[math-fun] Decorating a pseudosphere
Can anyone suggest a good way to decorate a pseudosphere? Just putting a grid on it, as in the image at the top of Paolo Lazzarini's page http://www.paololazzarini.it/geometria_sulla_sfera/modelli_noneu_start.htm, isn't bad, but I'm looking for something snazzier to serve as the visual "brand" for my blog. Tim Hoffman's image http://www.math.kobe-u.ac.jp/DIVERSIONS/TIMH1.JPG appeals to me more, but I don't know what curves are being approximated. Can any of you figure out what's going on in this image? I like Figure 6' in I. Tadao's "Hyperbolic Non-Euclidean World and Figure-8 Knot" site http://web1.kcn.jp/hp28ah77/us20_pseu.htm, but I'm suspicious. Tadao writes "Fig. 6' shows that the red concentric circles are certainly at regular intervals and that all the white circles are surely the same in size and placed at normal, regular intervals." But is this really true? It's unclear to me why the even spacing assertion holds when one crosses the cut-line. In any case, what I'm hoping for ultimately is an animated gif showing a decorated half-pseudosphere rotating about a vertical symmetry axis (looking something like a sorcerer's cap). I imagined at first that one of Escher's circle-limit pictures could be transported from the Poincare disk to the pseudosphere, but I don't think that that can work. The pseudosphere is obtained from the Poincare disk by cutting out a part of it and then gluing parts of the boundary together; this gluing doesn't play nicely with the symmetry groups of reflection tilings of the hyperbolic plane. (Or maybe it does, and I'm not thinking clearly.) Ditto for Apollonian gaskets and the like. Jim Propp
Since the pseudosphere has constant negative curvature, it ought to be possible to tile it with one of the "excessive" tilings like 3^7 or 4^5. On Tue, Jun 16, 2015 at 10:02 AM, James Propp <jamespropp@gmail.com> wrote:
Can anyone suggest a good way to decorate a pseudosphere?
Just putting a grid on it, as in the image at the top of Paolo Lazzarini's page http://www.paololazzarini.it/geometria_sulla_sfera/modelli_noneu_start.htm , isn't bad, but I'm looking for something snazzier to serve as the visual "brand" for my blog.
Tim Hoffman's image http://www.math.kobe-u.ac.jp/DIVERSIONS/TIMH1.JPG appeals to me more, but I don't know what curves are being approximated. Can any of you figure out what's going on in this image?
I like Figure 6' in I. Tadao's "Hyperbolic Non-Euclidean World and Figure-8 Knot" site http://web1.kcn.jp/hp28ah77/us20_pseu.htm, but I'm suspicious. Tadao writes "Fig. 6' shows that the red concentric circles are certainly at regular intervals and that all the white circles are surely the same in size and placed at normal, regular intervals." But is this really true? It's unclear to me why the even spacing assertion holds when one crosses the cut-line.
In any case, what I'm hoping for ultimately is an animated gif showing a decorated half-pseudosphere rotating about a vertical symmetry axis (looking something like a sorcerer's cap).
I imagined at first that one of Escher's circle-limit pictures could be transported from the Poincare disk to the pseudosphere, but I don't think that that can work. The pseudosphere is obtained from the Poincare disk by cutting out a part of it and then gluing parts of the boundary together; this gluing doesn't play nicely with the symmetry groups of reflection tilings of the hyperbolic plane. (Or maybe it does, and I'm not thinking clearly.) Ditto for Apollonian gaskets and the like.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Allan, The pseudosphere is obtained from the Poincare disk by cutting out a piece of it and then gluing parts of the boundary together; my impression is that this gluing doesn't play nicely with the symmetry groups of reflection tilings of the hyperbolic plane. Analogy: Take a sector of the plane with angle theta and glue together the two bounding rays. The resulting cone has constant curvature zero, but if the angle theta is irrational, you'll have trouble turning a doubly-periodic tiling of the plane into a nice tiling of the cone. (Or am I missing something?) Jim On Tue, Jun 16, 2015 at 11:36 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Since the pseudosphere has constant negative curvature, it ought to be possible to tile it with one of the "excessive" tilings like 3^7 or 4^5.
