# 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.)

What if applet does not run? |

### Reference

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

### Plane Filling Curves

- Plane Filling Curves
- Plane Filling Curves: Hilbert's and Moore's
- Plane Filling Curves: Peano's and Wunderlich's
- Plane Filling Curves: all possible Peano curve
- Plane Filling Curves: the Lebesgue Curve
- Following the Hilbert Curve
- Plane Filling Curves: One of Sierpinski's Curves
- A Plane Filling Curve for the Year 2017

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

Copyright © 1996-2018 Alexander Bogomolny

63241425 |