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.
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.
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.
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