Some Algebra of Dimensions

Dimensions of Interesctions

To understand the dimension of the intersection of sets, we must understand the codimension of a set.
Now we use codimension to study the dimension of the intersection of Euclidean sets.
Although the proof is quite complicated, the intersection result extends to fractal shapes:
if A and B are subsets of n-dimensional space, then
codim(A ∩ B) = codim(A) + codim(B)
That is,
n - dim(A ∩ B) = (n - dim(A)) + (n - dim(B))
giving
dim(A ∩ B) = dim(A) + dim(B) - n
 
Example Suppose A is a gasket with dimension d(A) = log(3)/log(2) = 1.58496, and B is a Cantor set with dimension d(B) = log(2)/log(3) = 0.63093.
Certainly, some placements of the Cantor set will miss the gasket completely. However, typically
d(A ∩ B) = 1.58496 + 0.63093 - 2 = 0.21589.
 
Example Suppose A and B are Cantor sets in the (1-dimensional) line, and both dim(A) and dim(B) are less than 1/2.
Then typically (that is, for almost all placements of A and B in the line),
codim(A ∩ B) = codim(A) + codim(B) > 1
hence dim(A ∩ B) < 0. In other words, almost all placements of A and B are disjoint.

Return to the algebra of dimensions.