
#1
Mar705, 11:09 PM

P: 1,636

here is an interesting link at mathworld that describes a hypersphere and the formula for determining the surface area of a sphere in higher dimensions. If you look at the graph, the maximum area is in 7th dimension and diverges to 0 for higher dimensions.
Although some people speculate the universe might have infinite dimensions, could this be proof that an infinite dimensional universe may not exist. http://mathworld.wolfram.com/Hypersphere.html 



#2
Mar1605, 08:13 PM

P: 147

No, just that a spherical universe with infinite dimensions has surface area 0 (probably because it wouldn't have a surface).
That's interesting, though. Does a surface area max occur with other hypershapes, as n approaches infinity? 



#3
Mar1705, 10:35 AM

P: 83

Apart from the hypersphere and the hypercube (area = 2n, which ofcourse diverges for n going to infinity) I think there are not really a lot of shapes that can be intuitively generalised to an arbitrary number of dimensions.




#4
Mar1705, 02:00 PM

P: 147

infinitie dimensions?
What about the hypertorus?
Mathworld outputs no results and Google isn't much help either. That's too bad... isn't it supposed to be one of the leadings candidates for the shape of the universe? I know Kaku mentions the hypertorus in "Hyperspace," but I'm not sure he mentioned anything about the surface area in relation to n. Oh, for those using Mac OS X, I found a screensaver of a rotating 4D Torus. Trippy! 



#5
Mar1705, 06:52 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,879

Mathematics cannot prove anything about physics it can only prove statements about particular models for physics perhaps that a particular model is not valid.



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