Sierpinski Sieve

A Fractal described by Sierpinski in 1915. It is also called the Sierpinski Gasket or Sierpinski Triangle. The curve can be written as a Lindenmayer System with initial string "FXF-FF-FF", String Rewriting rules "F" -> "FF", "X" -> "-FXF++FXF++FXF-", and angle 60°.

Let $N_n$ be the number of black triangles after iteration $n$, $L_n$ the length of a side of a triangle, and $A_n$ the fractional Area which is black after the $n$th iteration. Then

$$N_n = 3^n$$ (1)

$$L_n = ({\textstyle{1\over 2}})^n=2^{-n}$$ (2)

$$A_n = {L_n}^2N_n = ({\textstyle{3\over 4}})^n.$$ (3)

The Capacity Dimension is therefore

$$d_{\rm cap} = -\lim_{n\to\infty}{\ln N_n\over\ln L_n}=-\lim_{n\to\infty}{\ln(3^n)\over\ln(2^{-n})}={\ln 3\over\ln 2}$$

$$= 1.584962500\ldots.$$ (4)

In Pascal's Triangle, coloring all Odd numbers black and Even numbers white produces a Sierpinski sieve.

See also Lindenmayer System, Sierpinski Arrowhead Curve, Sierpinski Carpet, Tetrix

References

Crownover, R. M. Introduction to Fractals and Chaos. Sudbury, MA: Jones & Bartlett, 1995.

Dickau, R. M. ``Two-Dimensional L-Systems.'' http://forum.swarthmore.edu/advanced/robertd/lsys2d.html.

Dickau, R. M. ``Typeset Fractals.'' Mathematica J. 7, 15, 1997.

Dickau, R. ``Sierpinski-Menger Sponge Code and Graphic.'' http://www.mathsource.com/cgi-bin/MathSource/Applications/Graphics/0206-110.

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 13-14, 1991.

Peitgen, H.-O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, pp. 78-88, 1992.

Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, p. 282, 1988.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 108 and 151-153, 1991.

Wang, P. ``Renderings.'' http://www.ugcs.caltech.edu/~peterw/portfolio/renderings/.

Weisstein, E. W. ``Fractals.'' Mathematica notebook Fractal.m.