Pi is the number you get when you divide any circle's edge length by its width. One way to close in on that ratio is to wrap polygons tighter and tighter around a circle. You start with a square, then double the sides to 8, then 16, and so on. Each polygon's perimeter gets closer to the circle's circumference, producing a value that approaches pi from above.

A polygon with more sides hugs a circle more tightly — each added straight segment brings the flat outline closer to a curve. When you wrap a square around a circle of radius 1 and double its sides to 8, then 16, the perimeter shrinks toward the circle's circumference. That shrinking value approaches pi from above, producing a recursive equation that matches a famous expression published by François Viète centuries ago.

The circumference of a circle can be approached step by step using polygons. In this project, you draw regular polygons around a circle of radius 1 and keep doubling the number of sides. Each polygon's perimeter gets closer to the circle's circumference, showing that the distance around a circle is a fixed value that polygons can approximate but never quite match.

Recursive sequences are patterns where each new number is built from the ones before it. This experiment puts that idea to work: you start with a square drawn around a circle of radius 1, then double the number of sides to 8, then 16, and so on. Each polygon's perimeter feeds the next calculation, with every step creeping closer to pi from above. As the sides multiply, the upper-bound expression converges — and it turns out to match a famous formula published by François Viète centuries ago. Finally, you derive an algebraic polynomial whose roots include pi itself, along with new numbers called the Pi Associates.

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.

Drawing shapes around a circle gives an upper estimate of its size. You start with a square drawn around a circle of radius 1, then double the number of sides to 8, then 16, and so on. As each polygon gains more sides, its perimeter drops closer to the true circumference. This process produces a recursive equation that approaches pi from above — and it turns out to match a famous upper-bound expression published by Francois Viete centuries ago.