@Sandra I found a paper that gives a different explanation of how he arrived at it, using continued fractions. It is surprisingly simple.
- bound pi between 3 and 3.5
- using the continued fraction (7+3x)/(2+x)=pi, solve for x and substitute an estimate for pi. Take the closest integer to x, and do this iteratively.

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/41934/1/1257_14.pdf

@Sandra I had a closer look at that paper and I think it is not sound. The only reason why the presented calculation gives a close fraction for pi is because of the provided approximations for pi (3.14 and 3.1416). With any other value you simply get a different fraction.
So what it does is only explain how, given the approximations 22/7 and 3.1416, you can arrive at 355/113. The value of 3.1416 is introduced through handwaving.
So the only point to take from this paper is that Zu Chongzhi might have used continued fractions purely as a calculation technique. It's not an algorithm to approximate pi. So it's back to the 12288-gon (this paper mentions a 24576-gon but no explanation for that either).
@csepp

@Sandra @csepp
Raku implementation:

sub approx_pi_h (Rat \pi_lb, Rat \pi_hb, Rat \pi_ --> Rat ) {
my Real \x =
(pi_lb.numerator - pi_*pi_lb.denominator) /
(pi_*pi_hb.denominator - pi_hb.numerator);
my Int \m = floor(x);
(pi_hb_t*m + pi_lb_t) / (pi_hb_n*m + pi_lb_n);
}

#rakulang