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Three Dimensional Space-Filling Curves

Some of these curves naturally extend to higher dimensions. For example, the Hilbert curve can be naturally adjusted to give us a curve that fills a three dimensional cube. This is indicated in the next figure. The analytic description of the eight functions required (but without the following shrinking we use here to get fractals) can be found in Sagan ([Sag, p. 28]). The functions are also described in the source code, where it's 24 real-valued functions since one is needed for each coordinate.

4kb dim 1.4 5kb dim 2.0
Figure 14. Two fractal versions of 3-dimensional approximations to a Hilbert curve filling a cube. Also available is an animated version with more frames, a speed controlled version, a VRML version of the first picture (1300kb), and a VRML version of the second picture (warning: 10.7 mb file). Use the browser back button to return. Not all browsers will support VRML. Source code is available.

Similarly, the Peano curve can be so generalized. We show this in Figure 15. An analytic description (but without the shrinking for fractals or the figures) was given in Peano's original paper ([Pe, pp.159-160]). There are 27 functions from space to itself - doing each coordinate separately makes 81 - but some of these coordinate functions turn out to be the same so that only 18 different real-valued functions are needed. The details can be found in the source code.

4kb dim 1.2 5kb dim 2.0
Figure 15. Two fractal versions of 3-dimensional approximations to a Peano curve filling a cube. Also available is an animated version with more frames, a speed controlled version, a VRML version of the first picture (790 Kb), and a VRML version of the second picture (a rough approximation to keep the file down to 766 kb). Use the browser back button to return. Not all browsers will support VRML. Source code is available.

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Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 18 Aug 1998 --- Last modified: Sep 30, 2003 6:45:12 PM
Comments to: CVM@maa.org