[math-fun] No such thing as a self-avoiding spacefill
On 2015-12-23 12:00, Warren D Smith wrote:
J?rg, these recursive arcs are beautiful and ingenious, but in the limit, they all describe the same area-filling function, where in this case, the area filled is 1/3 of a "France Flake" island. All such area-fills map closed intervals onto closed sets, hitting *all* the points at least once, uncountably many at least twice, and at least countably many at least thrice.
--it seems to me, there must be some value in an area-filling curve that is the "limit" of a sequence curve1, curve2, ..., in which every curveN never self-intersects and "stays away from intersecting itself" by at least some natural distance f(N). Since in any real application, N will be finite.
Possible way to make that precise: Each point of curve N is distance >= f(N) away from any other point of curveN that is not at ArcDistance <= 2*f(N).
So then the question is: which area-filling curves can be manufactured in this way, and which cannot? One might argue they all can be done (proof: local surgery as needed) but perhaps not in a "nice" way (simple nice definition).
Sequences of curves are beguiling and misleading, by concealing that something singular happens as D reaches 2. The textures and designs that seem to distinguish these curves are just superficial treatments applied to the bottommost recursion. Recall that traditional pictures of Heighway's Dragon show whole grids of self-contacts. But simply joining the midpoints of its segments produces a completely self-avoiding "median curve": gosper.org/twindrag.png . Alternatively, sampling the limiting "curve" at a frequency slightly off from a power of 2 produces beat frequencies: gosper.org/insuck.png showing whole intervals of phases where the polygonal approximations self-avoid. Finally, recall the plain old square grid four-around-one (base i+2, digits 0, i^(0..3)): gosper.org/erez2.png and how different it looks when quarter-circles replace the line segments at the bottom level: gosper.org/erez7.PNG Bottom line: There is no bottom level. --rwg An interesting rendition of a spacefill might be to use Julian's inverter on a big bunch of double and triple points, darkening each one in proportion to the separation of its inverse images. I expect it would resemble leaf veins.
Has anyone created an image that approximates the graph in [0,1]^3 of a parametric space-filling curve viewed as a map from [0,1] to [0,1]^2? It would be especially nice if the image were dynamically rotatable. Jim Propp On Thursday, December 24, 2015, Bill Gosper <billgosper@gmail.com <javascript:_e(%7B%7D,'cvml','billgosper@gmail.com');>> wrote:
On 2015-12-23 12:00, Warren D Smith wrote:
J?rg, these recursive arcs are beautiful and ingenious, but in the limit, they all describe the same area-filling function, where in this case, the area filled is 1/3 of a "France Flake" island. All such area-fills map closed intervals onto closed sets, hitting *all* the points at least once, uncountably many at least twice, and at least countably many at least thrice.
--it seems to me, there must be some value in an area-filling curve that is the "limit" of a sequence curve1, curve2, ..., in which every curveN never self-intersects and "stays away from intersecting itself" by at least some natural distance f(N). Since in any real application, N will be finite.
Possible way to make that precise: Each point of curve N is distance >= f(N) away from any other point of curveN that is not at ArcDistance <= 2*f(N).
So then the question is: which area-filling curves can be manufactured in this way, and which cannot? One might argue they all can be done (proof: local surgery as needed) but perhaps not in a "nice" way (simple nice definition).
Sequences of curves are beguiling and misleading, by concealing that something singular happens as D reaches 2. The textures and designs that seem to distinguish these curves are just superficial treatments applied to the bottommost recursion.
Recall that traditional pictures of Heighway's Dragon show whole grids of self-contacts. But simply joining the midpoints of its segments produces a completely self-avoiding "median curve": gosper.org/twindrag.png .
