anderscolingustafson wrote:I also found that for triangle like shapes in which all the sides and angles are the same it seems like the more dimensions it has the smaller the height is in comparison to the edges. I wonder if this property applies to all polytopes.
As for whether it applies to all polytopes, obviously it doesn't apply to the n-cube. The height of the n-cube stays constant as n increases.
anderscolingustafson wrote:As for whether it applies to all polytopes, obviously it doesn't apply to the n-cube. The height of the n-cube stays constant as n increases.
Actually what I was asking is does the property in which the ratio of the largest distance across to the smallest distance across increases with the number of dimensions apply to all polytopes? I mean does the lonest distance across a polytope get longer in comparison to the shortest distance as the number of dimensions increases no matter what the polytope?
V_0 = 1
V_1 = 2
V_n = (2 pi / n) . V_(n-2)
V_(2k) = pi^k / k!
V_(2k+1) = 2 . k! . (4 pi)^k / (2k+1)!
V_n = pi^(n/2) / Gamma(n/2 + 1)
Klitzing wrote:You either can calculate the volume of the unit hyperball recursively
or explicitely, but separate for even and odd numbers, by means of
- Code: Select all
V_(2k) = pi^k / k!
V_(2k+1) = 2 . k! . (4 pi)^k / (2k+1)!
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