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isaacein
exp(ix) = cosx + isinx

Registered: 05/21/08
Posts: 1,141
Last seen: 15 years, 8 months

Re: Math

[Re: Coaster]
#9341964 - 12/01/08 04:16 AM (17 years, 10 months ago)

Coaster said:

so if its a 3-dimensional ball then the distance is 3/4

That is right. In three dimensions, the points in the sphere are on average 3/4 away from the center. Note that as the number of dimensions increases we approach 1.

Can you show it?

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Helixx
Mood:Fragglerocked

Registered: 06/08/07
Posts: 1,623
Last seen: 12 years, 11 months

Re: Math

[Re: Noteworthy]
#9341982 - 12/01/08 04:23 AM (17 years, 10 months ago)

Thanks

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Coaster
Baʿal

Registered: 05/22/06
Posts: 33,501
Loc: Deep in the Valley
Last seen: 15 years, 28 days

Re: Math

[Re: isaacein]
#9341986 - 12/01/08 04:24 AM (17 years, 10 months ago)

so ur saying
in an infinite dimensional sphere all points are on the surface?

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isaacein
exp(ix) = cosx + isinx

Registered: 05/21/08
Posts: 1,141
Last seen: 15 years, 8 months

Re: Math

[Re: Coaster]
#9342017 - 12/01/08 04:32 AM (17 years, 10 months ago)

Coaster said:

so ur saying
in an infinite dimensional sphere all points are on the surface?

Yes.

Or if you want, as the number of dimensions increases, "almost all" of the volume ends up in a thin crust on the surface. For instance a circle of radius 1/2 contains less than a ring of radius 1/2 around it.

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isaacein
exp(ix) = cosx + isinx

Registered: 05/21/08
Posts: 1,141
Last seen: 15 years, 8 months

Re: Math

[Re: isaacein]
#9342022 - 12/01/08 04:33 AM (17 years, 10 months ago)

Well I say yes but no. Obviously not all points are on the surface; but those closer to the surface account for an arbitrarily big proportion of the total volume

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