Background: The Euclidean distance between two points in $n$ dimensions, where $n$ is a positive integer, and position can be described by a vector is given by... $$D_E=\left(\sum_{k=1}^n \left((x_k)^2 \right) \right)^{1/2}$$ where $x_k$ is an entry in the vector. Now, strictly speaking fractional dimensionality is not strictly speaking Euclidean, it has more to do with fractals, however it seems as though converting the summation to an integral might allow for some kind of generalization of the metric. My vision is to see Euclidean metric extended in much the same way the Lebesgue measure is extended to become the Hausdorff measure.
Question: Now that you know the background, the first question isn't really that difficult.
How do you convert this summation to an integral so that n may now be a fractional number? (Obviously don't tell me to floor, ceiling, etc. the n so it becomes a sum again)
The result may be not be a metric, unphysical, but I'm would still interested in an answer with an example. The second question is more involved.
How would one define a coordinate system that has a fractional number of coordinates?
By this I literally mean defining coordinates with non-integral numbers of parameters. Of course, this would be less than useful without a reference, a way to represent typical fractals, and the possibility to define fractional functions.
I'll accept references for this second question. Also, any suggestions to extend metrics along the same lines as the Hausdorff measure extends uhmm... measure would be appreciated.
Preemptive Strike: Don't tell me this isn't possible unless you can provide proof, but since these are definitions, I highly doubt you'd be able to provide one. Also, I've read all of the relevant posts pertaining to this for MSE, so you don't feel compelled to link me those sources.
Above all else, whatever answer you do provide has to be able to create and find the distance spanned by a line embedded in your presented fractional space, just as a line can have distance in 3,4, or more dimensions.