On Tue, Jun 16, 2015 at 10:02 AM, James Propp <jamespropp@gmail.com> wrote:
Can anyone suggest a good way to decorate a pseudosphere?
Just putting a grid on it, as in the image at the top of Paolo Lazzarini's page
http://www.paololazzarini.it/geometria_sulla_sfera/modelli_noneu_start.htm
, isn't bad, but I'm looking for something snazzier to serve as the visual "brand" for my blog.
Tim Hoffman's image http://www.math.kobe-u.ac.jp/DIVERSIONS/TIMH1.JPG appeals to me more, but I don't know what curves are being approximated. Can any of you figure out what's going on in this image?
I like Figure 6' in I. Tadao's "Hyperbolic Non-Euclidean World and Figure-8 Knot" site http://web1.kcn.jp/hp28ah77/us20_pseu.htm, but I'm suspicious. Tadao writes "Fig. 6' shows that the red concentric circles are certainly at regular intervals and that all the white circles are surely the same in size and placed at normal, regular intervals." But is this really true? It's unclear to me why the even spacing assertion holds when one crosses the cut-line.
In any case, what I'm hoping for ultimately is an animated gif showing a decorated half-pseudosphere rotating about a vertical symmetry axis (looking something like a sorcerer's cap).
I imagined at first that one of Escher's circle-limit pictures could be transported from the Poincare disk to the pseudosphere, but I don't think that that can work. The pseudosphere is obtained from the Poincare disk by cutting out a part of it and then gluing parts of the boundary together; this gluing doesn't play nicely with the symmetry groups of reflection tilings of the hyperbolic plane. (Or maybe it does, and I'm not thinking clearly.) Ditto for Apollonian gaskets and the like.
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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For purposes of my analogy, one should delete the cone point itself, to avoid the infinite curvature there. The resulting manifold then has zero curvature everywhere, but it has fewer symmetries than the plane that gave rise to it by cutting and gluing. (How should one wallpaper the inside of a teepee for maximal mathematical elegance? Unclear.) Jim Propp On Tuesday, June 16, 2015, James Propp <jamespropp@gmail.com> wrote:
Allan,
The pseudosphere is obtained from the Poincare disk by cutting out a piece of it and then gluing parts of the boundary together; my impression is that this gluing doesn't play nicely with the symmetry groups of reflection tilings of the hyperbolic plane. Analogy: Take a sector of the plane with angle theta and glue together the two bounding rays. The resulting cone has constant curvature zero, but if the angle theta is irrational, you'll have trouble turning a doubly-periodic tiling of the plane into a nice tiling of the cone.
(Or am I missing something?)
Jim
On Tue, Jun 16, 2015 at 11:36 AM, Allan Wechsler <acwacw@gmail.com <javascript:_e(%7B%7D,'cvml','acwacw@gmail.com');>> wrote:
Since the pseudosphere has constant negative curvature, it ought to be possible to tile it with one of the "excessive" tilings like 3^7 or 4^5.
On Tue, Jun 16, 2015 at 10:02 AM, James Propp <jamespropp@gmail.com <javascript:_e(%7B%7D,'cvml','jamespropp@gmail.com');>> wrote:
Can anyone suggest a good way to decorate a pseudosphere?
Just putting a grid on it, as in the image at the top of Paolo Lazzarini's page
http://www.paololazzarini.it/geometria_sulla_sfera/modelli_noneu_start.htm
, isn't bad, but I'm looking for something snazzier to serve as the visual "brand" for my blog.
Tim Hoffman's image http://www.math.kobe-u.ac.jp/DIVERSIONS/TIMH1.JPG appeals to me more, but I don't know what curves are being approximated. Can any of you figure out what's going on in this image?
I like Figure 6' in I. Tadao's "Hyperbolic Non-Euclidean World and Figure-8 Knot" site http://web1.kcn.jp/hp28ah77/us20_pseu.htm, but I'm suspicious. Tadao writes "Fig. 6' shows that the red concentric circles are certainly at regular intervals and that all the white circles are surely the same in size and placed at normal, regular intervals." But is this really true? It's unclear to me why the even spacing assertion holds when one crosses the cut-line.
In any case, what I'm hoping for ultimately is an animated gif showing a decorated half-pseudosphere rotating about a vertical symmetry axis (looking something like a sorcerer's cap).
