I guess I'll mention something interesting I read in
Elegant Universe by
Brian Greene when he was covering
relativity. As long as you're sticking to two
dimensions,
pi is
constant. However, in the real world,
pi can
appear to be
variable. The trick is that space compresses in the direction of
momentum.
Greene's example used two people measuring
circumference and
diameter of that spinning room at the
amusement park. Both guys will get the same diameter, but because the room is moving
relative to the guy outside and space compresses in the direction of the motion, the circumference measured by the person outside the ride will appear to be smaller than the circumference measured by the person inside the ride. If they both calculate pi (circumference over diameter) using their measurements, the guy who was inside the room will get a larger value of pi than the other.
As an example you can actually try at home, draw a circle on a
deflated balloon. If an object is rotating, the center isn't moving but the outside is so when space-time bends you get a spherical shape much like the balloon when it is
inflated. If you measure circumference and diameter of the balloon when it's deflated, and then again when it's inflated, you'll see that pi will appear to be smaller when the balloon is inflated.
But
no worries, pi really is constant. It's caused by measuring something with equipment that can't deal with all the dimensions involved.