Metrics on the Reals

It has been something of a prime activity of my recent life to try and make a coordinate system for non integer dimensional spaces work. Although I have tried rather a lot of different approaches I have never really come up with anything satisfactory. For instance, at one point I considered a sort of pseudo euclidean quotient space which had the appropriate scaling law for the "volumes" of spheres by radius. However the space was less well behaved for concave sets. In fact the "volume" of a sub set of a concave set might actually be greater than the "volume" of the whole. Which at the very least means that the volume scaling law doesn't hold the same for all shapes in that space which was pretty much the kiss of death for that idea.

Other ideas have had a much longer and less clear history, for instance I have been mucking about thinking about using random coordinate systems and fuzzy logic. How random coordinate systems or fuzzy logic might be used in a concrete way to create a coordinate system I don't know. Which of course is why I still give it so much thought. One idea using random coordinate systems is to assume an infinite set of random vectors which have a particular probability distribution for the value of their dot products. Assuming uniform distribution of the vectors around the surface of the appropriate dimensional sphere gives a very specific expected dot product distribution for each dimension. If we assume a distribution somewhat in between say the 2 and 3 dimensional distributions then perhaps we would have a consistent coordinate system, albeit one with a necessarily infinite number of coordinates. This idea while pretty is something that I have not really gotten very far with. I really should put some sweat into it and see if I can make it work.

All of this is not really the point I was trying to make though (perhaps I should just rename this post and skip what I was trying to say) What I was thinking about recently is the fact that because the cardinality of the real numbers is the same as the cardinality of any euclidean space of any dimension (at least integer dimensional ones and presumably non integer dimensional ones too) you can find a bijective mapping from a space of any dimension onto the interval (0,1).

In other words as long as the cardinality of the point set of a space of non integer dimension is the same as the cardinality of the integer dimensional ones then lurking somewhere in the set of functions on the interval between 0 and 1 is the metric for any dimensional space you care to think of.

For this reason I have been thinking that perhaps the best way to try and think about non integer dimensional spaces is to think about real number theory. The kind of stuff where you talk about recursive function mappings of the real numbers for instance the mapping which we use as the basis of the decimal system. The decimal system can be thought of as the output of an algorithm which maps the interval from zero to one to itself. Say you want a decimal representation of any number. You begin by taking its integer part and then you minus that part out and multiply by 10 and then take the integer part of that and then rinse and repeat.

Perhaps by considering mappings of the unit square to itself we might come up with a suitable metric.