I imagined at first that one of Escher's circle-limit pictures could be transported from the Poincare disk to the pseudosphere, but I don't think that that can work. The pseudosphere is obtained from the Poincare disk by cutting out a part of it and then gluing parts of the boundary together; this gluing doesn't play nicely with the symmetry groups of reflection tilings of the hyperbolic plane. (Or maybe it does, and I'm not thinking clearly.) Ditto for Apollonian gaskets and the like.
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On Tue, Jun 16, 2015 at 9:40 AM, James Propp <jamespropp@gmail.com> wrote:
For purposes of my analogy, one should delete the cone point itself, to avoid the infinite curvature there. The resulting manifold then has zero curvature everywhere, but it has fewer symmetries than the plane that gave rise to it by cutting and gluing. (How should one wallpaper the inside of a teepee for maximal mathematical elegance? Unclear.)
Cut out either 90 degrees for a square tiling (or 60 degrees for an equilateral triangular tiling). Then at the one defect, you'll have three squares (or five triangles) meeting at a vertex, but everywhere else you'll have four (or six). Something similar should work on the hyperbolic plane. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
I confess to being puzzled over whether Jim intends his math blog to reflect his own taste or the taste of math-fun members. —Dan
On Jun 16, 2015, at 9:40 AM, James Propp <jamespropp@gmail.com> wrote:
For purposes of my analogy, one should delete the cone point itself, to avoid the infinite curvature there. The resulting manifold then has zero curvature everywhere, but it has fewer symmetries than the plane that gave rise to it by cutting and gluing. (How should one wallpaper the inside of a teepee for maximal mathematical elegance? Unclear.)
Jim Propp
On Tuesday, June 16, 2015, James Propp <jamespropp@gmail.com> wrote:
Allan,
The pseudosphere is obtained from the Poincare disk by cutting out a piece of it and then gluing parts of the boundary together; my impression is that this gluing doesn't play nicely with the symmetry groups of reflection tilings of the hyperbolic plane. Analogy: Take a sector of the plane with angle theta and glue together the two bounding rays. The resulting cone has constant curvature zero, but if the angle theta is irrational, you'll have trouble turning a doubly-periodic tiling of the plane into a nice tiling of the cone.
(Or am I missing something?)
Jim
On Tue, Jun 16, 2015 at 11:36 AM, Allan Wechsler <acwacw@gmail.com <javascript:_e(%7B%7D,'cvml','acwacw@gmail.com');>> wrote:
Since the pseudosphere has constant negative curvature, it ought to be possible to tile it with one of the "excessive" tilings like 3^7 or 4^5.
On Tue, Jun 16, 2015 at 10:02 AM, James Propp <jamespropp@gmail.com <javascript:_e(%7B%7D,'cvml','jamespropp@gmail.com');>> wrote:
Can anyone suggest a good way to decorate a pseudosphere?
Just putting a grid on it, as in the image at the top of Paolo Lazzarini's page
http://www.paololazzarini.it/geometria_sulla_sfera/modelli_noneu_start.htm
, isn't bad, but I'm looking for something snazzier to serve as the visual "brand" for my blog.
Tim Hoffman's image http://www.math.kobe-u.ac.jp/DIVERSIONS/TIMH1.JPG appeals to me more, but I don't know what curves are being approximated. Can any of you figure out what's going on in this image?
I like Figure 6' in I. Tadao's "Hyperbolic Non-Euclidean World and Figure-8 Knot" site http://web1.kcn.jp/hp28ah77/us20_pseu.htm, but I'm suspicious. Tadao writes "Fig. 6' shows that the red concentric circles are certainly at regular intervals and that all the white circles are surely the same in size and placed at normal, regular intervals." But is this really true? It's unclear to me why the even spacing assertion holds when one crosses the cut-line.
In any case, what I'm hoping for ultimately is an animated gif showing a decorated half-pseudosphere rotating about a vertical symmetry axis (looking something like a sorcerer's cap).
