From: Ken Clements (Ken@Innovation-On-Demand.com)
Date: Sat Mar 09 2002 - 18:51:06 MST
ABlainey@aol.com wrote:
>
> But is this only true for the outside observer ? From within the
> curvature Pi would still relate ?.
>
Pi is a constant that does not change no matter where you are, how much
you weigh, or how fast you are going. It says some things about flat
Euclidean space, and to the extent your observations of the space around
do not match, pi helps you figure out the nature of the curvature.
Here is an example: suppose you are at the south pole of the earth and
you have 100 m of steel wire. You can put one end of the wire at the
pole and stretch it out so as to walk a circular path around the pole at
a constant distance of 100 m. If you divide the distance you walked by
200 m, you will get a number very close to pi. Next, get a boat and
travel around antarctica at a constant distance of about 3000 km. If
you divide the distance you go around by 6000 km, you will find the
result is very much lower than pi. If you keep going north to the
equator and try this, the ratio of circumference to diameter will be
exactly 2, and if you keep going to within a couple meters of the north
pole, this ratio get very close to zero. So, pi did not change, but the
ratio of the circumference to diameter of a path through the locus of
points equidistant to a fixed point on a spherical shell is dependent on
the length chosen for the radius, although the curvature of the space is
constant.
More of the same: scratch a triangle in the dirt and measure the sum of
the angles, and you will get something close to 180 degrees. But if you
start at the north pole and go to the equator, then turn right 90
degrees and go 1/4 the way around, and then turn right again 90 degrees,
you will arrive back at the north pole on a line 90 degrees from start.
So in this case a triangle has three 90 degree angles that sum to 270
degrees. Again, the sum of the angles of a triangle in a space with
spherical curvature depends on its size in relation to the radius of
space curvature.
Don't mess with pi, you will get 2300 years of Greeks and Geeks all over
you.
-Ken
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