Re: [math-fun] Space-filling curves again
On 2016-09-26 11:51, James Propp wrote:
I just watched the "Space-Filling Curves" video on Numberphile (featuring Henry Segerman and his interpolating surfaces), and it rekindled my desire to see something similar that's a bit more canonical somehow.
It kindles my desire to shout "Henry, how can you perpetrate such baloney?!" Most spacefilling apologists just resort to circular reasoning, but this isn't even reasoning! It's clear to me that Segerman fully understands this business, but his model is too hairy to articulate in terms sufficiently elementary for Numberphile. The key lemma: A continuous function is completely defined by its values on a dense set. If we choose for our dense set the dyadic rationals, then a simple finite state machine will convert two bits of t to one bit of x and one bit of y in x(t) + i y(t) = Hilbert(t). It is rather easy to show continuity. And that a dense subset of [0,1]⨉[0,i] is covered *four* times! If poor Henry had asserted this, no-one would believe him. If, instead, the dense set is the rationals, then the four-way branching recursive definiton of H(t) will inevitably loop some point, yielding an equation that we can solve for that fixed point.
Is there a way to relax an approximation to a space-filling curve in continuous time so that it works out its kinks and regresses to simpler approximations?
(No interim self-intersections please!)
Jim Propp
http://gosper.org/FDrags128.mp4 If people really want to see it, Julian's recent Fourier matrix product can produce the analogous animation for Hilbert's "curve". But it is a seductive thought crime to view a spacefilling function as some kind of limit of "spacefilling curves"! Those curves Henry sketches are mere schematics, of no mathematical consequence. This leads inevitably to embarrassed hemming and hawing about how the area jumps from 0 to 1 at the very last moment, when both interior and exterior suddenly disappear and become boundary. Successive partial sums of the Fourier series are even more seductive. But no matter how many terms you take, you're still infinitely far from the end. --rwg
On Monday, December 28, 2015, James Propp <jamespropp@gmail.com> wrote:
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?
I hope Bill will write a lively document (or create a lively video) that explains what's wrong with so many accounts of spacefilling curves. (I myself wish that more accounts started in a "Not Knot"-ish vein, explaining why the "kindergartner's space-filling curve" --- scribbling one's crayon back and forth until the square is filled --- isn't a solution to the mathematician's problem.)
Is there a way to relax an approximation to a space-filling curve in continuous time so that it works out its kinks and regresses to simpler approximations?
(No interim self-intersections please!)
Jim Propp
http://gosper.org/FDrags128.mp4
If people really want to see it, Julian's recent Fourier matrix product can produce the analogous animation for Hilbert's "curve".
I would like to see it. But it is a seductive thought crime to view a spacefilling function as some
kind of limit of "spacefilling curves"! Those curves Henry sketches are mere schematics, of no mathematical consequence. This leads inevitably to embarrassed hemming and hawing about how the area jumps from 0 to 1 at the very last moment, when both interior and exterior suddenly disappear and become boundary. Successive partial sums of the Fourier series are even more seductive. But no matter how many terms you take, you're still infinitely far from the end. --rwg
That's one way to look at it. But it depends on the notion of path-space you use and what metric you put on it, doesn't it? In particular, it's crucial to look at paths equipped with a parametrization. Then you really can get convergence to a limit. And the limiting object is a continuous function from a line segment onto a square. If you just look at the range of a path and not the parametrization, you can't describe the square as the limit in any meaningful way. Which I gather is part of Bill's point. Jim Propp
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Bill Gosper -
James Propp