π = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631...

## What do we know about π?

- It has to do with circles - something something 2πr
- It tastes delicious
- It's infinite
- It goes on forever
- It has infinitely many decimal digits
- It's random
- There's no pattern to its digits
- It's irrational
- It's transcendental

## Measure a pendulum's swing

\[ \pi = \frac{T}{2} \sqrt{\frac{L}{g}} \]

## Embrace the noise

\[ x^2 + y^2 \leq 1 \]

π ≈ 4 × 0 / 0 ≈ ?

### Draw a polygon which looks like a circle

\[ P_{2n} = \frac{2p_nP_n}{p_n+P_n}, \\[1.5em] p_{2n} = \sqrt{p_nP_{2n}} \]

## Give up and legislate it

"the ratio of the diameter and circumference is as five-fourths to four."

## Machin's formula

\[ \frac{\pi}{4} = 4 \arctan \left(\frac{1}{5}\right) - \arctan \left(\frac{1}{239}\right) \]

Rewrite as an infinite series

\[ \pi = 4 \sum_{n=0}^{\infty} \frac{(-1)^n)}{2n+1} \left( 4 \left(\frac{1}{5}\right)^{2n+1} - \left(\frac{1}{239}\right)^{2n+1} \right) \]

## William Shanks, Local Hero

William Shanks spent over 20 years calculating π to 707 decimal places.

But he made an error at the 528th digit.

## William Shanks, Local Hero

William Shanks spent over 20 years calculating π to 707 decimal places.

But he made an error at the 528th digit.

## William Shanks, Local Hero

William Shanks spent over 20 years calculating π to 707 decimal places.

But he made an error at the 528th digit.

## Use a machine

D.F. Ferguson spotted Shanks's error and broke his record with a mechanical desk calculator.

Since then, computers have gone on to compute a stupid number of decimal places of π.

(Image by Wikipedia user Nageh)

# Spigot algorithms

(A bit like this bear, but for digits of π)

## Spigot algorithms

Iterative algorithms which produce digits one at a time, and never reuse a digit in a later step.

And they only use a constant amount of working memory.

## The Bailey-Borwein-Plouffe formula

\[ \pi = \sum_{k=0}^{\infty} \frac{1}{16^k}\left( \frac{4}{8k+1} - \frac{2}{8k+4} - \frac{1}{8k+5} - \frac{1}{8k+6} \right) \]

You can extract *any* digit of π without working out any of the previous ones.