Alternatively, sampling the limiting "curve" at a frequency slightly off from a power of 2 produces beat frequencies: gosper.org/insuck.png showing whole intervals of phases where the polygonal approximations self-avoid. Finally, recall the plain old square grid four-around-one (base i+2, digits 0, i^(0..3)): gosper.org/erez2.png and how different it looks when quarter-circles replace the line segments at the bottom level: gosper.org/erez7.PNG Bottom line: There is no bottom level. --rwg An interesting rendition of a spacefill might be to use Julian's inverter on a big bunch of double and triple points, darkening each one in proportion to the separation of its inverse images. I expect it would resemble leaf veins. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Not sure (because of your "as a map from [0,1] to [0,1]^2", the very last '2' is what puzzles me) if this is what you want: I can create approximations of the 3D Hilbert curve as STL file (use Blender to whirl it around). Best regards, jj * James Propp <jamespropp@gmail.com> [Dec 26. 2015 17:03]:
Has anyone created an image that approximates the graph in [0,1]^3 of a parametric space-filling curve viewed as a map from [0,1] to [0,1]^2? It would be especially nice if the image were dynamically rotatable.
Jim Propp
[...]
A parametric curve from [0,1] (time) to [0,1]^2 (space) has a graph in [0,1]^3 (spacetime). Does that explain my point of view (and the "2")? Jim On Saturday, December 26, 2015, Joerg Arndt <arndt@jjj.de> wrote:
Not sure (because of your "as a map from [0,1] to [0,1]^2", the very last '2' is what puzzles me) if this is what you want: I can create approximations of the 3D Hilbert curve as STL file (use Blender to whirl it around).
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Dec 26. 2015 17:03]:
Has anyone created an image that approximates the graph in [0,1]^3 of a parametric space-filling curve viewed as a map from [0,1] to [0,1]^2? It would be especially nice if the image were dynamically rotatable.
Jim Propp
[...]
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Um, so I just take a curve in the plane (coordinates x,y) and add a z-coordinate that ticks up by one with every stroke? jj * James Propp <jamespropp@gmail.com> [Dec 26. 2015 18:56]:
A parametric curve from [0,1] (time) to [0,1]^2 (space) has a graph in [0,1]^3 (spacetime).
Does that explain my point of view (and the "2")?
Jim
On Saturday, December 26, 2015, Joerg Arndt <arndt@jjj.de> wrote:
Not sure (because of your "as a map from [0,1] to [0,1]^2", the very last '2' is what puzzles me) if this is what you want: I can create approximations of the 3D Hilbert curve as STL file (use Blender to whirl it around).
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Dec 26. 2015 17:03]:
Has anyone created an image that approximates the graph in [0,1]^3 of a parametric space-filling curve viewed as a map from [0,1] to [0,1]^2? It would be especially nice if the image were dynamically rotatable.
Jim Propp
[...]
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On Saturday, December 26, 2015, Joerg Arndt <arndt@jjj.de> wrote:
Um, so I just take a curve in the plane (coordinates x,y) and add a z-coordinate that ticks up by one with every stroke?
I'm not sure what constitutes a "stroke" in the continuum limit. A space-filling curve as a limit object is not a polygonal approximation or a sequence of such approximations; it's a continuous nowhere-differentiable function from [0,1] to the plane (constructed as the limit of such approximations). By way of comparison, consider the unit circle, parametrized at constant speed. The graph is {(t, cos t, sin t): t in [0, 2 pi]}. Projected onto the x,y plane, it's a circle; projected onto the t,x plane or the t,y plane, it's a sinusoidal arch. I'd like to see (among other things) the space-filling-curve analogues of those sinusoids. Something like Bolzano's everywhere-continuous-but-nowhere-differentiable function? Jim Propp
http://www.shapeways.com/product/3MF7L6QKA/developing-hilbert-curve-large On Sat, Dec 26, 2015 at 3:45 PM, James Propp <jamespropp@gmail.com> wrote:
On Saturday, December 26, 2015, Joerg Arndt <arndt@jjj.de> wrote:
Um, so I just take a curve in the plane (coordinates x,y) and add a z-coordinate that ticks up by one with every stroke?
I'm not sure what constitutes a "stroke" in the continuum limit. A space-filling curve as a limit object is not a polygonal approximation or a sequence of such approximations; it's a continuous nowhere-differentiable function from [0,1] to the plane (constructed as the limit of such approximations).
