Following the Hilbert Curve

The standard polygonal approximation to the Hilbert plane filling curve is constructed following a simple algorithm that is best expressed in terms of L-Systems [e.g., Chaos and Fractals, Ch. 7]. The standard construction starts with a square U shape. Each step combines four smaller copies of the result of the pervious step: the two middle ones are identical to the previous shape, the other two are obtained by turning the latter left and right.

The applet bellow applies that algorithm to a semicircle and a tent function, instead of the original U shape. Both construction serve exactly the same purpose as the Hilbert curve, viz., filling a square with a continuous curve. In fact, Hilbert himself considered his curve only as a sample resulting from a general procedure described in a 1891 paper.

(To see the progress of polygonal approximations keep clicking in the applet area.)


If you are reading this, your browser is not set to run Java applets. Try IE11 or Safari and declare the site https://www.cut-the-knot.org as trusted in the Java setup.

Following the Hilbert Curve


What if applet does not run?

Reference

  1. Heinz-Otto Peitgen et al, Chaos and Fractals: New Frontiers of Science , Springer, 2nd edition, 2004
  2. H. Sagan, Space-Filling Curves, Springer-Verlag, 1994

Plane Filling Curves

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny

72022132