Plane Filling Curves:
One of Sierpinski's Curves

W. Sierpinski introduced a plane filling curve in a 1912 article. The curve follows the subsquares in the same sequence as Hilbert's curve. As [H. Sagan, p. 50] writes

Sierpinski showed that there is a bounded, continuous, and even function \(f\) of a real variable \(t\) which satisfies the functional equation

\( f(t) + f(t+1/2)=0 \)

for all real \(t\) and

\( 2f(t/4)+f(t+1/8)=1 \)

for all \(t\in [0,1]\) and

\( (f(t), f(t-1/4)), 0\le t\le 1 \)

passes through every point of the square \([-1,1]^2\).

He then constructed such a function and showed that the the resulting curve is the limit of a uniformly convergent sequence of closed polygons several first member of which are illustrated by the applet below.

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

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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Pólya later (1913) observed that a diagonal of the square neatly divides the curve into two pieces each contained in one half of the square.

Reference

H. Sagan, Space-Filling Curves, Springer-Verlag, 1994