Re: [math-fun] Moebius-like torus name?
Steve Witham <sw@tiac.net> wrote:
What does one call a tube looped around as if to make a torus, but connected clockwise-to-counterclockwise, as in this sewing pattern?
z y>--C-->z y +---------+ x|w>--A-->x|w v|^ v|^ ||| ||| B|D B|D ||| ||| v|^ v|^ y|z<--C--<y|z +---------+ w x<--A--<w x
If I'm understanding your diagram correctly, that's a Purse of Fortunatus. If you start with a square, and attach the top to the bottom without any twists and attach the left side to the right side without any twists, you get a torus. If, instead, you attach either the top to the bottom or the left to the right with a twist, i.e. reverse the direction, you get a Klein Bottle. If you reverse the directions of both connections, you get a Purse of Fortunatus. None of these surfaces have any intrinsic curvature (i.e. triangles have the same sum of angles, and circles have the same ratio of circumference to diameter, as in a flat plane). Their embeddings in 3-space, however, have extrinsic curvature. And, except for the torus, are self-intersecting unless embedded in 4-space or higher. The Klein Bottle and the Purse of Fortunatus are non-orientable, i.e. the chirality of a figure can change by moving the figure. I don't know what the analogous results would be in higher dimensions, i.e. gluing opposite sides of a cube together, flipping none, some, or all of them. It would be interesting if one of those was the topology of our universe. I've read that some astronomers have done searches for repeating patterns of galactic clusters in an attempt to test this. If our space is non-orientable, then if you travel far enough in a straight line, you'll return to Earth swapped left-to-right. That might mean that you'd be converted to anti-matter. Or it might just mean that pine trees would smell like lemons and you couldn't digest the food. It would be a good defense against viruses, however.
Is there a non-Archimedean way of measuring extrinsic curvature that allows infinite values as well as finite ones, so that the Gauss-Bonnet theorem remains true for polyhedra (and for self-intersecting polyhedral models of Klein bottles and projective planes)? It should be something like a combination of differential geometry and distribution theory. Jim Propp On Sun, May 3, 2020 at 1:51 PM Keith F. Lynch <kfl@keithlynch.net> wrote:
Steve Witham <sw@tiac.net> wrote:
What does one call a tube looped around as if to make a torus, but connected clockwise-to-counterclockwise, as in this sewing pattern?
z y>--C-->z y +---------+ x|w>--A-->x|w v|^ v|^ ||| ||| B|D B|D ||| ||| v|^ v|^ y|z<--C--<y|z +---------+ w x<--A--<w x
If I'm understanding your diagram correctly, that's a Purse of Fortunatus.
If you start with a square, and attach the top to the bottom without any twists and attach the left side to the right side without any twists, you get a torus.
If, instead, you attach either the top to the bottom or the left to the right with a twist, i.e. reverse the direction, you get a Klein Bottle.
If you reverse the directions of both connections, you get a Purse of Fortunatus.
None of these surfaces have any intrinsic curvature (i.e. triangles have the same sum of angles, and circles have the same ratio of circumference to diameter, as in a flat plane). Their embeddings in 3-space, however, have extrinsic curvature. And, except for the torus, are self-intersecting unless embedded in 4-space or higher.
The Klein Bottle and the Purse of Fortunatus are non-orientable, i.e. the chirality of a figure can change by moving the figure.
I don't know what the analogous results would be in higher dimensions, i.e. gluing opposite sides of a cube together, flipping none, some, or all of them. It would be interesting if one of those was the topology of our universe. I've read that some astronomers have done searches for repeating patterns of galactic clusters in an attempt to test this.
If our space is non-orientable, then if you travel far enough in a straight line, you'll return to Earth swapped left-to-right. That might mean that you'd be converted to anti-matter. Or it might just mean that pine trees would smell like lemons and you couldn't digest the food. It would be a good defense against viruses, however.
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<< Purse of Fortunatus >> Or, rather more prosaically, (real) projective plane. WFL On 5/3/20, James Propp <jamespropp@gmail.com> wrote:
Is there a non-Archimedean way of measuring extrinsic curvature that allows infinite values as well as finite ones, so that the Gauss-Bonnet theorem remains true for polyhedra (and for self-intersecting polyhedral models of Klein bottles and projective planes)?
