“Sierpinski Sieve”的概念、定义、习题解题方法及翻译-数学辞典

单词 Sierpinski Sieve

释义

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 Systemwith initial string "FXF-FF-FF", String Rewriting rules "F" -> "FF", "X" -> "-FXF++FXF++FXF-", andangle 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 thiteration. 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.

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.

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