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μ-seminar

David Widmann
david.widmann@it.uu.se
Uppsala University
16 June 2022
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π-seminar

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Why?

Well, I’m not sure. Is there really a reason needed? In any case, there are some possible explanations:
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  • Had no other idea 🤔
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  • Not yet another seminar about calibration or Julia 🥱
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  • Something “fun” before the summer 🌴
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  • Good opportunity to try Mathematica presentations 💻
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  • At least one μ-seminar with pi(e)
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    Many mathematical formulas contain the term 2π, e.g.,
    - Gaussian distribution
    - Fourier transformation
    - Cauchy integral formula
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    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.
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    There is at least one publication that uses τ, and many programming languages (e.g., Python and Julia) support it.
    Notπbut2πisspecial:
    Taumanifesto
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    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
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    π is irrational

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

    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.
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    Estimation

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    Ramanujan’s formulae

    Ramanujan was an Indian mathematician, who lived from 1887 to 1920 (died at the age of 32).
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    Had almost no formal training in pure mathematics and developed his own research in isolation.
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    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.
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    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.
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    There are multiple movies portraying his life (a recent one is The Man Who Knew Infinity from 2015), plays, and books.
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    Rate of convergence

    Convergence is extremely fast.
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    In 1985, William Gosper used this formula to calculate the first 17 million digits of π.
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    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.
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    This (and similar formulas) have been used in projects (e.g. PiHex) for calculating many digits of π using distributed computing.
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    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.
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    Similar formulas have been found for many other irrational numbers as well.

    Bailey-Borwein-Plouffe formula (1995)

    Plouffe published a formula that can be used for computing the nth decimal digit in January 2022.
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    100 trillion digits of pi

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    #ThrowbackThursday: Borwein integral

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    Take home message

    Have a great summer and celebrate τ day!