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Archimedes' Polygon Cage Method
- Archimedes trapped the circle between inscribed and circumscribed polygons to bound π and refined those bounds up to 96-sided polygons.
- That exhaustion method let him prove π lies between 3 1/7 and 3 10/71 without algebra or trigonometry, changing how numbers were rigorously approached.
Archimedes' Sliced Sphere Thought Experiment
- Archimedes imagined slicing a sphere into infinitely many thin slices and balanced them conceptually to relate sphere and cylinder areas and volumes.
- A rediscovered Vatican manuscript preserves his non-rigorous mental 'balance' argument and the diagram he famed on his tomb.
Madhava's Infinite Series For Pi
- Madhava introduced infinite series for π centuries before Europe, giving π as 4 times (1 - 1/3 + 1/5 - 1/7 + ...).
- The series shows π as a numerical infinite sum so truncating it yields ever-improving approximations but never exactness.