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Hilbert's two-dimensional space-filling curve is appreciated for its goodlocality-preserving properties and easy implementation for many applications.However, Hilbert did not describe how to generalize his construction to higherdimensions. In fact, the number of ways in which this may be done ranges fromzero to infinite, depending on what properties of the Hilbert curve oneconsiders to be essential.  In this work we take the point of view that a Hilbert curve should at leastbe self-similar and traverse cubes octant by octant. We organize and explorethe space of possible three-dimensional Hilbert curves and the potentiallyuseful properties which they may have. We discuss a notation system that allowsus to distinguish the curves from one another and enumerate them. This systemhas been implemented in a software prototype, available from the author'swebsite. Several examples of possible three-dimensional Hilbert curves arepresented, including a curve that visits the points on most sides of the unitcube in the order of the two-dimensional Hilbert curve; curves of which notonly the eight octants are similar to each other, but also the four quarters; acurve with excellent locality-preserving properties and endpoints that are notvertices of the cube; a curve in which all but two octants are each other'simages with respect to reflections in axis-parallel planes; and curves that canbe sketched on a grid without using vertical line segments. In addition, wediscuss several four-dimensional Hilbert curves.

How many three-dimensional Hilbert curves are there?