Sierpinski curve

fractal

last updated: 2005-03-27

The Sierpinski curve is a base motif fractal where the base is a square. After subdivision in 3x3 equal squares the motif is to remove the middle square:

The curve is also known as the Sierpinski (universal plane) curve, Sierpinski square or the Sierpinski carpet.
It has named after the Polish mathematician Waclaw Sierpinski (1882-1969), but it was Stefan Mazurkiewicz who found the curve (in 1913).

The curve is the only plane locally connected one-dimensional continuum S such that the boundary of each complementary domain of S is a simple closed curve and no two of these complementary domain boundaries intersect.
Wow!
In other words: the Sierpinski curve contains a topologically equivalent copy of any compact one-dimensional object in the plane.

The fractal dimension of the curve is equal to log 8/ log 3, i.e. about 1.8928 1).
The curve is a two-dimensional generalization of the Cantor set.

Some kind of shells (conus textilus, conus gloriatnatis) have patterns that resemble the Sierpinksi square.

Professor Gerda de Vries of the University of Alberta designed a quilt named ´Sierpinksi Meets Mondrian´, based on the Sierpinski curve. The quilt was made in 2002 in response to the Edmonton & District Quilters' Guild challenge to create a quilt in the theme ´Voices in Cloth´. It was made for the entry category ´A Picture is Worth a Thousand Words´.

 


notes

1) Fractal dimension = log N / log e, where N is the number of line segments and e the magnification.
For the Sierpinski curve: N=8, e=3.