array9

 It seems so obvious that the number of dimensions is an integer.  But I am never sure how one would ever measure it.  One thinks of 'flatland' and it just confuses one.

But at least flatland only ever had integer numbers of dimensions.

I also don't like the presentation of the 'laws' of gravity with a simple 2 as the exponent of the distance.  would it be more helpful to have "number of dimensions' -1".  Then at least one is forced to think about it.

Now I prefer a system that does not start with a euclidean space as its dominant structure.  I am happy with counting, points, distance, and 'spheres' .  I am happy with different sizes of spheres.  Now one can measure the dimensionality.  Basically  build a big sphere and then pack as many small spheres inside this as you can. See what the answer is for the smallest spheres you can create or imagine.  and from this you can work out the dimensionality of that bit of space.  

I have no idea how one would do such an experiment, particularly on the grand scale.  Sounds a bit like measuring gravity waves, and how many can fit in a box.  Or measuring the volume of a black hole, which might be zero for all I know.

The formula for the normalised number is  effectively pi.r.r  for the plane.  pi.r.r.r/3 or 2.pi.r.r.r/3!  for 3 dimensional space and 2.pi.r^n/n! for n dimensional space.  Now one can obviously generalise to a continuous variable for dimensionality, and then do the experiment and find the dimensionality, which I will label d.  One should not worry about how things are actually squashed in.  This is not euclidean space.

This probably gives one the formula for gravity ( or whatever) as f = G.m1.m2.r^(d-1).

I guess that the best way of thinking about fractional dimensions is to think of them as not independent of others. Indeed that some space has an integer dimensionality of n, does not imply that there are n independent dimensions.  Again black holes might be worth consideration as one seems to be able to pour a considerable mass into such small spaces.

have fun trying to think about it

martin

# posted by martinW @ 3:30 pm