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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

f0(z) = rs(1+i)z*  and  f1(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/log2s. 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.

2kb frame 1/22 2kb frame 12/22
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.

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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