Wrapping polygons around a circle reveals geometric convergence in action. You start with a square drawn around a circle of radius 1, then double the sides to 8, then 16, and beyond. As the side count grows, each polygon's perimeter closes in on the circle's circumference. This process produces a recursive equation that approaches pi from above — and it turns out to match a famous upper-bound formula published by François Viète centuries ago. From there, you derive an algebraic polynomial where one root is pi itself, with the remaining roots being new numbers called the Pi Associates, whose properties remain unexplored.

Ancient mathematicians estimated pi by drawing polygons inside and around circles — and the more sides a polygon has, the closer the estimate gets. You calculate pi by averaging the perimeters of polygons inscribed in and circumscribed about a circle, then track how the error changes as the side count grows. Testing six different pairs of polygons, a clear pattern emerges: the error ratio approaches the square of the inverse of the side-count ratio. That means the error shrinks in a predictable way as you add more sides, giving you a precise measure of how fast the shape converges toward a perfect circle.