Comment by arethuza
14 years ago
I wouldn't have said Pi was a physical constant - it's not like we actually measure circles to derive it empirically.
14 years ago
I wouldn't have said Pi was a physical constant - it's not like we actually measure circles to derive it empirically.
True, but it certainly is a dimensionless number that you can't get rid of (and that actually shows up in physical formulae, such as http://en.wikipedia.org/wiki/Coulomb%27s_law). Come to think of it, so is 2.
I guess if you tried to measure π by constructing circles, you'd actually be measuring the curvature tensor of space (which is an experimental observation), not π (which is just π).
π being, simply, the measurement of the curvature tensor of an ideal Euclidean plane. If we lived in a universe that was non-Euclidean at macro-scale, π would be just another irrational number, and some other quantity would be exalted as "fundamental."
Pi would still be fundamental in mathematics whatever the geometry of our universe. (Examples of where it would turn up: consider the differential equation f''=-f; all its solutions are periodic with period 2pi. The series 1-1/3+1/5-1/7... has sum pi/4. The series 1+1/4+1/9+1/16+... has sum pi^2/6. exp(pi sqrt(163)) is ridiculously close to being an integer. There are deep reasons for all these things, and they wouldn't go away if the universe were very far from spatially flat.)