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# Cantor Square Fractal

A fractal which can be constructed using string rewriting beginning with a cell [1] and iterating the rules

 (1)

The size of the unit element after the th iteration is

 (2)

and the number of elements is given by the recurrence relation

 (3)

where , and the first few numbers of elements are 5, 65, 665, 6305, ... (OEIS A118004). Expanding out gives

 (4)

The capacity dimension is therefore

 (5) (6)

Since the dimension of the filled part is 2 (i.e., the square is completely filled), Cantor's square fractal is not a true fractal.

Box Fractal, Cantor Dust, Haferman Carpet, Sierpiński Carpet

## Explore with Wolfram|Alpha

More things to try:

## References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 82-83, 1991.Sloane, N. J. A. Sequence A118004 in "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Cantor Square Fractal

## Cite this as:

Weisstein, Eric W. "Cantor Square Fractal." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CantorSquareFractal.html