Wednesday, August 29, 2012

Differential Geometry without continuity or derivatives

A great deal can be accomplished using chi squared comparison to uniform distributions when we create distributions by projecting all data points, lying within a sphere, perpendicularly onto a plane through the origin of the sphere. We know how many points to expect in each cell if the points are uniformly distributed inside the sphere so
xhiS= sum { (measured - expected )^2/expected }
(over all cells)
measures the deviation from uniformity. It varies as we vary the projection direction. The direction with minimum xhiS serves as a surface normal. The direction of maximum xhiS is that of minimal curvature. Interesting eh?

I read that physicists and mathematicians are pursuing efforts to define space as foamy. It seems to me that the diverse possibilities of this xhiS metric (including the relation to the cell size, including the frangible concepts of "expected" within a sphere) make xhiS a good candidate for handling physical diversities. There could be multiple tangent planes, multiple curvatures, and a spectrum of ideas around the fact that minima and maxim of xhiS can be local rather than global. For example space around a point could be defined as "2D" if the tangent and minimum curvature are global extremes and there are no local extremes to confuse the issue.

However, space around a subatomic particle or around an atom might have multiple directions that were local minima/maxima. As such the dimension is in question. 

Update 2: Probably this all is nonsense. You would need to already be "in" 3D to measure the proposed dimension.  
Update 3: No, you could be in 3D, making direction dependent measurements there, and still uncover a higher dimension entity within your data.