diminishing returns from adding spatial dimensions?

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diminishing returns from adding spatial dimensions?

Postby DMattMooney » Fri Jun 21, 2013 1:24 am

I sometimes use the Excel graphing capability to get a quick look at functions of interest. One time I plotted the force-distance power law for gravity or electromagnetism using a set of curves to show the effect of varying the number of spatial dimensions. The curves were scaled to one at a distance of 0.2 and the vertical asymptotes were truncated to a height of one. There was a huge drop in the tail going from two to three dimensions, then a smaller drop going to four dimensions, then smaller still going to five, etc. The process seems to have gone to completion by only the sixth dimension. There were obvious diminishing returns on linear-linear axes. Apparently high dimensionality destroys physics. I know this diminishing returns picture could have been reduced by using log-log axes, but I still think it's interesting. Here's a thought experiment. Take 100 spheres that attract each other and only allow them freedom of motion in 1-D. They form a line. Now give them another dimension: the line contracts until the spheres fill a circle. Now add the third dimension: the disc of spheres rounds up into a glob of spheres. Can you see diminishing returns here in the linear amount of contraction allowed by the third dimension versus the second? The diminishing returns possibility suggests that 4-D space is more like 3-D space than we would assume by comparing 3-D space to 2-D space.
Last edited by DMattMooney on Tue Jul 09, 2013 2:29 am, edited 1 time in total.
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Re: diminishing returns from adding spatial dimensions?

Postby wendy » Fri Jun 21, 2013 7:16 am

The usual model for radiant fields like gravity and magnetism is to use flux. Flux drops off with the surface of the sphere.

On the other hand, one should note that uniformly dense points increases with the surface of the sphere. One can use this to prove a black sky implies a finite universe, since the light from any given shell will give the same number of lumens, and that an infinite number of populated shells gives an infinite number of lumens.

Gravity in 4d, works, but the nature of an inverse-cube law is that elliptic orbits or any non-circular orbit, is not stable, and planets on a non-stable orbit, would be thrown from the sun.
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