Thoughts about Pi

Diameters might appear in drawings. But they NEVER appear in physics. (The distance between particles determines forces. Diameters are a boring parameter about the size of something.) Machine tool programming is mostly radii in the paths, and diameters in the setup parameters.

The statement about the diameter being the interesting parameter is kind of exactly wrong. This is true for all the same reasons Tau is a better constant than Pi.

For example, how does one derive the area of a circle? The most natural way is to integrate with respect to radius. When using Tau and radius, you get A = (1/2)Taur^2, which is nice since it looks a lot like the integral of, say, momentum, which gives you kinetic energy. If you use Pi or diameter or both, then weird factors of 2 start to creep in. It's also much more natural to think about defining the radial extent at each angle, rather than the diameter, especially if you're dealing with something other than a circle that has changing radius.

Yeah but the radius of the moon's orbit is sure interesting.

Your conjecture as to the surface area of a hypersphere embedded in 4 dimensions (a "3-sphere") is not borne out :) The answer is in fact d^3 pi^2 / 4. See https://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_ar...