Re: Pi in non-euclidiean spaces
- From: Narasimham <mathma18@xxxxxxxxxxx>
- Date: Thu, 30 Aug 2007 02:12:41 -0700
On Aug 29, 9:50 pm, Robert Israel
<isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
benigr...@xxxxxxxxx writes:
Whereas pi is defined as the ratio of the circumference of a circle to
it's diameter, it is constant and irrational on a euclidiean plane.
It seems logical to me to suppose that the value of pi would be
different on curved surfaces (like a sphere or hyperbola). This led
me to suppose that there were curvatures where the value of pi is
rational. From there I wondered if there were some characteristics of
those surfaces wherein pi ir irrational that were shared in common.
To a mathematician, pi is always the same number. On a curved surface,
you can talk about the ratio of circumference to diameter of a circle, but
don't call it pi. For a surface that isn't flat, this ratio will not be
constant. For example, on a sphere of radius R, for a circle of radius r
the ratio will be pi R/r sin(r/R). This can take any value between 0
(which is the limit as r/R -> pi) and pi (the limit as r/R -> 0).
--
Robert Israel isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Another example from non-euclidean hyperbolic space on a pseudosphere
of cuspidal radius R for a circle of radius r, this ratio is pi R/r
sinh(r/R). This takes largest value between Infinity (which is the
limit as r/R -> Infinity) and smallest value pi (the limit as r/R ->
0).In the former case we have very deep wavy frills/convolutions and
in the latter it is flat, just as for the 2-D circle case.
Narasimham
.
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- Pi in non-euclidiean spaces
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