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.

Shapes drawn inside and around a circle act as lower and upper bounds for pi — one flat outline sits just within the curve, the other just outside it. As you test six different pairs and track how the error ratio changes, a clear pattern emerges: the gap between the two estimates shrinks in a predictable way. It approaches the square of the inverse of the side-count ratio, meaning each doubling of sides yields a reliably smaller error.

Some shapes can tile a flat surface without gaps — hexagons, squares, and triangles all do this. What sets hexagons apart is efficiency: using algebra and geometry, you can prove that hexagons use less boundary material to enclose the same area than squares or triangles do. Bees exploit exactly this property, building hexagonal honeycombs rather than square or triangular ones to minimize the wax they need.