Answer and Discussion

The fractal constructed as described above is the union of N=2 disjoint copies of the original unit interval that are scaled by a factor of r=1/3. Hence, the dimension is given by

Input := 

dimension[2,1/3]

Output =

0.63093

This fractal is called the middle-thirds Cantor set or the Cantor dust. The fractal properties of the Cantor dust are extremely important. It turns out that the sum of the lengths of the open intervals removed in the construction of the dust is exactly 1, yet the remaining set has the same cardinality as the unit interval [0,1]. It's as though we took the whole line out and were left with just as many points as we started with.