More Geometries

Generally I have been considering one particle for geometries here, in which infinity is at the origin, and magnitudes decrease with distance; -1/x, basically.

Considering the space between two particles, we have two infinities.  But it has finite extent (the proper distance between the two particles), and curvature remains.

We arrive at an interesting parallel to hyperbolic spaces, and our metric regains something like positive-dimensional properties, considered from an outside perspective.  However, the curvature isn't constant in this model, so it isn't actually hyperbolic.

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Pi^3 probably doesn't work, although it provides nice values on the scale of the solar system.  What it utterly fails to do is provide a half-period between the nuclear force and gravity, which is a significant failure.  Even if relativistic corrections might recover the cosmological constant at appropriate scales, the maximum period that is feasible is probably around 10^15.  I can potentially recover some orders of magnitude in relativistic corrections when the resulting force is attractive, but I have to lose some when it is repulsive, and I need a repulsive phase between the nuclear force and gravity for any of this to work.

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I think I might have missed an important element in the logarithmic relationship - I think r may need to be the complex dimension.  This makes things somewhat easier for me to conceptualize.  Still trying to wrap my head around a brick there, though.