It's continued Pi Day - again!

Oh. My. Gawd - it’s Pi Day, again! I’ve written a few times about Pi already. No idea why this number fascinates me so much. I even made a blog post exactly 10 years ago. Unfortunately, back then I used Silverlight. Had I just used Javascript, I would still have running code.

I have also written about continued fractions once before. They are a fascinating construct which can be used to build interesting numbers. One interesting number is…Pi. You can see on wikipedia some examples for Pi expansions in the form of continued fractions.

x=3+126+326+526+726+92x = 3 + \cfrac{1^2}{6 + \cfrac{3^2}{6 + \cfrac{5^2}{6 + \cfrac{7^2}{6 + \cfrac{9^2}{\ddots}} } } }
Continued fraction of Pi as taken from Wikipedia

If we take one of those and slap it into js, we get sth like this:

function pi(steps) {
    const result = 3 + (1 / fraction(3, steps));
    return result;

function fraction(numerator, remainingSteps) {
    if (remainingSteps === 0)
      return 6;
    return 6 + ((numerator * numerator) / 
               fraction(numerator + 2, remainingSteps - 1));

The steps is the input as to how far you want to go down the rabbit hole on the continued fraction.

Let’s give it a shot:

> pi(40)

> pi(400)

> pi(4000)

> pi(8631)

8631 was then incidentally the highest recursion depth before node on the cli would tell me… RangeError: Maximum call stack size exceeded. Checking against the number Pi as found on the internet I see that the first 12 digits after the decimal point are correct. Good enough for most calculations, I assume.

Happy Pi day everyone!