By way of comparison, consider the unit circle, parametrized at constant speed. The graph is {(t, cos t, sin t): t in [0, 2 pi]}. Projected onto the x,y plane, it's a circle; projected onto the t,x plane or the t,y plane, it's a sinusoidal arch. I'd like to see (among other things) the space-filling-curve analogues of those sinusoids. Something like Bolzano's everywhere-continuous-but-nowhere-differentiable function?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
The image Mike has provided a link to is lovely and perhaps better (i.e., more informative) than what I asked for, but not quite what I wanted. One might stop the iterative construction at some finite stage, and plot the graph of the resulting piecewise-linear parametric curve in 3D. Indeed, some of you will rightly point out that such a graph is all I'm entitled to, given that any given image has only finite resolution. Then again, one can imagine an interactive program that allows one to dynamically zoom in on the graphical object by a factor of a googol or more; this interactivity would create as much of the illusion of infinite precision as a finite human being could ask for. Jim On Sunday, December 27, 2015, Mike Stay <metaweta@gmail.com> wrote:
http://www.shapeways.com/product/3MF7L6QKA/developing-hilbert-curve-large
On Sat, Dec 26, 2015 at 3:45 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
On Saturday, December 26, 2015, Joerg Arndt <arndt@jjj.de <javascript:;>> wrote:
Um, so I just take a curve in the plane (coordinates x,y) and add a z-coordinate that ticks up by one with every stroke?
I'm not sure what constitutes a "stroke" in the continuum limit. A space-filling curve as a limit object is not a polygonal approximation or a sequence of such approximations; it's a continuous nowhere-differentiable function from [0,1] to the plane (constructed as the limit of such approximations).
By way of comparison, consider the unit circle, parametrized at constant speed. The graph is {(t, cos t, sin t): t in [0, 2 pi]}. Projected onto the x,y plane, it's a circle; projected onto the t,x plane or the t,y plane, it's a sinusoidal arch. I'd like to see (among other things) the space-filling-curve analogues of those sinusoids. Something like Bolzano's everywhere-continuous-but-nowhere-differentiable function?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com <javascript:;> http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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Is there anything like this surface that has constant negative curvature? That is: Can a topological disk embedded in 3-space with constant negative curvature have a (2-)space-filling curve as its boundary? Jim Propp On Sunday, December 27, 2015, Mike Stay <metaweta@gmail.com> wrote:
http://www.shapeways.com/product/3MF7L6QKA/developing-hilbert-curve-large
On Sat, Dec 26, 2015 at 3:45 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
On Saturday, December 26, 2015, Joerg Arndt <arndt@jjj.de <javascript:;>> wrote:
Um, so I just take a curve in the plane (coordinates x,y) and add a z-coordinate that ticks up by one with every stroke?
I'm not sure what constitutes a "stroke" in the continuum limit. A space-filling curve as a limit object is not a polygonal approximation or a sequence of such approximations; it's a continuous nowhere-differentiable function from [0,1] to the plane (constructed as the limit of such approximations).
By way of comparison, consider the unit circle, parametrized at constant speed. The graph is {(t, cos t, sin t): t in [0, 2 pi]}. Projected onto the x,y plane, it's a circle; projected onto the t,x plane or the t,y plane, it's a sinusoidal arch. I'd like to see (among other things) the space-filling-curve analogues of those sinusoids. Something like Bolzano's everywhere-continuous-but-nowhere-differentiable function?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com <javascript:;> http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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But any computer can only do approximations. Good enough to appear "the limit" to the human eye, though. So: choose a curve and level of approximation. Example: terdragon, iterate 8: Data: http://jjj.de/tmp-xmas/propp-terdragon.dat Script: http://jjj.de/tmp-xmas/propp-gnuplot.plt On you command line issue gnuplot propp-gnuplot.plt Whirl around with mouse. Is this (modulo perfection) what you meant? Best regards, jj * James Propp <jamespropp@gmail.com> [Dec 27. 2015 07:48]:
On Saturday, December 26, 2015, Joerg Arndt <arndt@jjj.de> wrote:
Um, so I just take a curve in the plane (coordinates x,y) and add a z-coordinate that ticks up by one with every stroke?
I'm not sure what constitutes a "stroke" in the continuum limit. A space-filling curve as a limit object is not a polygonal approximation or a sequence of such approximations; it's a continuous nowhere-differentiable function from [0,1] to the plane (constructed as the limit of such approximations).
