# μ-seminar

μ-seminar

David Widmann

david.widmann@it.uu.se

david.widmann@it.uu.se

Uppsala University

16 June 2022

## π-seminar

π-seminar

## Why?

Why?

Well, I’m not sure. Is there really a reason needed? In any case, there are some possible explanations:

Had no other idea 🤔

Not yet another seminar about calibration or Julia 🥱

Something “fun” before the summer 🌴

Good opportunity to try Mathematica presentations 💻

At least one μ-seminar with pi(e)

Many mathematical formulas contain the term 2π, e.g.,

- Gaussian distribution

- Fourier transformation

- Cauchy integral formula

Therefore many people such as Bob Palais argue that “π is wrong” and one should use a special constant for 2π instead.

- Gaussian distribution

- Fourier transformation

- Cauchy integral formula

Therefore many people such as Bob Palais argue that “π is wrong” and one should use a special constant for 2π instead.

π is wrong! Robert Palais

Physicist Michael Hartl suggested to call it τ in the τ manifesto in 2010.

There is at least one publication that uses τ, and many programming languages (e.g., Python and Julia) support it.

There is at least one publication that uses τ, and many programming languages (e.g., Python and Julia) support it.

## Pilish

Pilish

But a time I spent wandering in gloomy night;Yon tower, tinkling chimewise, loftily opportune.Out, up, and together came sudden to Sunday rite,The one solemnly off to correct plenilune. Joseph Shipley

More information (e.g., about different dialects): http://www.cadaeic.net/pilish.htm

## π is irrational

π is irrational

Unfortunately, there are infinitely many non-repeating digits of π .

### Proof due to Ivan Niven (1947)

Proof due to Ivan Niven (1947)

and

Hence the integral is a positive integer as well. However, for 0 < x < π we have

That is a contradiction.

## Estimation

Estimation

## Ramanujan’s formulae

Ramanujan’s formulae

Ramanujan was an Indian mathematician, who lived from 1887 to 1920 (died at the age of 32).

Had almost no formal training in pure mathematics and developed his own research in isolation.

His work included solutions to mathematical problems considered unsolvable.

G.H. Hardy (University of Cambridge) recognised his extraordinary work and arranged for him to travel to Cambridge.

Became one of the youngest Fellows of the Royal Society, only second Indian member, and first Indian to be elected a Fellow of Trinity College, Cambridge.

There are multiple movies portraying his life (a recent one is The Man Who Knew Infinity from 2015), plays, and books.

Had almost no formal training in pure mathematics and developed his own research in isolation.

His work included solutions to mathematical problems considered unsolvable.

G.H. Hardy (University of Cambridge) recognised his extraordinary work and arranged for him to travel to Cambridge.

Became one of the youngest Fellows of the Royal Society, only second Indian member, and first Indian to be elected a Fellow of Trinity College, Cambridge.

There are multiple movies portraying his life (a recent one is The Man Who Knew Infinity from 2015), plays, and books.

## Rate of convergence

Rate of convergence

Convergence is extremely fast.

In 1985, William Gosper used this formula to calculate the first 17 million digits of π.

In 1985, William Gosper used this formula to calculate the first 17 million digits of π.

## Computing the nth digit

Computing the nth digit

The BBP formula can be used to construct an algorithm to compute nth base-16 (hexadecimal) digit of π without computing the preceding digits.

This (and similar formulas) have been used in projects (e.g. PiHex) for calculating many digits of π using distributed computing.

Formula was a surprise, before it had been widely believed that computing the nth digit is just as hard as computing the first n digits. It was discovered by Plouffe, and he published it together with Bailey and Borwein.

Similar formulas have been found for many other irrational numbers as well.

This (and similar formulas) have been used in projects (e.g. PiHex) for calculating many digits of π using distributed computing.

Formula was a surprise, before it had been widely believed that computing the nth digit is just as hard as computing the first n digits. It was discovered by Plouffe, and he published it together with Bailey and Borwein.

Similar formulas have been found for many other irrational numbers as well.

### Bailey-Borwein-Plouffe formula (1995)

Bailey-Borwein-Plouffe formula (1995)

Plouffe published a formula that can be used for computing the nth decimal digit in January 2022.

## 100 trillion digits of pi

100 trillion digits of pi

## #ThrowbackThursday: Borwein integral

#ThrowbackThursday: Borwein integral

## Take home message

Take home message

Have a great summer and celebrate τ day!