Some Algebra of Dimensions Exercises

Dimension of Unions, Exercise 1 Answer

We select squares of side lengths 1/3, 1/9, 1/27, ... , 1/3n, ... noting the scaling of the Cantor set.

N(1/3) = 2 + 3

N(1/32) = 22 + 32

N(1/33) = 23 + 33

and in general

N(1/3n) = 2n + 3n

Knowing N(1/3n) we can compute the box-counting dimension by factoring out the larger term from the sum:

d = limn→∞Log(N(1/3n)) / Log(1/(1/3n)) = limn→∞Log(2n + 3n) / Log(3n) = limn→∞Log(3n(1 + (2/3)n)) / Log(3n) = limn→∞(Log(3n) + Log((1 + (2/3)n))) / Log(3n) = 1 + limn→∞Log((1 + (2/3)n))) / n⋅Log(2) = 1

Return to Dimension Algebra Exercises.