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I am seeking articles where a space filling curve has been used as a theoretical application, such as in the study of general orthogonal polynomials.

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    $\begingroup$ The Cannon--Thurston map very naturally associates a space-filling curve to any fibred hyperbolic 3-manifold. $\endgroup$ – HJRW Apr 23 '16 at 15:50
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    $\begingroup$ John Bartholdi III at Georgia Tech has discovered some really cool applications of space-filling curves as heuristics for the travelling salesman problem, which have been applied to a number of real-world problems. See www2.isye.gatech.edu/~jjb/research/mow/mow.html and www2.isye.gatech.edu/~jjb/research/mow/mow.pdf , for example. $\endgroup$ – John Gunnar Carlsson Apr 23 '16 at 19:27
  • $\begingroup$ By "theoretical application," do you mean to specifically exclude practical applications? Among the latter are digital half-toning and nearest-neighbor searching. $\endgroup$ – Joseph O'Rourke Apr 23 '16 at 20:43
  • $\begingroup$ I applied, during the first half of 1985, the Hilbert curve to the image compression. $\endgroup$ – Włodzimierz Holsztyński Apr 24 '16 at 5:09
  • $\begingroup$ Also, this question should be community-wiki. $\endgroup$ – HJRW Apr 24 '16 at 10:29
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It's used in Google maps to introduce cache locality: When you move a little bit while viewing the map, you want to be moving only a little bit in the memory, which is after all arranged linearly. This is where continuous space filling curves find one of their uses.

Edit: You can find a non-academic references about Google's use of Hilbert curves.

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    $\begingroup$ Sounds cool. Can you add a reference so we can read more about it? $\endgroup$ – Amir Sagiv Apr 23 '16 at 22:34
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    $\begingroup$ That seems to imply if you happen to be in the wrong part of the map, like near $(1/2,1)$ for the Hilbert curve illustrated on Wikipedia, you have a massive jump in memory location. $\endgroup$ – AHusain Apr 24 '16 at 4:48
  • $\begingroup$ @AHusain, one does the best. It's never going to be perfect. $\endgroup$ – Włodzimierz Holsztyński May 8 '16 at 3:12
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In a very different style, N. Katz has studied "space filling curves" over finite fields (https://web.math.princeton.edu/~nmk/spacefill.pdf), and given some applications (e.g., every abelian variety over a finite field is a quotient of the jacobian of a curve, using ideas of Gabber).

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Space filling curves have been used a good deal by Katsuya Eda and his coauthors to compute singular homology groups of Peano continua (which are precisely the continuous images of $[0,1]$).

For example:

The singular homology of the Griffiths Twin cone is computed in this paper. A key part of this is Theorem 3.1 which requires the use of a space filling curve.

Similar techniques are used in

K. Eda, K. Kawamura, The surjectivity of the canonical homomorphism from singular homology to Cech homology, Proc. Amer. Math. Soc. 128 (1999), 1487–1495. (See Lemma 2.4)

K. Eda, Singular homology groups of one-dimensional Peano continua (see Lemma 4.1)

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If you are interested in signal processing, it seems that space filling curves are widely used there; see this article and this one.

It would be most helpful if you narrow down the field of application, or instead ask for a "big-list" of all possible applications.

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In a similar manner to the way they are used by Google maps, space filling curves are used for load balancing super computers. Since computational domains are sometimes more refined in certain locations, space filling curves can catch this (by refining the curve as needed). Another advantage of using these curves for load balancing is that communication between computation units is usually better (though this is only a heuristic).

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