It should be something like a combination of differential geometry and distribution theory.
Jim Propp
On Sun, May 3, 2020 at 1:51 PM Keith F. Lynch <kfl@keithlynch.net> wrote:
Steve Witham <sw@tiac.net> wrote:
What does one call a tube looped around as if to make a torus, but connected clockwise-to-counterclockwise, as in this sewing pattern?
z y>--C-->z y +---------+ x|w>--A-->x|w v|^ v|^ ||| ||| B|D B|D ||| ||| v|^ v|^ y|z<--C--<y|z +---------+ w x<--A--<w x
If I'm understanding your diagram correctly, that's a Purse of Fortunatus.
If you start with a square, and attach the top to the bottom without any twists and attach the left side to the right side without any twists, you get a torus.
If, instead, you attach either the top to the bottom or the left to the right with a twist, i.e. reverse the direction, you get a Klein Bottle.
If you reverse the directions of both connections, you get a Purse of Fortunatus.
None of these surfaces have any intrinsic curvature (i.e. triangles have the same sum of angles, and circles have the same ratio of circumference to diameter, as in a flat plane). Their embeddings in 3-space, however, have extrinsic curvature. And, except for the torus, are self-intersecting unless embedded in 4-space or higher.
The Klein Bottle and the Purse of Fortunatus are non-orientable, i.e. the chirality of a figure can change by moving the figure.
I don't know what the analogous results would be in higher dimensions, i.e. gluing opposite sides of a cube together, flipping none, some, or all of them. It would be interesting if one of those was the topology of our universe. I've read that some astronomers have done searches for repeating patterns of galactic clusters in an attempt to test this.
If our space is non-orientable, then if you travel far enough in a straight line, you'll return to Earth swapped left-to-right. That might mean that you'd be converted to anti-matter. Or it might just mean that pine trees would smell like lemons and you couldn't digest the food. It would be a good defense against viruses, however.
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Not sure whether this is what you're looking for, but you can patch up the Gauss-Bonnet theorem to include surfaces with vertices, that is, surfaces that are locally one of: 1. Smooth 2. Look like the edge of a polyhedron, that is, a straight edge that is the boundary of two smooth surfaces 3. Look like the vertex of a convex polyhedron, that is, a finite number of the edges from 2 meeting in a point. in order for the sum to be correct, you have to add to the integral of the curvature, a term for each vertex, where the total curvature in an infinitesimal region surrounding the vertex is the difference between tau and the sum of the angles between the edges. So if you want to state this as an integral, the curvature at the vertex would need to be a delta function times the angle defect. If the edges aren't straight, you also would need a term that's a line integral along each edge. I think the integrand would be something like the product of the curvature of the edge and some term that depends on the angle formed between the two surfaces, possibly pi minus the angle. Something that approaches some fixed finite value as the angle approaches 0, and 0 as the angle approaches pi. Andy Andy On Sun, May 3, 2020 at 2:14 PM James Propp <jamespropp@gmail.com> wrote:
Is there a non-Archimedean way of measuring extrinsic curvature that allows infinite values as well as finite ones, so that the Gauss-Bonnet theorem remains true for polyhedra (and for self-intersecting polyhedral models of Klein bottles and projective planes)?
It should be something like a combination of differential geometry and distribution theory.
Jim Propp
On Sun, May 3, 2020 at 1:51 PM Keith F. Lynch <kfl@keithlynch.net> wrote:
Steve Witham <sw@tiac.net> wrote:
What does one call a tube looped around as if to make a torus, but connected clockwise-to-counterclockwise, as in this sewing pattern?
z y>--C-->z y +---------+ x|w>--A-->x|w v|^ v|^ ||| ||| B|D B|D ||| ||| v|^ v|^ y|z<--C--<y|z +---------+ w x<--A--<w x
If I'm understanding your diagram correctly, that's a Purse of Fortunatus.
If you start with a square, and attach the top to the bottom without any twists and attach the left side to the right side without any twists, you get a torus.
If, instead, you attach either the top to the bottom or the left to the right with a twist, i.e. reverse the direction, you get a Klein Bottle.
If you reverse the directions of both connections, you get a Purse of Fortunatus.
