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.)

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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

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