The Sierpinski-Knopp Curve
As a last example of using a continuum of IFS's to get a sequence of
fractal curves that converges to a planar space-filling curve, we return
to the Sierpinski-Knopp curve that was described as the limit of snowflake
curves in Section 4. We give
an alternative approach here that requires just two IFS functions but
forces us to include connecting segments. Starting with an isosceles
right triangle, the functions shrink toward the acute angles and reflect
in the angle bisectors. Analytically, with complex numbers, for r
equal to the square root of 1/2 and s in the interval
[0,r] (and * still meaning conjugation) we have
f_{0}(z) = rs(1+i)z* and
f_{1}(z) = rs(1-i)z*+(1-rs)+rsi
In Figure 11 we show two frames from the movie of the graphs we get.
Note that the Cantor set attractor will have dimension given by
-1/log_{2}s. Adding the connecting segments will
make the actual dimension the larger of this dimension and one. The two
shown frames have Cantor sets (once the connecting segments are removed)
with similarity dimension 1 and 1.5.
Figure 11. Converging to the
Sierpinski-Knopp
space-filling curve.
Also available is an animated
(140kb) version and a
speed controlled version.
Use the browser back button to return.
Source code is available.
Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998,
The Mathematical Association of America.
All rights reserved.
Created: 18 Aug 1998 ---
Last modified: Sep 30, 2003 6:34:33 PM
Comments to: CVM@maa.org