By way of comparison, consider the unit circle, parametrized at constant speed. The graph is {(t, cos t, sin t): t in [0, 2 pi]}. Projected onto the x,y plane, it's a circle; projected onto the t,x plane or the t,y plane, it's a sinusoidal arch. I'd like to see (among other things) the space-filling-curve analogues of those sinusoids. Something like Bolzano's everywhere-continuous-but-nowhere-differentiable function?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I suspect it's what I want, if the third dimension is really the time-parameter along the curve (and not some sort of interpolated "development" parameter, as in the image Mike pointed me to). Unfortunately my access to my laptop is limited (for family reasons) and my iPhone isn't up to doing gnuplot images, so I won't know for sure until I get a chance to try it. In any case, thanks for creating the graphic; I'm sure it'll tell me something I don't know. Jim On Sunday, December 27, 2015, Joerg Arndt <arndt@jjj.de> wrote:
But any computer can only do approximations. Good enough to appear "the limit" to the human eye, though.
So: choose a curve and level of approximation.
Example: terdragon, iterate 8: Data: http://jjj.de/tmp-xmas/propp-terdragon.dat Script: http://jjj.de/tmp-xmas/propp-gnuplot.plt On you command line issue gnuplot propp-gnuplot.plt Whirl around with mouse.
Is this (modulo perfection) what you meant?
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Dec 27. 2015 07:48]:
On Saturday, December 26, 2015, Joerg Arndt <arndt@jjj.de <javascript:;>> wrote:
Um, so I just take a curve in the plane (coordinates x,y) and add a z-coordinate that ticks up by one with every stroke?
I'm not sure what constitutes a "stroke" in the continuum limit. A space-filling curve as a limit object is not a polygonal approximation or a sequence of such approximations; it's a continuous nowhere-differentiable function from [0,1] to the plane (constructed as the limit of such approximations).
By way of comparison, consider the unit circle, parametrized at constant speed. The graph is {(t, cos t, sin t): t in [0, 2 pi]}. Projected onto the x,y plane, it's a circle; projected onto the t,x plane or the t,y plane, it's a sinusoidal arch. I'd like to see (among other things) the space-filling-curve analogues of those sinusoids. Something like Bolzano's everywhere-continuous-but-nowhere-differentiable function?
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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* James Propp <jamespropp@gmail.com> [Dec 27. 2015 19:07]:
I suspect it's what I want, if the third dimension is really the time-parameter along the curve (and not some sort of interpolated "development" parameter, as in the image Mike pointed me to).
That is what I have done. It was just one line to add in the program. As it already has the pairs (x,y), putting print x, y, tick has done the trick.
Unfortunately my access to my laptop is limited (for family reasons) and my iPhone isn't up to doing gnuplot images, so I won't know for sure until I get a chance to try it. In any case, thanks for creating the graphic; I'm sure it'll tell me something I don't know.
If you need more data or in other form(s), just holler! Best regards, jj P.S.: If anyone has the idea to play these curves as audio, x-coord for left channel, y-coord for right, don't. It's boring by definition, just a low pitch hum.
Jim
On Sunday, December 27, 2015, Joerg Arndt <arndt@jjj.de> wrote: [...]
This sounded very appealing, but now I wonder if such a thing would be so stretched-out in 3D along the R^1 direction that the only interesting view of it might be *along* the R^1 direction, in which case you'd just see the usual graph in the plane. Still, this seems definitely worth trying. * * * Is there a nice sequence of self-avoiding curves — approaching an area-filling curve — that are defined by simple Fourier series? I also wonder if there are some particularly nice geometric conditions that one might ask of such a sequence, such that there is essentially only one solution. (Maybe there's a way to make the idea of such curves being "simple" rigorous.) I'd guess there's a better chance of finding such a thing if instead of the usual n-cube, the codomain of the curve were instead a cubical n-torus R^n / Z^n. —Dan
On Dec 26, 2015, at 7:42 AM, James Propp <jamespropp@gmail.com> wrote:
Has anyone created an image that approximates the graph in [0,1]^3 of a parametric space-filling curve viewed as a map from [0,1] to [0,1]^2? It would be especially nice if the image were dynamically rotatable.
