Accelerating Archimedes
One way to approximate is to find the areas of polygons inscribed inside a circle and polygons circumscribed outside the circle. The approximation improves as the number of sides in the polygons increases. This idea goes back at least as far as Archimedes (287-212 BC). Maybe you've tried this. It's a lot of work.
In 1667 James Gregory came up with a more efficient way to carry out Archimedes' approach. Tom Edgar and David Richeson cite Gregory's theorem in [1] as follows.
Let Ik and Ck denote the areas of regular k-gons inscribed in and circumscribed around a given circle. Then I2n is the geometric mean of In and Ck, and C2n is the harmonic mean of I2n and Cn; that is,
We can start with n = 4. Obviously the square circumscribed around the unit circle has vertices at (1, 1) and area 4. A little thought finds that the inscribed square has vertices at (2/2, 2/2) and area 2.
The following Python script finds to six decimal places.
n = 4I, C = 2, 4 # inscribed area, circumscribed areadelta = 2while delta > 1e-6: I = (I*C)**0.5 C = 2*C*I / (C + I) delta = C - I n *= 2 print(n, I, C, delta)
The script stops when n = 8192, meaning that the difference between the areas of an inscribed and circumscribed 8192-gon is less than 10-6. The final output of the script is.
8192 3.1415923 3.1415928 4.620295e-07
If we average the upper and lower bound we get one more decimal place of accuracy.
Gregory's algorithm isn't fast, though it's much faster than carrying out Archimedes' approach by computing each area from scratch.
Gregory gives an interesting recursion. It has an asymmetry that surprised me: you update I, then C. You don't update them simultaneously as you do when you're computing the arithmetic-geometric mean.
Related posts[1] Tom Edgar and David Richeson. A Visual Proof of Gregory's Theorem. Mathematics Magazine, December 2019, Vol. 92, No. 5 (December 2019), pp. 384-386
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