Cantor dust


How do you make Cantor dust?
Take a black square (illustration 1).
Divide the black square into 16 equal squares. Using a special pattern leave 4 squares black and make 12 squares white (illustration 2). The white squares now are out of the game, because the fractal consists only of black squares.

Apply the same procedure to every black square that is left (illustration 3).
And again (illustration 4). Endlessly.

What's the end result?
If the surface area of the first black square is 16, the total surface of the 4 small squares (illustration 2) is only 4. And the surface of the 16 smaller squares (illustration 3) is 1.
After every operation the surface of the black areas is 4 times smaller.
In the long run the surface must approach 0.

So ultimately we have nothing left?
The squares disappear (because each square is divided further and further), but what is left consists of points. That's what we call Cantor dust.

What is the dimension of Cantor dust?
If you measure the surface area of the this fractal "4 times as accurate" (i.e. the side of the square that you use to measure is a fourth of the original) , the number that represents the surface area each time is 4 times as large.
As for the dimension: 4^dimension = 4.
So the dimension of the Cantor dust is 1.

But how can we explain that surprising answer, honored master?
Imagine that the first square is a kind of tray.
What happens if you incline the plane of that tray a little bit?
All black squares slide to the bottom line.
And all squares fit together perfectly.
If the Cantor dust consists only of dust (a collection of points),
you have a line with a dimension of 1.


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