I imagined at first that one of Escher's circle-limit pictures could be transported from the Poincare disk to the pseudosphere, but I don't think that that can work. The pseudosphere is obtained from the Poincare disk by cutting out a part of it and then gluing parts of the boundary together; this gluing doesn't play nicely with the symmetry groups of reflection tilings of the hyperbolic plane. (Or maybe it does, and I'm not thinking clearly.) Ditto for Apollonian gaskets and the like.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:_e(%7B%7D,'cvml','math-fun@mailman.xmission.com');> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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The blog is intended as a reflection of my own taste. But I'm eager to draw on the contributions of others, especially math-funsters, who share my tastes more than the average mathematician. (Did "Mathematical Recreations" reflect Gardner's tastes? or did it reflect the tastes of Graham, Diaconis, Conway, et al.? Yes and yes.) Jim Propp On Tue, Jun 16, 2015 at 3:03 PM, Dan Asimov <asimov@msri.org> wrote:
I confess to being puzzled over whether Jim intends his math blog to reflect his own taste or the taste of math-fun members.
—Dan
On Jun 16, 2015, at 9:40 AM, James Propp <jamespropp@gmail.com> wrote:
For purposes of my analogy, one should delete the cone point itself, to avoid the infinite curvature there. The resulting manifold then has zero curvature everywhere, but it has fewer symmetries than the plane that gave rise to it by cutting and gluing. (How should one wallpaper the inside of a teepee for maximal mathematical elegance? Unclear.)
Jim Propp
On Tuesday, June 16, 2015, James Propp <jamespropp@gmail.com> wrote:
Allan,
The pseudosphere is obtained from the Poincare disk by cutting out a piece of it and then gluing parts of the boundary together; my impression is that this gluing doesn't play nicely with the symmetry groups of reflection tilings of the hyperbolic plane. Analogy: Take a sector of the plane with angle theta and glue together the two bounding rays. The resulting cone has constant curvature zero, but if the angle theta is irrational, you'll have trouble turning a doubly-periodic tiling of the plane into a nice tiling of the cone.
(Or am I missing something?)
Jim
On Tue, Jun 16, 2015 at 11:36 AM, Allan Wechsler <acwacw@gmail.com <javascript:_e(%7B%7D,'cvml','acwacw@gmail.com');>> wrote:
Since the pseudosphere has constant negative curvature, it ought to be possible to tile it with one of the "excessive" tilings like 3^7 or 4^5.
On Tue, Jun 16, 2015 at 10:02 AM, James Propp <jamespropp@gmail.com <javascript:_e(%7B%7D,'cvml','jamespropp@gmail.com');>> wrote:
Can anyone suggest a good way to decorate a pseudosphere?
Just putting a grid on it, as in the image at the top of Paolo Lazzarini's page
http://www.paololazzarini.it/geometria_sulla_sfera/modelli_noneu_start.htm
, isn't bad, but I'm looking for something snazzier to serve as the visual "brand" for my blog.
Tim Hoffman's image http://www.math.kobe-u.ac.jp/DIVERSIONS/TIMH1.JPG appeals to me more, but I don't know what curves are being approximated. Can any of you figure out what's going on in this image?
I like Figure 6' in I. Tadao's "Hyperbolic Non-Euclidean World and Figure-8 Knot" site http://web1.kcn.jp/hp28ah77/us20_pseu.htm, but I'm suspicious. Tadao writes "Fig. 6' shows that the red concentric circles are certainly at regular intervals and that all the white circles are surely the same in size and placed at normal, regular intervals." But is this really true? It's unclear to me why the even spacing assertion holds when one crosses the cut-line.
In any case, what I'm hoping for ultimately is an animated gif showing a decorated half-pseudosphere rotating about a vertical symmetry axis (looking something like a sorcerer's cap).
I imagined at first that one of Escher's circle-limit pictures could be transported from the Poincare disk to the pseudosphere, but I don't think that that can work. The pseudosphere is obtained from the Poincare disk by cutting out a part of it and then gluing parts of the boundary together; this gluing doesn't play nicely with the symmetry groups of reflection tilings of the hyperbolic plane. (Or maybe it does, and I'm not thinking clearly.) Ditto for Apollonian gaskets and the like.