Which all suggests this question, that I've wondered before in any case: Suppose we are given a complex Fourier series of form f(t) = Sum_{-oo < n < oo} c_n exp(nit) (with each c_n in C), defining a function that by abuse of notation we call f: R/(2pi)Z —> C . Then: How can you tell from the coefficients {c_n} when f is one-to-one (i.e., self-avoiding)? —Dan ___________________________________________ P.S. Random thought: Isn't it a bit vexing that if the basis functions are exp(it), then the period is 2pi; and if the period is 1, then the basis functions are {exp(2pi*it)} — always with the obnoxious 2pi in the way. Maybe it would be best if the basis functions were {exp(sqrt(2pi)*it} and so the period would be sqrt(2pi), so at least there's some symmetry.
Some well-known mathematician was reported to ahve announced at the start of a course in complex analysis that he would set 2 pi i = 1 . History does not relate the outcome. WFL On 12/27/15, Dan Asimov <asimov@msri.org> wrote:
Which all suggests this question, that I've wondered before in any case:
Suppose we are given a complex Fourier series of form
f(t) = Sum_{-oo < n < oo} c_n exp(nit)
(with each c_n in C), defining a function that by abuse of notation we call
f: R/(2pi)Z —> C
. Then:
How can you tell from the coefficients {c_n} when f is one-to-one (i.e., self-avoiding)?
—Dan ___________________________________________
P.S. Random thought: Isn't it a bit vexing that if the basis functions are exp(it), then the period is 2pi; and if the period is 1, then the basis functions are {exp(2pi*it)} — always with the obnoxious 2pi in the way.
Maybe it would be best if the basis functions were {exp(sqrt(2pi)*it} and so the period would be sqrt(2pi), so at least there's some symmetry. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Is there a flavor of generalized dimensional analysis that predicts exponents of pi where they arise in various mathematical contexts? Jim Propp On Sunday, December 27, 2015, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Some well-known mathematician was reported to ahve announced at the start of a course in complex analysis that he would set 2 pi i = 1 . History does not relate the outcome.
WFL
On 12/27/15, Dan Asimov <asimov@msri.org <javascript:;>> wrote:
Which all suggests this question, that I've wondered before in any case:
Suppose we are given a complex Fourier series of form
f(t) = Sum_{-oo < n < oo} c_n exp(nit)
(with each c_n in C), defining a function that by abuse of notation we call
f: R/(2pi)Z —> C
. Then:
How can you tell from the coefficients {c_n} when f is one-to-one (i.e., self-avoiding)?
—Dan ___________________________________________
P.S. Random thought: Isn't it a bit vexing that if the basis functions are exp(it), then the period is 2pi; and if the period is 1, then the basis functions are {exp(2pi*it)} — always with the obnoxious 2pi in the way.
Maybe it would be best if the basis functions were {exp(sqrt(2pi)*it} and so the period would be sqrt(2pi), so at least there's some symmetry. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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On 2015-12-27 00:15, Dan Asimov wrote:
This sounded very appealing, but now I wonder if such a thing would be so stretched-out in 3D along the R^1 direction that the only interesting view of it might be *along* the R^1 direction, in which case you'd just see the usual graph in the plane.
Still, this seems definitely worth trying.
* * *
Is there a nice sequence of self-avoiding curves — approaching an area-filling curve — that are defined by simple Fourier series?
http://gosper.org/DDrag.mp4 But the only formula I found for the coefficients is an infinite product of 3x3 matrices. (There's a dimension parameter, with self-avoidance for D<2.) --rwg
I also wonder if there are some particularly nice geometric conditions that one might ask of such a sequence, such that there is essentially only one solution. (Maybe there's a way to make the idea of such curves being "simple" rigorous.)
I'd guess there's a better chance of finding such a thing if instead of the usual n-cube, the codomain of the curve were instead a cubical n-torus R^n / Z^n.
—Dan
On Dec 26, 2015, at 7:42 AM, James Propp <jamespropp@gmail.com> wrote:
Has anyone created an image that approximates the graph in [0,1]^3 of a parametric space-filling curve viewed as a map from [0,1] to [0,1]^2? It would be especially nice if the image were dynamically rotatable.
Nice one! My browser refuses to play the file, saying it is corrupt. So wget http://gosper.org/DDrag.mp4 and vlc DDrag.mp4 to the rescue. Best regards, jj * rwg <rwg@sdf.org> [Dec 28. 2015 07:18]:
On 2015-12-27 00:15, Dan Asimov wrote:
This sounded very appealing, but now I wonder if such a thing would be so stretched-out in 3D along the R^1 direction that the only interesting view of it might be *along* the R^1 direction, in which case you'd just see the usual graph in the plane.
