No, really, pi is wrong: The Tau Manifesto by Michael Hartl
I continue to be impressed with how rich this subject is, and my understanding of π and τ continues to evolve. On Half Tau Day, 2012, I believed I identified exactly what is wrong with π. My argument hinged on an analysis of the surface area and volume of an n-dimensional sphere, which (as shown below) makes clear that π doesn’t have any fundamental geometric significance. My analysis was incomplete, though—a fact brought to my attention in a remarkable message from Tau Manifesto reader Jeff Cornell. As a result, this section is an attempt not only to definitively debunk π, but also to articulate the truth about τ, a truth that is deeper and subtler than I had imagined. Note: This section is more advanced than the rest of the manifesto and can be skipped without loss of continuity. 5.1 Surface area and volume of a hypersphere We start our investigations with the generalization of a circle to arbitrary dimensions. which consists of the two points ±r. is the line segment from −r to r. n!!