The Quarter-Turn and the Ball

Abstract

In his Tau Manifesto, Michael Hartl presents a formula, credited to Jeff Cornell, for the volume of the unit $n$-ball in terms of the quarter-turn constant $\eta =\pi /2$. We read the formula as a set of clues to a geometric proof, leading through orthant geometry and a higher-dimensional version of Lambert's equal-area projection. We conclude with some additional thoughts on the circle constant.