From: Alex Ramonsky (alex@ramonsky.com)
Date: Fri Jul 05 2002 - 12:58:35 MDT
I confess I haven't read Diaspora, but the sphere thing seems fairly
straightforward to me.
A 2-D sphere rotates about a point; a 3-D sphere rotates about a line; a
4-D sphere rotates about a plane, and a 5-D sphere rotates about a
hyperplane. Assuming the sphere to have its mass distributed with
spherical symmetrry, this N-2-plane would pass through the centre of the
N-sphere, intersecting it in an N-2-sphere of the same radius. This
intersection is the "pole". When the N-sphere rotates about this pole,
every point in the N-sphere will describe a circle about some point on
the pole (in fact, the closest point to it).
On the Earth, our pole is a line, but since we normally concern
ourselves only with points on the surface, we perceive only the North
and South poles - two points. But two points is just the surface of a
1-sphere. On an N-sphere, the pole would intersect the surface in an
N-3-sphere. So, in 5-D space, the pole is a 3-sphere, whose surface is
a 2-sphere.
So doesn't this mean Poincare was right all along? ...or is it too hard
to visualise...could it be that wer'e not intoxicated enough?
Votes?
Damien Broderick wrote:
>A pal writes:
>
>< Greg Egan has a vital correction to his novel _Diaspora_,
>whose 17th chapter laughably refers to `Poincare's rotational "pole" --
>the two-dimensional sphere on the hypersurface that stayed fixed in space
>as the star rotated.' The author's website warns that this is in fact an
>unlikely situation in 5-dimensional space, and that it's more probable
>that there would be `two single-point rotational poles, as in 3
>dimensions.' >
>
>I knew you'll all want this news rushed to the list as swiftly as possible.
>
>Damien Broderick
>
>
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