Reply-to: rwg@osots.com
I'm going to cut a circular cookie into thirds (3 equal areas) by making two vertical cuts. Where should they be made?
Of course, relaxing the "vertical" (or parallel) restriction buys nothing. In light of the seemingly contradictory identities cos(a) / [ 2 %pi I 2 sqrt(1 - x ) dx = --- - 2 a, ] 2 / sin(a) a >= 0 cos(a) / [ 2 %pi I 2 sqrt(1 - x ) dx = sin(2 a) + ---, ] 2 / - sin(a) we have in closed form the chords delimiting an area of pi/3 or anything we want. Unfortunately, the leftover "heel" segments have nondescript areas. Note that for area 2 pi/3, we need the peculiar quantity a = (asin pi/6)/2. Some transcendental equations at least have nice series solutions, e.g. y e^y = x -> inf ==== i - 1 i i + 1 \ (i + 1) (- 1) x y = > -------------------------- . / i! ==== i = 0 Can we similarly trisect a cookie with an infinite sum of a closed form? Or is this why Triscuits were square? --rwg