Re: Mystery of High Dimensions [NOT OT]

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Sujet : Re: Mystery of High Dimensions [NOT OT]
De : fsquared (at) *nospam* fsquared.linux (Farley Flud)
Groupes : comp.os.linux.advocacy
Date : 03. Jan 2025, 15:20:02
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Organisation : UsenetExpress - www.usenetexpress.com
Message-ID : <181734969de0eede$47707$3091964$802601b3@news.usenetexpress.com>
References : 1 2
User-Agent : Pan/0.146 (Hic habitat felicitas; d7a48b4 gitlab.gnome.org/GNOME/pan.git)
On Thu, 02 Jan 2025 16:55:28 -0600, Physfitfreak wrote:


 
Sounds like something one gets tempted to ask AI about it.
>

There's no need for AI (even it could give an answer).

The answer is in the "corners."

A hypercube of n dimensions has 2^n corners, and it is these corners
that contain most of the volume.

An inscribed hypersphere will always touch each face of the cube but
it always curves away from the corners.

Furthermore, the distance from the center of the hypercube (and its
inscribed hypersphere) to each corner is sqrt(n) * r, where n is the
number of dimensions and r is the radius of the hypersphere and also
the length of 1/2 side of the hypercube.

Thus as n increases, so too does the distance to each corner from
the center of the hypersphere.

N-D spaces are fascinating places.

For n=100, the cube has 2^100 corners with each corner located
10 units from the center of a hypersphere of radius = 1 unit.

That's LOT of hypervolume outside of the hypersphere but still
inside the hypercube.


--
Hail Linux!  Hail FOSS!  Hail Stallman!


Date Sujet#  Auteur
1 Jan 25 * Mystery of High Dimensions [NOT OT]12Farley Flud
2 Jan 25 `* Re: Mystery of High Dimensions [NOT OT]11Physfitfreak
3 Jan 25  +* Re: Mystery of High Dimensions [NOT OT]2Farley Flud
4 Jan 25  i`- Re: Mystery of High Dimensions [NOT OT]1Physfitfreak
3 Jan 25  `* Re: Mystery of High Dimensions [NOT OT]8Farley Flud
4 Jan 25   +* Re: Mystery of High Dimensions [NOT OT]4Physfitfreak
4 Jan 25   i`* Re: Mystery of High Dimensions [NOT OT]3Farley Flud
4 Jan 25   i +- Re: Mystery of High Dimensions [NOT OT]1DFS
5 Jan 25   i `- Re: Mystery of High Dimensions [NOT OT]1Physfitfreak
4 Jan 25   `* Re: Mystery of High Dimensions [NOT OT]3RonB
4 Jan 25    `* Re: Mystery of High Dimensions [NOT OT]2Farley Flud
5 Jan 25     `- Re: Mystery of High Dimensions [NOT OT]1Physfitfreak

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