Miraculously, you can estimate π just by counting the number of iterations for a point near the neck to escape. The method has a rigorous error bound.
Python, 13 lines
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Copy to clipboard1 2 3 4 5 6 7 8 9 10 11 12 13 | def pi(err, abs=abs):
'Approximation of pi +/- err using the Mandlebrot set'
n = 0
z = c = complex(-0.75, err)
while abs(z) < 2.0:
n += 1
z = z * z + c
return n * err
if __name__ == '__main__':
for i in range(1, 8):
err_bound = 10 ** (-i)
print(pi(err_bound), 'within', err_bound)
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See proof at:
