Circumscribe a triangle about a circle, another circle around the triangle, a square outside the circle, another circle outside the square, and so on. The circumradius and inradius for an n-gon are then related by r = R cos(π/n), so an infinitely nested set of circumscribed polygons and circles has K | congruent | r_(final circle)/r_(initial circle) | = | sec(π/3) sec(π/4) sec(π/5)... | = | product_(n = 3)^∞ sec(π/n).

infinite product | nested polygon | polygon inscribing | whirl