## 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 be the number of black triangles after iteration , the length of a side of a triangle, and the fractional Area which is black after the th iteration. Then

 (1) (2) (3)

The Capacity Dimension is therefore
 (4)

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

References

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

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