Don Herbison-Evans ,
donherbisonevans@yahoo.com

26 August 2014

Mathematicians calculate the theoretical value of pi as 3.141592653... etc.

Has anyone actually measured it accurately? To measure pi, one needs to measure the length along the circumference of a circle, and also its diameter, and then divide one by the other.

If one thought that the earth was flat and happened to use a circle drawn around the equator for the measurement, one would however discover that pi was close to 2.0. This measured value of pi informs us we are on a curved surface.

The General Theory of Relativity suggests that the mass of the earth is creating a three dimensional curvature in space in its vicinity which we feel as gravity. Maybe we could measure this curvature by measuring pi.

One can draw an arbitrary circle and measure its diameter as maximum distance between an arbitrary point on the circle and all the other points on the circumference. But measuring the length along the circumference is difficult.

A constructive solution is to measure the length around the circumference of a circular cylinder by rolling it along a ruler till one got back to the starting point. A cylinder can be made accurately round by creating it on a lathe. Its diameter can be measured as the distance between two flat plates touching opposite sides of the cylinder, and aligned to be accurately parallel.

So if we made a cylinder 1 meter long say with a diameter of 1 metre. and used it to measure our local value of pi, what number would we get?

Would we get different values for pi if the axis of the cylinder was vertical compared with if the axis was horizontal? Would we get different values if the axis was nort-south from that with its axis east-west?

The angle that circular cross-section of the cylinder
would subtend at the centre of the earth
(say approx 10^{7} metres away) would be about 10^{-7} radians
(ignoring minor factors of 2 and pi etc),
so if space were curved and spherical, and say centred at the centre of the earth,
our measured value of pi might differ from its theoretical value
by only about 1 part in 10^{14}.

The cylinder would need to be made of a material that is very hard
and resistant to scratching and deformation.
Do you think de Beer's would make a 1 metre diameter diamond cylinder for me?
Could we measure its circumference and diameter to accuracies of 10^{-14}?