Cantor Dust
المؤلف:
Dickau, R. M.
المصدر:
"Cantor Dust." http://mathforum.org/advanced/robertd/cantor.html.
الجزء والصفحة:
...
13-9-2021
1250
Cantor Dust
Cantor dust is a fractal that can be constructed using string rewriting beginning with a cell [0] and iterating the rules
![<span style=]() {0->[0 0 0; 0 0 0; 0 0 0],1->[1 0 1; 0 0 0; 1 0 1]}. " src="https://mathworld.wolfram.com/images/equations/CantorDust/NumberedEquation1.gif" style="height:54px; width:192px" /> |
(1)
|
The 
th iteration of Cantor dust is implemented in the Wolfram Language as CantorMesh[n, 2].
Let 
be the number of black boxes, 
the length of a side of a box, and 
the fractional area of black boxes after the 
th iteration, then
The number of black squares after 
, 1, 2, ... iterations is therefore 1, 4, 16, 64, 256, 1024, 4096, 16384, ... (OEIS A000302). The capacity dimension is therefore
REFERENCES:
Dickau, R. M. "Cantor Dust." http://mathforum.org/advanced/robertd/cantor.html.
Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, p. 80, 1983.
Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 103-104, 1993.
Sloane, N. J. A. Sequences A000302/M3518 and A100831 in "The On-Line Encyclopedia of Integer Sequences."
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