### Abstract

In this work we take the point of view that a Hilbert curve should at least be self-similar and traverse cubes octant by octant. We organize and explore the space of possible three-dimensional Hilbert curves and the potentially useful properties which they may have. We discuss a notation system that allows us to distinguish the curves from one another and enumerate them. This system has been implemented in a software prototype, available from the author's website.

Several examples of possible three-dimensional Hilbert curves are presented, including a curve that visits the points on most sides of the unit cube in the order of the two-dimensional Hilbert curve; curves of which not only the eight octants are similar to each other, but also the four quarters; a curve with excellent locality-preserving properties and endpoints that are not vertices of the cube; a curve in which all but two octants are each other's images with respect to reflections in axis-parallel planes; and curves that can be sketched on a grid without using vertical line segments. In addition, we discuss several four-dimensional Hilbert curves.

Original language | English |
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Pages (from-to) | 206–281 |

Journal | Journal of Computational Geometry |

Volume | 8 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 |

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### Cite this

*Journal of Computational Geometry*,

*8*(1), 206–281. https://doi.org/10.20382/jocg.v8i1a10

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*Journal of Computational Geometry*, vol. 8, no. 1, pp. 206–281. https://doi.org/10.20382/jocg.v8i1a10

**How many three-dimensional Hilbert curves are there?** / Haverkort, H.J.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - How many three-dimensional Hilbert curves are there?

AU - Haverkort, H.J.

PY - 2017

Y1 - 2017

N2 - Hilbert's two-dimensional space-filling curve is appreciated for its good locality-preserving properties and easy implementation for many applications. However, Hilbert did not describe how to generalize his construction to higher dimensions. In fact, the number of ways in which this may be done ranges from zero to infinite, depending on what properties of the Hilbert curve one considers to be essential.In this work we take the point of view that a Hilbert curve should at least be self-similar and traverse cubes octant by octant. We organize and explore the space of possible three-dimensional Hilbert curves and the potentially useful properties which they may have. We discuss a notation system that allows us to distinguish the curves from one another and enumerate them. This system has been implemented in a software prototype, available from the author's website.Several examples of possible three-dimensional Hilbert curves are presented, including a curve that visits the points on most sides of the unit cube in the order of the two-dimensional Hilbert curve; curves of which not only the eight octants are similar to each other, but also the four quarters; a curve with excellent locality-preserving properties and endpoints that are not vertices of the cube; a curve in which all but two octants are each other's images with respect to reflections in axis-parallel planes; and curves that can be sketched on a grid without using vertical line segments. In addition, we discuss several four-dimensional Hilbert curves.

AB - Hilbert's two-dimensional space-filling curve is appreciated for its good locality-preserving properties and easy implementation for many applications. However, Hilbert did not describe how to generalize his construction to higher dimensions. In fact, the number of ways in which this may be done ranges from zero to infinite, depending on what properties of the Hilbert curve one considers to be essential.In this work we take the point of view that a Hilbert curve should at least be self-similar and traverse cubes octant by octant. We organize and explore the space of possible three-dimensional Hilbert curves and the potentially useful properties which they may have. We discuss a notation system that allows us to distinguish the curves from one another and enumerate them. This system has been implemented in a software prototype, available from the author's website.Several examples of possible three-dimensional Hilbert curves are presented, including a curve that visits the points on most sides of the unit cube in the order of the two-dimensional Hilbert curve; curves of which not only the eight octants are similar to each other, but also the four quarters; a curve with excellent locality-preserving properties and endpoints that are not vertices of the cube; a curve in which all but two octants are each other's images with respect to reflections in axis-parallel planes; and curves that can be sketched on a grid without using vertical line segments. In addition, we discuss several four-dimensional Hilbert curves.

U2 - 10.20382/jocg.v8i1a10

DO - 10.20382/jocg.v8i1a10

M3 - Article

VL - 8

SP - 206

EP - 281

JO - Journal of Computational Geometry

JF - Journal of Computational Geometry

SN - 1920-180X

IS - 1

ER -