Re: [math-fun] Julian has constructed a spacefill dense with sextuple points
On 2016-08-23 12:51, James Propp wrote:
Take the graph of a square-filling parametrized path from [0,1] onto [0,1]x[0,1], as an object in [0,1]^3; does it in any meaningful sense have a Fourier transform?
Yes! gosper.org/fst.pdf gives the Fourier series for various dragons, snowflakes, and carpets. I'm pretty sure I used those techniques on the Hilbert, but I'll probably have to rederive it.
If so, perhaps the anisotropy of the graph would show up in the transform in some more localized fashion.
Jim Propp
Thank you for thinking of this. I've been wondering what happens to the area enclosed by, say, the closed loop of Hilbert's "curve" between preimages 1/6 and 1/2 as the sampling frequency → ∞. gosper.org/prf.pdf For finite samplings and finite Fourier approximations, the enclosed area approaches half the area enclosed by the progression of quadrants, = ½ (¼.+ 1/16+ 1/64+ ...) = 1/6. But in the limit, the interior and exterior suddenly disappear, and everything becomes boundary. There is a formula for the area enclosed by a curve described by a Fourier series. It will be interesting to see if it converges, almost certainly to ⅙, as the sampling frequency → ∞. Infinite sums are by definition limits. They can't suddenly jump when they reach ∞. OtOH, Fourier approximations can be unfaithful. Is there some fancy measure theory, applied to the actual Hilbert function, that gets a different answer? Say, 0?
On Tue, Aug 23, 2016 at 1:43 PM, Joerg Arndt <arndt@jjj.de> wrote:
* Bill Gosper <billgosper@gmail.com> [Aug 23. 2016 19:35]:
[...] Technically, these functions _are_ curves, being continuous images of one- dimensional sets. But people confuse curves with their graphs, and
their
graphs are blobs. But ironically, nonblobs should be invisible. "If you can see a line segment, you've drawn it wrong." The only accurate picture I can think of of a connected 1D set is as the boundary of a 2D area. But what if it's an unclosed arc? For B&W, a nice solution might be to shade one side of the curve, fading from black to white with distance from the boundary. (Not always easily computed.) More stylish: Gray background. Fade from white to gray on one side; black to gray on the other. --rwg [...]
Isn't that a non-problem resolved by differentiating between the "finite approximations" (certainly curves!) and the limits? Note how I dance around that by saying "shapes" (limits) and "iterates" (curves) in my arXiv paper.
Best regards, jj
Not surprisingly, Bill's identities and pictures are beautiful. But unless I skimmed overhastily, none of the pictures depict the Fourier transforms themselves in a fashion that would give an immediate visual answer to my question about anisotropy. I'm also not sure how Bill is defining the Fourier transform (due to lazy skimming on my part, I'm sure). Is he integrating a delta-function concentrated on the set, whose "dimensionality of delta-ness" is calibrated to ensure that the relevant integrals neither vanish nor blow up? If not, how is the transform defined? Jim Propp On Wed, Aug 24, 2016 at 6:28 AM, Bill Gosper <billgosper@gmail.com> wrote:
On 2016-08-23 12:51, James Propp wrote:
Take the graph of a square-filling parametrized path from [0,1] onto [0,1]x[0,1], as an object in [0,1]^3; does it in any meaningful sense have a Fourier transform?
Yes! gosper.org/fst.pdf gives the Fourier series for various dragons, snowflakes, and carpets. I'm pretty sure I used those techniques on the Hilbert, but I'll probably have to rederive it.
If so, perhaps the anisotropy of the graph would show up in the transform in some more localized fashion.
Jim Propp
Thank you for thinking of this. I've been wondering what happens to the area enclosed by, say, the closed loop of Hilbert's "curve" between preimages 1/6 and 1/2 as the sampling frequency → ∞. gosper.org/prf.pdf For finite samplings and finite Fourier approximations, the enclosed area approaches half the area enclosed by the progression of quadrants, = ½ (¼.+ 1/16+ 1/64+ ...) = 1/6. But in the limit, the interior and exterior suddenly disappear, and everything becomes boundary.
There is a formula for the area enclosed by a curve described by a Fourier series. It will be interesting to see if it converges, almost certainly to ⅙, as the sampling frequency → ∞. Infinite sums are by definition limits. They can't suddenly jump when they reach ∞. OtOH, Fourier approximations can be unfaithful. Is there some fancy measure theory, applied to the actual Hilbert function, that gets a different answer? Say, 0?
On Tue, Aug 23, 2016 at 1:43 PM, Joerg Arndt <arndt@jjj.de> wrote:
* Bill Gosper <billgosper@gmail.com> [Aug 23. 2016 19:35]:
[...] Technically, these functions _are_ curves, being continuous images of one- dimensional sets. But people confuse curves with their graphs, and
their
graphs are blobs. But ironically, nonblobs should be invisible. "If you can see a line segment, you've drawn it wrong." The only accurate picture I can think of of a connected 1D set is as the boundary of a 2D area. But what if it's an unclosed arc? For B&W, a nice solution might be to shade one side of the curve, fading from black to white with distance from the boundary. (Not always easily computed.) More stylish: Gray background. Fade from white to gray on one side; black to gray on the other. --rwg [...]
Isn't that a non-problem resolved by differentiating between the "finite approximations" (certainly curves!) and the limits? Note how I dance around that by saying "shapes" (limits) and "iterates" (curves) in my arXiv paper.
Best regards, jj
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