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Cut the cone, open it up into a sector of angle alpha, and give the sector complex coordinates w = r exp(i theta). Let L be a lattice of translations of the z-plane such that there exists an integer n so that for each translation t in L, Im(t) is a multiple of alpha/n. Under the conformal map w = exp(z), the z-plane translation t sends w to w' = exp(z+t) with r' = Re(t) r, theta' = theta + alpha/n. For example, suppose L is the square lattice generated by t1 = i alpha/n, t2 = alpha/n. Then the cone is tessellated by conformal squares of the form r1 < r < exp(alpha/n) r1, theta1 < theta < theta + alpha/n. -- Gene From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Tuesday, June 16, 2015 9:40 AM Subject: Re: [math-fun] Decorating a pseudosphere For purposes of my analogy, one should delete the cone point itself, to avoid the infinite curvature there. The resulting manifold then has zero curvature everywhere, but it has fewer symmetries than the plane that gave rise to it by cutting and gluing. (How should one wallpaper the inside of a teepee for maximal mathematical elegance? Unclear.) Jim Propp
If I ever move into a teepee (and hey, with house-prices being what they are where I am, that's probably all that I can afford!), I'll definitely decorate it according to Gene's scheme. Thanks! As for decorating a hemi-pseudosphere, I'll probably go with http://jamespropp.org/pseudosphere.jpg for now. It's a bit of a black-and-white herring, since the coloring wrongly suggests that I'm going to teach readers of my blog how to play pseudosphere chess. :-) Jim On Tue, Jun 16, 2015 at 3:57 PM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
Cut the cone, open it up into a sector of angle alpha, and give the sector complex coordinates w = r exp(i theta). Let L be a lattice of translations of the z-plane such that there exists an integer n so that for each translation t in L, Im(t) is a multiple of alpha/n. Under the conformal map w = exp(z), the z-plane translation t sends w to w' = exp(z+t) with r' = Re(t) r, theta' = theta + alpha/n. For example, suppose L is the square lattice generated by t1 = i alpha/n, t2 = alpha/n. Then the cone is tessellated by conformal squares of the form r1 < r < exp(alpha/n) r1, theta1 < theta < theta + alpha/n. -- Gene
From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Tuesday, June 16, 2015 9:40 AM Subject: Re: [math-fun] Decorating a pseudosphere
For purposes of my analogy, one should delete the cone point itself, to avoid the infinite curvature there. The resulting manifold then has zero curvature everywhere, but it has fewer symmetries than the plane that gave rise to it by cutting and gluing. (How should one wallpaper the inside of a teepee for maximal mathematical elegance? Unclear.)
Jim Propp
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On Jun 16, 2015, at 4:13 PM, James Propp <jamespropp@gmail.com> wrote:
As for decorating a hemi-pseudosphere, I'll probably go with http://jamespropp.org/pseudosphere.jpg for now. It's a bit of a black-and-white herring, since the coloring wrongly suggests that I'm going to teach readers of my blog how to play pseudosphere chess. :-)
Why not decorate it with geodesics in a way that shows multiple “lines”, all passing through the same point and yet parallel to another? For example, mark off four equally spaced points along the rim, say at N, E, S and W. Pass single lines through N and S that go to the tip of the hat, and fans of lines that pass through E and W and don’t intersect the former pair of lines. Try to avoid referring to it as a “pseudosphere” —that word just doesn’t make sense. -Veit
The decorated hemipseudosphere / tratrichoid is now on view at http://mathenchant.wordpress.com (though I haven't put up my zeroeth post yet). Jim On Wednesday, June 17, 2015, Veit Elser <ve10@cornell.edu> wrote:
On Jun 16, 2015, at 4:13 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
As for decorating a hemi-pseudosphere, I'll probably go with http://jamespropp.org/pseudosphere.jpg for now. It's a bit of a black-and-white herring, since the coloring wrongly suggests that I'm going to teach readers of my blog how to play pseudosphere chess. :-)
Why not decorate it with geodesics in a way that shows multiple “lines”, all passing through the same point and yet parallel to another? For example, mark off four equally spaced points along the rim, say at N, E, S and W. Pass single lines through N and S that go to the tip of the hat, and fans of lines that pass through E and W and don’t intersect the former pair of lines.
Try to avoid referring to it as a “pseudosphere” —that word just doesn’t make sense.
-Veit _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (6)
-
Allan Wechsler -
Dan Asimov -
Eugene Salamin -
James Propp -
Mike Stay -
Veit Elser