None of these surfaces have any intrinsic curvature (i.e. triangles have the same sum of angles, and circles have the same ratio of circumference to diameter, as in a flat plane). Their embeddings in 3-space, however, have extrinsic curvature. And, except for the torus, are self-intersecting unless embedded in 4-space or higher.
The Klein Bottle and the Purse of Fortunatus are non-orientable, i.e. the chirality of a figure can change by moving the figure.
I don't know what the analogous results would be in higher dimensions, i.e. gluing opposite sides of a cube together, flipping none, some, or all of them. It would be interesting if one of those was the topology of our universe. I've read that some astronomers have done searches for repeating patterns of galactic clusters in an attempt to test this.
If our space is non-orientable, then if you travel far enough in a straight line, you'll return to Earth swapped left-to-right. That might mean that you'd be converted to anti-matter. Or it might just mean that pine trees would smell like lemons and you couldn't digest the food. It would be a good defense against viruses, however.
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-- Andy.Latto@pobox.com
Georg, Brian: I have confidence in you two contributors. If you agree on the changes that need to be made, both the sequence that is wrong, and the incorrect A-numbers, and anything else that you discover, then please go ahead and fix things without me. (I am rather short of time right now) Thanks Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sun, May 3, 2020 at 3:26 PM Andy Latto <andy.latto@pobox.com> wrote:
Not sure whether this is what you're looking for, but you can patch up the Gauss-Bonnet theorem to include surfaces with vertices, that is, surfaces that are locally one of:
1. Smooth 2. Look like the edge of a polyhedron, that is, a straight edge that is the boundary of two smooth surfaces 3. Look like the vertex of a convex polyhedron, that is, a finite number of the edges from 2 meeting in a point.
in order for the sum to be correct, you have to add to the integral of the curvature, a term for each vertex, where the total curvature in an infinitesimal region surrounding the vertex is the difference between tau and the sum of the angles between the edges.
So if you want to state this as an integral, the curvature at the vertex would need to be a delta function times the angle defect.
If the edges aren't straight, you also would need a term that's a line integral along each edge. I think the integrand would be something like the product of the curvature of the edge and some term that depends on the angle formed between the two surfaces, possibly pi minus the angle. Something that approaches some fixed finite value as the angle approaches 0, and 0 as the angle approaches pi.
Andy
Andy
On Sun, May 3, 2020 at 2:14 PM James Propp <jamespropp@gmail.com> wrote:
Is there a non-Archimedean way of measuring extrinsic curvature that
allows
infinite values as well as finite ones, so that the Gauss-Bonnet theorem remains true for polyhedra (and for self-intersecting polyhedral models of Klein bottles and projective planes)?
It should be something like a combination of differential geometry and distribution theory.
Jim Propp
On Sun, May 3, 2020 at 1:51 PM Keith F. Lynch <kfl@keithlynch.net> wrote:
Steve Witham <sw@tiac.net> wrote:
What does one call a tube looped around as if to make a torus, but connected clockwise-to-counterclockwise, as in this sewing pattern?
z y>--C-->z y +---------+ x|w>--A-->x|w v|^ v|^ ||| ||| B|D B|D ||| ||| v|^ v|^ y|z<--C--<y|z +---------+ w x<--A--<w x
If I'm understanding your diagram correctly, that's a Purse of Fortunatus.
If you start with a square, and attach the top to the bottom without any twists and attach the left side to the right side without any twists, you get a torus.
If, instead, you attach either the top to the bottom or the left to the right with a twist, i.e. reverse the direction, you get a Klein Bottle.
If you reverse the directions of both connections, you get a Purse of Fortunatus.
None of these surfaces have any intrinsic curvature (i.e. triangles have the same sum of angles, and circles have the same ratio of circumference to diameter, as in a flat plane). Their embeddings in 3-space, however, have extrinsic curvature. And, except for the torus, are self-intersecting unless embedded in 4-space or higher.
The Klein Bottle and the Purse of Fortunatus are non-orientable, i.e. the chirality of a figure can change by moving the figure.
I don't know what the analogous results would be in higher dimensions, i.e. gluing opposite sides of a cube together, flipping none, some, or all of them. It would be interesting if one of those was the topology of our universe. I've read that some astronomers have done searches for repeating patterns of galactic clusters in an attempt to test this.
