An inventory of three-dimensional Hilbert space-filling curves

Abstract

Hilbert's two-dimensional space-filling curve is appreciated for its good locality properties for many applications. However, it is not clear what is the best way to generalize this curve to filling higher-dimensional spaces. We argue that the properties that make Hilbert's curve unique in two dimensions, are shared by 10694807 structurally different space-filling curves in three dimensions. These include several curves that have, in some sense, better locality properties than any generalized Hilbert curve that has been considered in the literature before.

arXiv comment25 pages, 13 figures. Addition Oct 2016: Appendix C explaining what parts of this article have been superseded by the more recent manuscript "How many three-dimensional Hilbert curves are there?" (http://arxiv.org/abs/1610.00155), and pointing out a minor error on page 7

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