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.)
| 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
- 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
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