If our space is non-orientable, then if you travel far enough in a straight line, you'll return to Earth swapped left-to-right. That might mean that you'd be converted to anti-matter. Or it might just mean that pine trees would smell like lemons and you couldn't digest the food. It would be a good defense against viruses, however.
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-- Andy.Latto@pobox.com
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I think gmail just tricked me into replying to the wrong email (by shifting the email queue under my cursor). Please ignore that email beginning "Georg, Brian" Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sun, May 3, 2020 at 8:10 PM Neil Sloane <njasloane@gmail.com> wrote:
Georg, Brian:
I have confidence in you two contributors. If you agree on the changes that need to be made, both the sequence that is wrong, and the incorrect A-numbers, and anything else that you discover, then please go ahead and fix things without me.
(I am rather short of time right now)
Thanks
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Sun, May 3, 2020 at 3:26 PM Andy Latto <andy.latto@pobox.com> wrote:
Not sure whether this is what you're looking for, but you can patch up the Gauss-Bonnet theorem to include surfaces with vertices, that is, surfaces that are locally one of:
1. Smooth 2. Look like the edge of a polyhedron, that is, a straight edge that is the boundary of two smooth surfaces 3. Look like the vertex of a convex polyhedron, that is, a finite number of the edges from 2 meeting in a point.
in order for the sum to be correct, you have to add to the integral of the curvature, a term for each vertex, where the total curvature in an infinitesimal region surrounding the vertex is the difference between tau and the sum of the angles between the edges.
So if you want to state this as an integral, the curvature at the vertex would need to be a delta function times the angle defect.
If the edges aren't straight, you also would need a term that's a line integral along each edge. I think the integrand would be something like the product of the curvature of the edge and some term that depends on the angle formed between the two surfaces, possibly pi minus the angle. Something that approaches some fixed finite value as the angle approaches 0, and 0 as the angle approaches pi.
Andy
Andy
On Sun, May 3, 2020 at 2:14 PM James Propp <jamespropp@gmail.com> wrote:
Is there a non-Archimedean way of measuring extrinsic curvature that
allows
infinite values as well as finite ones, so that the Gauss-Bonnet theorem remains true for polyhedra (and for self-intersecting polyhedral models of Klein bottles and projective planes)?
It should be something like a combination of differential geometry and distribution theory.
Jim Propp
On Sun, May 3, 2020 at 1:51 PM Keith F. Lynch <kfl@keithlynch.net> wrote:
Steve Witham <sw@tiac.net> wrote:
What does one call a tube looped around as if to make a torus, but connected clockwise-to-counterclockwise, as in this sewing pattern?
z y>--C-->z y +---------+ x|w>--A-->x|w v|^ v|^ ||| ||| B|D B|D ||| ||| v|^ v|^ y|z<--C--<y|z +---------+ w x<--A--<w x
If I'm understanding your diagram correctly, that's a Purse of Fortunatus.
If you start with a square, and attach the top to the bottom without any twists and attach the left side to the right side without any twists, you get a torus.
If, instead, you attach either the top to the bottom or the left to the right with a twist, i.e. reverse the direction, you get a Klein Bottle.
If you reverse the directions of both connections, you get a Purse of Fortunatus.
None of these surfaces have any intrinsic curvature (i.e. triangles have the same sum of angles, and circles have the same ratio of circumference to diameter, as in a flat plane). Their embeddings in 3-space, however, have extrinsic curvature. And, except for the torus, are self-intersecting unless embedded in 4-space or higher.
The Klein Bottle and the Purse of Fortunatus are non-orientable, i.e. the chirality of a figure can change by moving the figure.
I don't know what the analogous results would be in higher dimensions, i.e. gluing opposite sides of a cube together, flipping none, some, or all of them. It would be interesting if one of those was the topology of our universe. I've read that some astronomers have done searches for repeating patterns of galactic clusters in an attempt to test this.
If our space is non-orientable, then if you travel far enough in a straight line, you'll return to Earth swapped left-to-right. That might mean that you'd be converted to anti-matter. Or it might just mean that pine trees would smell like lemons and you couldn't digest the food. It would be a good defense against viruses, however.
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-- Andy.Latto@pobox.com
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participants (5)
-
Andy Latto -
Fred Lunnon -
James Propp -
Keith F. Lynch -
Neil Sloane