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[Phys-l] n-spheres



In my last message, I chose my words carefully. I spoke of
_radial_ symmetry. The radius is denoted r.

-- In D=1, r = sqrt(x^2)
this is also called bilateral or left/right symmetry
-- In D=2, r = sqrt(x^2 + y^2)
this is also called cylindrical symmetry
-- In D=3, r = sqrt(x^2 + y^2 + z^2)
this is also called spherical symmetry
-- In D=4, r = sqrt(w^2 + x^2 + y^2 + z^2)
this is also called hyperspherical symmetry

In every case, these have the symmetry of the n-sphere, where
either n = D-1 or n = D, depending on whether you listen to
the topologists or the geometers.
http://mathworld.wolfram.com/Hypersphere.html

If anybody feels these are anything less than profound and
elegant, my only advice is to lie down until the feeling
goes away. It's often said I tend to explain the obvious,
and I usually don't mind doing that, but there are limits.