The Cantor square fractal is a construction on a unit square using a string rewriting system. Begin with an unfilled cell labeled [0], let 1 denote a filled cell, and iterate the rules
![{0->[0 1 0; 1 1 1; 0 1 0],1->[1 1 1; 1 1 1; 1 1 1]}.](/images/equations/CantorSquareFractal/NumberedEquation1.svg) |
(1)
|
Each replacement subdivides a square into nine congruent squares. Their side length after 
iterations, where 
is a nonnegative integer,
is
 |
(2)
|
and the number 
of filled cells is given by the recurrence relation
 |
(3)
|
for 
,
where 
.
The first few numbers of filled cells, beginning at 
, are 5, 65, 665, 6305, ... (OEIS A118004).
Expanding out gives
 |
(4)
|
The capacity dimension is therefore
 |  |  |
(5)
|
 |  |  |
(6)
|
The unfilled cells follow the Cantor dust construction. Their total area is 
, while the filled area is 
and tends to 1. The union
of all filled cells contains an open square and has capacity dimension 2. This does not mean that
every point is filled. For example, the four corners of the initial unit
square remain unfilled.
