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.