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User:Waldyrious/Tau

< User:Waldyrious
For the mainspace article, see Tau (mathematical constant), which redirects to Turn (angle)#Tau proposals.

Historical usage of 2π as a constant

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  • 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 π.
  • William Oughtred used ⁠π/δ⁠ to represent ⁠perimeter/diameter⁠.
  • David Gregory used ⁠π/ρ⁠ to represent ⁠perimeter/radius⁠.
  • William Jones first used π 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 ⁠2π/4⁠ to ⁠π/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.

Mathematical publications about τ

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  • Palais, Robert (2001). "π Is Wrong!" (PDF). The Mathematical Intelligencer. 23 (3): 7–8. doi:10.1007/BF03026846.
  • Abbott, Stephen (April 2012). "My Conversion to Tauism" (PDF). Math Horizons. 19 (4): 34. doi:10.4169/mathhorizons.19.4.34.

Notable endorsements

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People
Organizations
Published mathematicians

Celebration of 2π day before Hartl's manifesto (2010)

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Support in tools and programming languages

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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.

Included by default

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Language Name Value
.NET Framework
(issue #24678, PR #37517, docs; released in v5.0, 10 Nov 2020) 		Tau 6.28318 53071 79586 476925
Java
(issue, docs, value; released in v19, 20 Sep 2022) 		TAU 6.28318 53071 79586
Python
(PEP, issue, commit, docs; released in v3.6, 23 Dec 2016) 		tau 6.28318 53071 79586 47692 52867 66559 00576 83943
Rust
(docs, issue, PR, tweet; released in 1.47, 8 Oct 2020) 		TAU 6.28318 53071 79586 47692 52867 66559 00577
Modula-2
(source, commit 760ac3b from 18 Dec 2013) 		tau 6.28318 53071 79586 47692 52867 66559 0
Nim
(source, commit, docs; released in v0.14, 7 Jun 2016) 		TAU 2 * PI
Processing
(docs, issue 1, issue 2, changelog; released in v2.0, 3 Jun 2013) 		TAU 6.28318 53071 79586 47693
Zig
(source, commit; released in v0.6, 13 Apr 2020) 		tau 2 * pi

Available as a non-default, third-party module

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Tool Name Value
JavaScript: Math.js tau 6.28318 53071 79586 (Math.PI*2)
C++: Boost[1] tau two_pi (alias to existing constant, which itself is set to: 6.28318 53071 79586 47692 52867 66559 00576 8e+00)
Haskell: tau module[2] τ / tau 6.28318 53071 79586 (2*pi)
Julia: Tau.jl package[3] τ / tau 6.28318 53071 79586 47692
Ruby: math-tau gem[4] TAU 6.28318 53071 79586 (PI * 2.0)

Explicit two pi constant (with no tau alias)

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Tool/Language Name Value
Fortran TWOPI
OGRE TWO_PI
OpenGUI TWO_PI
Pascal TwoPI 6.28318 53071 79586
Wiring TWO_PI 6.28318 53071 79586 47693
Extreme Optimization Libraries TwoPi 6.28318 53071 79586 47692 52867 66558

Textbooks

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

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Tau conversion hubs

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Neat stuff

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  • Circle is formally defined as all points at same distance —radius, not diameter— of a center point.
  • Tau day is a perfect day, because 6 and 28 are the two first perfect numbers.[5][6]
    • 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 τ: starts earlier (761 digits after the radix mark[7] rather than 762 in π), is longer (7 nines[8] rather than 6 nines in π), and thus more improbable (0.008% vs. 0.08% in π[9])
  • Decimal expansion of 2*Pi and related links at the On-Line Encyclopedia of Integer Sequences
  • "You can't eat pie on Tau Day!"
    1. First of all, the pun is not that strong of an argument: it only works because π 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.
    2. 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." [1]
  • 3Blue1Brown's "Euler's formula with introductory group theory" shows the significance of   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   special is that when the exponential   maps vertical slides to rotations, a vertical slide of one unit, corresponding to  , 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   units up, corresponding to the input   maps to a rotation of exactly   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.

References

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  1. ^ PR #566; announcement tweet. Apparently it was not included in release v1.76 (the version of the math library in that tag doesn't include the tau lines in include/boost/math/constants/constants.hpp). It is present in the 1.77 version though.
  2. ^ In Haskell, pi is the only constant defined by default in the standard prelude; pretty much everything else, including e, isn't; like tau, they require extra modules, e.g. science-constants (tau is not included in this one, btw).
  3. ^ See also PR #4864 (rejected).
  4. ^ See also issue #4897 (rejected), and associated patch.
  5. ^ Marcus du Sautoy (1 July 2009). "Perfect Numbers". The Times. Archived from the original on 2011-08-12. Retrieved 2011-08-12.
  6. ^ Dave Richeson (1 July 2009). "Last Sunday was a perfect day". Division by Zero. Retrieved 2011-07-24.
  7. ^ Michael Hartl. "100,000 digits of τ". Retrieved 6 July 2011.. See also a file with 1 billion digits available here.
  8. ^ Harremoes
  9. ^ Arndt, J.; Haenel, C. (2001), Pi — Unleashed, Berlin: Springer, p. 3, ISBN 3540665722 {{citation}}: Unknown parameter |lastauthoramp= ignored (|name-list-style= suggested) (help).

TODO

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See also

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