User:Waldyrious/Tau


 * For the mainspace article, see Tau (mathematical constant), which redirects to Turn (angle).

Historical usage of 2&pi; as a constant

 * Islamic mathematicians like Jamshīd al-Kāshī (c. 1380–1429) focused on the circle constant 6.283... although they were fully aware of the work of Archimedes focusing on the circle constant that is nowadays called &pi;.
 * William Oughtred used $\pi⁄$δ$$ to represent $perimeter⁄diameter$.
 * David Gregory used $\pi⁄$ρ$$ to represent $perimeter⁄radius$.
 * William Jones first used $\pi$ as it is used today to represent $perimeter⁄diameter$ (Synopsis palmariorum matheseos (London, 1706), p.263.)
 * Leonhard Euler adopted the same definition as William Jones, which helped popularized it into the standard it is today.
 * Paul Matthieu Hermann Laurent, though never explaining why, treated 2π as if it were a single symbol in Traité D'Algebra by consistently not simplifying expressions like $2\pi⁄4$ to $\pi⁄2$.
 * Fred Hoyle, in Astronomy, A history of man's investigation of the universe, proposed using centiturns (hundredths of a turn) and milliturns (thousandths of a turn) as units for angles.

Notable endorsements

 * People
 * Sal Khan of Khan Academy: Tau versus Pi (2011) and Happy Tau Day! (2012)
 * Vi Hart: Pi Is (still) Wrong (2011) and A Song About A Circle Constant (2012)
 * Robert Dixon: Pi ain't all that
 * Michael Cavers of Spiked Math (and The Pi Manifesto): Math Fact Tuesday: Tau
 * Randall Munroe of xkcd: Comic #1292: Pi vs. Tau (18 November 2013; not really an endorsement, but an interesting acknowledgement/remark)
 * Zach Weiner of Saturday Morning Breakfast Cereal: Comic #3134 (4 October 2013)
 * Eric S. Raymond: Tau versus Pi
 * James Grime of Numberphile: Meet James Grime
 * Organizations
 * MITadmissions.org: I have SMASHING news! (2012)
 * University of Oxford: Tau vs Pi: Fixing a 250-year-old Mistake (2013)


 * Published mathematicians
 * Stanley Max: Radian Measurement: What It Is, and How to Calculate It More Easily Using τ Instead of π
 * Kevin Houston: Pi is wrong! Here comes Tau Day

Celebration of 2&pi; day before Hartl's manifesto (2010)

 * 1998:
 * 2009:
 * 2009:
 * 2009:

Support in tools and programming languages

 * See also: https://github.com/nschloe/tau#in-programming 
 * Note: although not a programming language, it's worth noting that tau is available in Google calculator.

Textbooks

 * T. Colignatus. Trigonometry reconsidered. Measuring angles in unit meter around and using the unit radius functions Xur and Yur. T. Colignatus, 2008.
 * T. Colignatus. Conquest of the Plane. Using The Economics Pack Applications of Mathematica for a didactic primer on Analytic Geometry and Calculus. Consultancy & Econometrics, March 2011. ISBN 978-90-804774-6-9.
 * John M. Lee. Axiomatic Geometry. American Mathematical Society, Apr 10, 2013

News (not published around pi day or tau day, or otherwise significant)

 * (they actually contacted Palais, and took a picture of him wearing a Tau Day T-shirt)
 * (they seem to have actually interviewed Hartl)
 * (interviewed Hartl, and wasn't published around Tau day)
 * (they reached out to Palais via email)

Tau conversion hubs

 * The Tau Manifesto by Michael Hartl (since 28 June 2010)
 * Al-Kashi’s constant τ by Peter Harremöes (since... 2010? confirm)
 * Pi is Wrong! by Robert Palais (since September 2001)
 * Tau Before It Was Cool by Joseph Lindenberg (since... 2010? confirm)

Neat stuff

 * Circle is formally defined as all points at same distance —radius, not diameter— of a center point.
 * Main conventions regarding circles are based in radius: unit circle, radians, standard circle formulas.
 * Tau day is a perfect day, because 6 and 28 are the two first perfect numbers.
 * 6:28 is a more convenient time to start celebrating than 3:15 (besides being after the actual start of the day rather than midnight)
 * Feynman point better in &tau;: starts earlier (761 digits after the radix mark rather than 762 in &pi;), is longer (7 nines rather than 6 nines in &pi;), and thus more improbable (0.008% vs. 0.08% in &pi; )
 * Decimal expansion of 2*Pi and related links at the On-Line Encyclopedia of Integer Sequences
 * "You can't eat pie on Tau Day!"
 * First of all, the pun is not that strong of an argument: it only works because &pi; is mispronounced "pie" in English, rather than "pea" as in the original Greek and most other languages. Even if people decided to eat peas instead, the pun would still only work for English speakers, which doesn't play well with the universality of a mathematical constant.
 * Second, pi radians is half a circle, not a full circle as most pies are, which weakens the association. If this inconvenience is ignored, then this ends up actually backfiring into favoring Tau, since on Tau day you can eat two pies!
 * An intriguing comment by Terence Tao: "It may be that 2*pi*i is an even more fundamental constant than 2*pi or pi. It is, after all, the generator of log(1). The fact that so many formulae involving pi^n depend on the parity of n is another clue in this regard."
 * 3Blue1Brown's "Euler's formula with introductory group theory" shows the significance of $$e^{i\pi}=-1$$ as highlighting the equivalence between multiplicative actions (rotations) and additive actions (translations) in the complex plane.
 * It might be interesting to consider what this means for the Tau Manifesto's arguments related to this equation.
 * Furthermore, from 20:08 onwards: "what makes the number $$e$$ special is that when the exponential $$e^x$$ maps vertical slides to rotations, a vertical slide of one unit, corresponding to $$i$$, maps to a rotation of exactly one radian — a walk around the unit circle covering a distance of exactly one. (...) and a vertical slide of exactly $$\pi$$ units up, corresponding to the input $$\pi * i$$ maps to a rotation of exactly $$\pi$$ radians, half way around the circle; and that's the multiplicative action associated with the number negative one."
 * This seems to be a special case of Euler's rotation theorem, which states that any affine transformation (TODO: confirm) can be represented as a single rotation around a given "half-vector" (origin point + direction).
 * The tau symbol having one leg (compared to pi's two) may be interpreted as the diameter (horizontal stroke of the character) over the radius (vertical stroke), while pi is the diameter over twice the radius
 * There's a formal proof of tau = 2pi in Metamath here. It's surprisingly more extensive than I'd expect. I wonder if other formal math systems/libraries (e.g. Lean's Mathlib, Coq's Mathematical Components, etc.) could have something equivalent, and whether they would choose different approaches to prove the fact.

TODO

 * Gather more stuff from Harremoës' and Palais' pages.