Hilbert and Moore 3D Fractal Curves
The Hilbert curve and the Moore curve are two famous plane-filling curves that can be extended to 3D space-filling curves. They have similar recursive constructions, here using L-systems. The curves shown in this Demonstration map points in three-dimensional space to points on a one-dimensional line, and thus have properties that make them useful for certain types of data manipulation, such as image processing.
Snapshot 1: viewing multiple iterations at once and rotating the resulting image can assist in making the recursive construction of the curves clearer
Snapshot 2: one way in which the Hilbert curve and Moore curve differ is in the relative positions of the start and end points of the curve. While the Hilbert curve begins and ends in adjacent corners of the bounding cube, the Moore curve begins and ends in adjacent points in the resulting three-dimensional grid.
Snapshot 3: in this Demonstration, the Hilbert curve and the Moore curve begin with the same initial shape; it can be illuminating to switch between the Hilbert and Moore figures for a given iteration level
The Lebesgue curve has similar properties; while it is easier to compute, the Hilbert curve is often desirable in that no two consecutive points are very far apart.