Paul wrote:Perhaps this is a nonsensical question...
I'm trying to think about how this might work. But, I keep running into
the Strong Law of Small Numbers. I was trying to think about what one would have to do with say, dimension a*b where both a and b are integers. But, I have to get up to dimension 2*3 to be useful at all since: 1*n = n and 2*2 = 2+2.
Anyhow, my first thought was to take the Cartesian product of the spaces. So, if (x,y) is one space and (z) is another space, then (x,y,z) is in the new space. Unfortunately, if the original spaces are of dimension a and b, the new space will be of dimension a+b rather than a*b.
And, taking all of the linear functions from a space of dimension a to a space of dimension b results in a space of dimension b<sup>a</sup> (if I'm thinking about that right).
So, I need something between addition and exponentiation here.
Of course, the formulas for fractal dimensions involve logarithms, so maybe exponentiation is a good way out.
But, as for some of the side questions in the meantime.... 'similar' would probably have to mean
homeomorphic while 'essentially similar' would mean that the homeomorphism is "natural" in some way. (Sorry, I couldn't find a good link for 'natural'.)