Π day

[robve]

(03-14-2022 01:55 PM)EdS2 Wrote:  There's an unusual one by Valentin... 3 years ago, which seems appropriate.

Here is the unusual pi computation in Haskell, which self-applies \( f\,(h,p)=(p\lceil\frac{h}{p}\rceil,p-1) \) iteratively \( n-1 \) times starting with \( (1,n) \) to return \( \frac{n^2}{h} \) as an approximation of \( \pi \):

unusual_pi n =
  let (h,p) = head (drop (n-2) (iterate (\(h,p) -> (p*ceiling (fromIntegral h/fromIntegral p), p-1)) (1,n)))
  in (fromIntegral (n*n))/(fromIntegral h)
unusual_pi 100000
3.141526183050601

The reason why this works is explained here: "Beginning with any positive integer \( n \), round up to the nearest multiple of \( n–1 \), then up to the nearest multiple of \( n–2 \), and so on, up to the nearest multiple of 1. Let \( f(n) \) denote the result. For example, \( f(10) = 34 \). Interestingly, the ratio \( n^2/f(n) \) approaches \( \pi \) (i.e., 3.14159...) as \( n \) increases."

(03-14-2022 01:55 PM)EdS2 Wrote:  It feels worthwhile also to showcase one of Gerson's [but see downthread!] pandigital approximations ("good to 17 digits"):
That is: (ln{[2×5!+(8-1)!]^(sqrt(9))+4!+(3!)!})/(sqrt(67))

Yes, very interesting and impressive work by Gerson.

- Rob