How many three-dimensional Hilbert curves are there?

TL;DR

This paper explores the diversity of three-dimensional Hilbert curves, establishing a systematic notation, enumerating possible variants, and presenting examples with various properties, including some extending to four dimensions.

Contribution

It introduces a notation system for classifying 3D Hilbert curves, explores their possible variations, and provides a software implementation with multiple example curves.

Findings

Enumerated the possible 3D Hilbert curves based on self-similarity and octant traversal

Presented examples with unique properties, such as non-vertex endpoints and high locality

Discussed potential extensions to four-dimensional Hilbert curves

Abstract

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.