Still, this seems definitely worth trying.
* * *
Is there a nice sequence of self-avoiding curves — approaching an area-filling curve — that are defined by simple Fourier series?
http://gosper.org/DDrag.mp4 But the only formula I found for the coefficients is an infinite product of 3x3 matrices. (There's a dimension parameter, with self-avoidance for D<2.) --rwg
I also wonder if there are some particularly nice geometric conditions that one might ask of such a sequence, such that there is essentially only one solution. (Maybe there's a way to make the idea of such curves being "simple" rigorous.)
I'd guess there's a better chance of finding such a thing if instead of the usual n-cube, the codomain of the curve were instead a cubical n-torus R^n / Z^n.
—Dan
On Dec 26, 2015, at 7:42 AM, James Propp <jamespropp@gmail.com> wrote:
Has anyone created an image that approximates the graph in [0,1]^3 of a parametric space-filling curve viewed as a map from [0,1] to [0,1]^2? It would be especially nice if the image were dynamically rotatable.
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On 2015-12-27 22:36, Joerg Arndt wrote:
Nice one! My browser refuses to play the file, saying it is corrupt. So wget http://gosper.org/DDrag.mp4 and vlc DDrag.mp4 to the rescue.
Best regards, jj
Thank you for persisting! Ahah, http://gosper.org/TDrag3c.mp4 is a Fudgeflake filled by three Triadic Dragons, m=3 in http://gosper.org/fst.pdf , p144 (actual page 16) et seq. The Fourier coefficients are infinite products of ordinary scalars. A technique similar to Julian's amazing little piecewiserecursivefractal function permits exact evaluation at any rational time value. Plugging said time into the Fourier series and equating produces closed forms for sums of infinite products which I think would have even gotten a rise out of Ramanujan. Earlier in the paper this is exhibited for the Snowflake family of fractals, leading to the pair of identities (d247) and (d246) in http://www.tweedledum.com/rwg/idents.htm , where the only difference on the lhs is changing 260 to 261. It's a shame that our unfamiliarity with matrix products makes it hard to appreciate the analogous identities for the dyadic (Heighway) Dragon and Sierpinski Gasket. As explained in fst.pdf, the Snowflake and Terdragon Fourier series were also derived with matrix products, which then miraculously telescoped. --rwg
* rwg <rwg@sdf.org> [Dec 28. 2015 07:18]:
On 2015-12-27 00:15, Dan Asimov wrote:
This sounded very appealing, but now I wonder if such a thing would be so stretched-out in 3D along the R^1 direction that the only interesting view of it might be *along* the R^1 direction, in which case you'd just see the usual graph in the plane.
Still, this seems definitely worth trying.
* * *
Is there a nice sequence of self-avoiding curves — approaching an area-filling curve — that are defined by simple Fourier series?
http://gosper.org/DDrag.mp4 But the only formula I found for the coefficients is an infinite product of 3x3 matrices. (There's a dimension parameter, with self-avoidance for D<2.) --rwg
I also wonder if there are some particularly nice geometric conditions that one might ask of such a sequence, such that there is essentially only one solution. (Maybe there's a way to make the idea of such curves being "simple" rigorous.)
I'd guess there's a better chance of finding such a thing if instead of the usual n-cube, the codomain of the curve were instead a cubical n-torus R^n / Z^n.
—Dan
On Dec 26, 2015, at 7:42 AM, James Propp <jamespropp@gmail.com> wrote:
Has anyone created an image that approximates the graph in [0,1]^3 of a parametric space-filling curve viewed as a map from [0,1] to [0,1]^2? It would be especially nice if the image were dynamically rotatable.
