Some Algebra of Dimensions Exercises

Dimension of Products, 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⋅2   N(1/32) = 22⋅22   N(1/33) = 23⋅23

and in general

N(1/3n) = 2n⋅2n = 4n

Knowing N(1/3n) we can compute the box-counting dimension:

d = limn→∞Log(N(1/3n)) / Log(1/(1/3n)) = limn→∞Log(4n) / Log(3n) = Log(4) / Log(3) = 2⋅Log(2) / Log(3)

Return to Dimension Algebra Exercises.