Hilbert Curve

$\begin{figure}\begin{center}\BoxedEPSF{hilbert_curve.epsf scaled 700}\end{center}\end{figure}$

A Lindenmayer System invented by Hilbert (1891) whose limit is a Plane-Filling Curve which fills a square. Traversing the Vertices of an -D Hypercube in Gray Code order produces a generator for the -D Hilbert curve (Goetz). The Hilbert curve can be simply encoded with initial string "L", String Rewriting rules "L" -> "+RF-LFL-FR+", "R"->"-LF+RFR+FL-", and angle 90° (Peitgen and Saupe 1988, p. 278).

$\begin{figure}\begin{center}\BoxedEPSF{HilbertIICurve.epsf scaled 700}\end{center}\end{figure}$

A related curve is the Hilbert II curve, shown above (Peitgen and Saupe 1988, p. 284). It is also a Lindenmayer System and the curve can be encoded with initial string "X", String Rewriting rules "X" -> "XFYFX+F+YFXFY-F-XFYFX", "Y" -> "YFXFY-F-XFYFX+F+YFXFY", and angle 90°.

References

Bogomolny, A. ``Plane Filling Curves.'' http://www.cut-the-knot.com/do_you_know/hilbert.html.

Dickau, R. M. ``Two-Dimensional L-Systems.'' http://forum.swarthmore.edu/advanced/robertd/lsys2d.html.

Dickau, R. M. ``Three-Dimensional L-Systems.'' http://forum.swarthmore.edu/advanced/robertd/lsys3d.html.

Goetz, P. ``Phil's Good Enough Complexity Dictionary.'' http://www.cs.buffalo.edu/~goetz/dict.html.

Hilbert, D. ``Über die stetige Abbildung einer Linie auf ein Flachenstück.'' Math. Ann. 38, 459-460, 1891.

Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, pp. 278 and 284, 1988.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 198-206, 1991.