Re: [Phys-l] a mathematical curiosity

[John Denker <jsd@av8n.com>]

On 01/09/2012 11:31 AM, Bob Sciamanda wrote:
> I would only remark that these are statements more about the manner in which 
> we have chosen to partition the area of a sphere rather than about  the 
> intrinsic topological properties of a sphere.  I think that the only 
> intrinsic topological content of these “equator –based” statements is simply 
> that a great circle (any great circle) is larger (in radius and 
> circumference)  than any other circle drawn on the surface of a 3-sphere.

We agree the result has got nothing to do with topology.
It does however depend on scaling and on dimensionality.

Suppose you pick two random points on a sphere, perhaps a sphere 
with no rotation and no coordinates (lat/lon or otherwise).  
Consider the angle those points subtend, as seen from the center 
of the sphere.  That angle is more likely to be within a few 
degrees of 90 than within a few degrees of zero.

That's a fact ... not a choice.  It has got nothing to do with
how "we have chosen to partition the area of a sphere".

If you find the result to be obvious, that's fine ... but there
are plenty of people who don't find it obvious.

Perhaps one reason why it's not obvious is that it's not true
for a circle: In fact, the angle subtended by two randomly chosen 
points on a circle is just as likely to be near 90 degrees as to 
be near zero.  Anybody who develops an intuition about D=2 and
tries to extrapolate to higher D is in for a surprise.

For large D, the angle is verrry likely to be verrry near 90 degrees.