From: Eliezer S. Yudkowsky (sentience@pobox.com)
Date: Fri Jul 05 2002 - 13:58:40 MDT
Alex Ramonsky wrote:
> 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?
I think it's because in 5-D space, the 5-sphere can rotate along
multiple axes, though I could be wrong.
-- Eliezer S. Yudkowsky http://singinst.org/ Research Fellow, Singularity Institute for Artificial Intelligence
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