types of basis "holes" in a torus: a longitude (the long way around) and a meridian (the short way).
-----Original Message-----
From: James Propp <jamespropp@gmail.com> Sent: Jul 14, 2016 1:27 PM To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Everting a torus
Compare the easy-to-follow choreography at https://www.youtube.com/watch?v=S4ddRPvwcZI (due to Greg McShane, I'm guessing) with the much subtler choreography at https://www.youtube.com/watch?v=jA86M6fdm_Q (due to Arnaud Cheritat, I'm guessing).
Is the former eversion honest? Note that a circular patch has been cut away from the torus to make the action easier to follow; this permits unintended cheating. After all, you could make a video of a sphere-with-a-hole-in-it being turned inside out, and it would look convincing, unless you think hard about what happens to the boundary of the patch that bounds the hole.
The same goes for a recent gif of Simon Pampena's (which I can view on Twitter but can't find on the web).
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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The two videos demonstrate different phenomena: as Dan said, the first shows _embedded_ inversion of a _punctured_ torus; the second shows _immersed_ inversion of a _complete_ torus (no hole, but self-intersections). WFL On 7/14/16, James Propp <jamespropp@gmail.com> wrote:
Let me be more precise: I don't think the video in any way corresponds to an eversion of an UNMUTILATED torus.
(It does show an interesting way to deform a mutilated torus with self-intersections allowed, but that's not the same.)
Am I right?
Jim
On Thursday, July 14, 2016, James Propp <jamespropp@gmail.com> wrote:
Actually, by "honest" I mean "free of pinch singularities".
And now that I'm looking more carefully and thinking harder, I'm pretty sure there IS a pinch at 0:04.
That is, I don't think the video shows what Smale would call an eversion of the torus.
Jim Propp
On Thursday, July 14, 2016, Dan Asimov <dasimov@earthlink.net <javascript:_e(%7B%7D,'cvml','dasimov@earthlink.net');>> wrote:
Sure it's honest. IF by that you mean:
----- Is it an isotopy* of the torus T^2 minus a disk D into R^3 that starts with a standard torus with a small round disk removed, and ends with the torus-with-a-hole occupying the same image in R^3 but with the generators** interchanged? -----
You can easily see that it's honest by letting it progress slowly (I have to start and stop it to do this; YMMV.) It's easy to follow either of the two types of holes** and see that it ends up as the other type.
(OK, the teensy cheat is only that the removed disk ends up elsewhere from where it started, but a rotation about the z-axis will easily fix that.)
—Dan
——————————————————————— * that is, a continuous family of homeomorphisms, or in technical terms, a map H:
H: (T^2 - D)x[0,1] —> R^3
** the conventional types of basis "holes" in a torus: a longitude (the long way around) and a meridian (the short way).
-----Original Message-----
From: James Propp <jamespropp@gmail.com> Sent: Jul 14, 2016 1:27 PM To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Everting a torus
Compare the easy-to-follow choreography at https://www.youtube.com/watch?v=S4ddRPvwcZI (due to Greg McShane, I'm guessing) with the much subtler choreography at https://www.youtube.com/watch?v=jA86M6fdm_Q (due to Arnaud Cheritat, I'm guessing).
Is the former eversion honest? Note that a circular patch has been cut away from the torus to make the action easier to follow; this permits unintended cheating. After all, you could make a video of a sphere-with-a-hole-in-it being turned inside out, and it would look convincing, unless you think hard about what happens to the boundary of the patch that bounds the hole.
The same goes for a recent gif of Simon Pampena's (which I can view on Twitter but can't find on the web).
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I should have read Dan's email more carefully. Jim On Thursday, July 14, 2016, Fred Lunnon <fred.lunnon@gmail.com> wrote:
The two videos demonstrate different phenomena: as Dan said, the first shows _embedded_ inversion of a _punctured_ torus; the second shows _immersed_ inversion of a _complete_ torus (no hole, but self-intersections). WFL
On 7/14/16, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
Let me be more precise: I don't think the video in any way corresponds to an eversion of an UNMUTILATED torus.
(It does show an interesting way to deform a mutilated torus with self-intersections allowed, but that's not the same.)
Am I right?
Jim
On Thursday, July 14, 2016, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
Actually, by "honest" I mean "free of pinch singularities".
And now that I'm looking more carefully and thinking harder, I'm pretty sure there IS a pinch at 0:04.
That is, I don't think the video shows what Smale would call an eversion of the torus.
Jim Propp
On Thursday, July 14, 2016, Dan Asimov <dasimov@earthlink.net <javascript:;> <javascript:_e(%7B%7D,'cvml','dasimov@earthlink.net <javascript:;>');>> wrote:
Sure it's honest. IF by that you mean:
----- Is it an isotopy* of the torus T^2 minus a disk D into R^3 that starts with a standard torus with a small round disk removed, and ends with the torus-with-a-hole occupying the same image in R^3 but with the generators** interchanged? -----
You can easily see that it's honest by letting it progress slowly (I have to start and stop it to do this; YMMV.) It's easy to follow either of the two types of holes** and see that it ends up as the other type.
(OK, the teensy cheat is only that the removed disk ends up elsewhere from where it started, but a rotation about the z-axis will easily fix that.)
—Dan
——————————————————————— * that is, a continuous family of homeomorphisms, or in technical terms, a map H:
H: (T^2 - D)x[0,1] —> R^3
** the conventional types of basis "holes" in a torus: a longitude (the long way around) and a meridian (the short way).
-----Original Message-----
From: James Propp <jamespropp@gmail.com <javascript:;>> Sent: Jul 14, 2016 1:27 PM To: math-fun <math-fun@mailman.xmission.com <javascript:;>> Subject: [math-fun] Everting a torus
Compare the easy-to-follow choreography at https://www.youtube.com/watch?v=S4ddRPvwcZI (due to Greg McShane, I'm guessing) with the much subtler choreography at https://www.youtube.com/watch?v=jA86M6fdm_Q (due to Arnaud Cheritat, I'm guessing).
Is the former eversion honest? Note that a circular patch has been cut away from the torus to make the action easier to follow; this permits unintended cheating. After all, you could make a video of a sphere-with-a-hole-in-it being turned inside out, and it would look convincing, unless you think hard about what happens to the boundary of the patch that bounds the hole.
The same goes for a recent gif of Simon Pampena's (which I can view on Twitter but can't find on the web).
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (3)
-
Dan Asimov -
Fred Lunnon -
James Propp