Re: Pi in non-euclidiean spaces

benigreen@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 israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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