Proof  on Cross Products of Two Fractals

By Shahin M Movafagh

Theorem:

Given A,B as two different Euclidean spaces and the fractal dimension of A and B as dim(A) and dim(B) respectively; it will then follow that the Cartesian product space dim(AxB) will be at least dim(A)+dim(B).


Proof:

[Graphics:Images/index_gr_4.gif]
Lets consider the specific example of A as the middle thirds Cantor set and B as the middle second and four fifth Cantor set.
Given AxB={(x,y) in
[Graphics:Images/index_gr_2.gif]:x in A,y in B}, then AxB is the Cantor product or Cantor dust consisting of those points in a plane with both coordinates in [Graphics:Images/index_gr_3.gif]:
 

 

[Graphics:Images/index_gr_24.gif]

[Graphics:Images/index_gr_33.gif]

[Graphics:Images/index_gr_37.gif]