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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Sierpinski Curve


What if applet does not run?

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

Plane Filling Curves

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