After staring at the videos repeatedly I now see that the Fourier series can be obtained mechanically for all the (tiles of the) curves. We have a complex basis B for the corresponding numeration system for our curve of order R, where abs(B) = sqrt(R) The lowest nontrivial approximation (let's call it "principal") comes from moving along the first iterate of the tile (Theta_{1} in my draft). The higher approximations are obtained by adding powers of (1/B) times the (possibly phase-shifted) principal played at R times the speed of the principal. Now that should be a geometric series, where telescoping is less of a miracle. Did I miss something? Best regards, jj P.S.: Looks like the arrangement of rods with rotating joints in the videos are the contributions of the principal (innermost rod) followed by the higher harmonics. Right? * rwg <rwg@sdf.org> [Dec 28. 2015 12:09]:
On 2015-12-27 22:36, Joerg Arndt wrote: [...]
Thank you for persisting!
Ahah, http://gosper.org/TDrag3c.mp4 is a Fudgeflake filled by three Triadic Dragons, m=3 in http://gosper.org/fst.pdf , p144 (actual page 16) et seq. The Fourier coefficients are infinite products of ordinary scalars. A technique similar to Julian's amazing little piecewiserecursivefractal function permits exact evaluation at any rational time value. Plugging said time into the Fourier series and equating produces closed forms for sums of infinite products which I think would have even gotten a rise out of Ramanujan. Earlier in the paper this is exhibited for the Snowflake family of fractals, leading to the pair of identities (d247) and (d246) in http://www.tweedledum.com/rwg/idents.htm , where the only difference on the lhs is changing 260 to 261.
It's a shame that our unfamiliarity with matrix products makes it hard to appreciate the analogous identities for the dyadic (Heighway) Dragon and Sierpinski Gasket. As explained in fst.pdf, the Snowflake and Terdragon Fourier series were also derived with matrix products, which then miraculously telescoped. --rwg
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On 2015-12-28 04:03, Joerg Arndt wrote:
After staring at the videos repeatedly I now see that the Fourier series can be obtained mechanically for all the (tiles of the) curves.
We have a complex basis B for the corresponding numeration system for our curve of order R, where abs(B) = sqrt(R)
The lowest nontrivial approximation (let's call it "principal") comes from moving along the first iterate of the tile (Theta_{1} in my draft). The higher approximations are obtained by adding powers of (1/B) times the (possibly phase-shifted) principal played at R times the speed of the principal.
Now that should be a geometric series, where telescoping is less of a miracle.
Did I miss something?
Maybe I did. Are you saying you can find infinite products of scalars to replace the twindragon 3x3 product_n of Sec[a]/4*{{Exp[-I*a], -Exp[I*(a - t)], 2*I*Sin[a + t/2]*Exp[-I*t/2]}, {-Exp[I*a], Exp[-I*(a + t)], 2*I*Sin[a - t/2]*Exp[-I*t/2]}, {0, 0, 4*Cos[a]*Cos[t/2]*Exp[-I*t/2]}} (t:= (m k + 1)/2^n, for the kth harmonic of m copies around an m-gon. a->π/4 for 2D) or the nontriangular 2x2 product for Sierpinski's Gasket http://gosper.org/gaskettalk.pdf ? I always wondered if it was possible. It would make for some amazing identities. --rwg
Best regards, jj
P.S.: Looks like the arrangement of rods with rotating joints in the videos are the contributions of the principal (innermost rod) followed by the higher harmonics. Right?
* rwg <rwg@sdf.org> [Dec 28. 2015 12:09]:
On 2015-12-27 22:36, Joerg Arndt wrote: [...]
Thank you for persisting!
Ahah, http://gosper.org/TDrag3c.mp4 is a Fudgeflake filled by three Triadic Dragons, m=3 in http://gosper.org/fst.pdf , p144 (actual page 16) et seq. The Fourier coefficients are infinite products of ordinary scalars. A technique similar to Julian's amazing little piecewiserecursivefractal function permits exact evaluation at any rational time value. Plugging said time into the Fourier series and equating produces closed forms for sums of infinite products which I think would have even gotten a rise out of Ramanujan. Earlier in the paper this is exhibited for the Snowflake family of fractals, leading to the pair of identities (d247) and (d246) in http://www.tweedledum.com/rwg/idents.htm , where the only difference on the lhs is changing 260 to 261.
It's a shame that our unfamiliarity with matrix products makes it hard to appreciate the analogous identities for the dyadic (Heighway) Dragon and Sierpinski Gasket. As explained in fst.pdf, the Snowflake and Terdragon Fourier series were also derived with matrix products, which then miraculously telescoped. --rwg
[...]
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rwg