From: Ken Clements (Ken@Innovation-On-Demand.com)
Date: Wed Mar 13 2002 - 00:25:05 MST
Mike Lorrey wrote:
> Ken Clements wrote:
> > I covered the spherical case in my post. Yes, the ratio changes with curvature, but pi does not change. Again, the ratio is simply no longer equal to pi.
>
> When pi is no longer equal to that ratio, how does pi deserve to be
> called pi? Pi is defined as that ratio, ergo its value is relative to
> the local time space curvature.
>
Suppose we were drawing circles on the surface of a big sphere (something like the Earth) with radius S. Let d be the diameter of such a circle. The circumference will be c = 2S pi sin( d / (2S)). Pi in this formula is the same old 3.14159265... constant
number that is used for ordinary geometry. Now, as you let S get bigger and bigger, the space gets flatter. If you let S approach infinity (flat space), or let d approach zero (tiny circles), the value of sin( d / (2S)) approaches d / (2S). So, the 2S in
the numerator cancels the 2S in the denominator, and you get the usual c = pi d. Pi is defined as a constant number, like 3 or 4, and it pops up all over math, not just in circles.
-Ken
This archive was generated by hypermail 2.1.5 : Sat Nov 02 2002 - 09:12:57 MST