Draft:Tau (Constant)

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In 2010, Michael Hartl proposed to use the Greek letter tau ( $τ$ ) to represent the circle constant for which: $π = .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}τ⁄2$ . He offered two reasons. First, $τ$ is the number of radians in one turn, which allows fractions of a turn to be expressed more directly: for instance, a 3/4 turn would be represented as $3 τ / 4$ rad instead of $3 π / 2$ rad. Second, $τ$ visually resembles $π$ , whose association with the circle constant is unavoidable.^[1] Hartl's Tau Manifesto^[2] gives many examples of formulas that are asserted to be clearer where $τ$ is used instead of $π$ ,^[3]^[4]^[5] such as a tighter association with the geometry of Euler's identity using $e iτ = 1$ instead of $e iπ = -1$ .

Initially, neither of these proposals received widespread acceptance by the mathematical and scientific communities.^[6] However, the use of $τ$ has become more widespread,^[7] for example:

In 2012, the educational website Khan Academy began accepting answers expressed in terms of $τ$ .^[8]
The constant $τ$ is made available in the Google calculator, Desmos graphing calculator^[9] and in several programming languages such as Python,^[10]^[11] Raku,^[12] Processing,^[13] Nim,^[14] Rust,^[15] Java,^[16]^[17] and .NET.^[18]^[19]
It has also been used in at least one mathematical research article,^[20] authored by the $τ$ -promoter Peter Harremoës.^[21]

The following table shows how various identities appear if $τ = 2 π$ was used instead of $π$ .^[22]^[23] For a more complete list, see List of formulae involving $π$ .


Formula	Using $π$	Using $τ$	Notes
Angle subtended by 1/4 of a circle	$π / 2$ rad	$τ / 4$ rad	$τ / 4 rad = 1 / 4 turn$
Circumference $C$ of a circle of radius $r$	$C = 2 π r$	$C = τ r$
Area of a circle	$A = π r 2$	$A = 1 / 2 τ r 2$	The area of a sector of angle $θ$ is $A = 1 / 2 θr 2$ .
Area of a regular $n$ -gon with unit circumradius	$A = n / 2 sin 2π / n$	$A = n / 2 sin τ / n$
$n$ -ball and $n$ -sphere volume recurrence relation	$V n (r) = r / n S n -1 (r)$ $S n (r) = 2 π r V n -1 (r)$	$V n (r) = r / n S n -1 (r)$ $S n (r) = τ r V n -1 (r)$	$V 0 (r) = 1$ $S 0 (r) = 2$
Cauchy's integral formula	$f(a)={\frac {1}{{\color {redorange}2\pi }i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$	$f(a)={\frac {1}{{\color {redorange}\tau }i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$
Standard normal distribution	$\varphi (x)={\frac {1}{\sqrt {\color {redorange}2\pi }}}e^{-{\frac {x^{2}}{2}}}$	$\varphi (x)={\frac {1}{\sqrt {\color {redorange}\tau }}}e^{-{\frac {x^{2}}{2}}}$
Stirling's approximation	$n!\sim {\sqrt {{\color {redorange}2\pi }n}}\left({\frac {n}{e}}\right)^{n}$	$n!\sim {\sqrt {{\color {redorange}\tau }n}}\left({\frac {n}{e}}\right)^{n}$
Euler's identity	0 $e i π = -1$ $e i π + 1 = 0$	0 $e i τ = 1$ $e i τ - 1 = 0$	For any integer $k$ , $e ikτ = 1$
$n$ th roots of unity	$e^{{\color {redorange}2\pi }i{\frac {k}{n}}}=\cos {\frac {{\color {redorange}2}k{\color {redorange}\pi }}{n}}+i\sin {\frac {{\color {redorange}2}k{\color {redorange}\pi }}{n}}$	$e^{{\color {redorange}\tau }i{\frac {k}{n}}}=\cos {\frac {k{\color {redorange}\tau }}{n}}+i\sin {\frac {k{\color {redorange}\tau }}{n}}$
Planck constant	$h={\color {redorange}2\pi }\hbar$	$h={\color {redorange}\tau }\hbar$	$ħ$ is the reduced Planck constant.
Angular frequency	$\omega ={\color {redorange}2\pi }f$	$\omega ={\color {redorange}\tau }f$

References[edit]

^ Cite error: The named reference Hartl_2019 was invoked but never defined (see the help page).
^ Cite error: The named reference Hartl_2010 was invoked but never defined (see the help page).
^ Cite error: The named reference Aron_2011 was invoked but never defined (see the help page).
^ Cite error: The named reference Landau_2011 was invoked but never defined (see the help page).
^ Cite error: The named reference Bartholomew_2014 was invoked but never defined (see the help page).
^ Cite error: The named reference Telegraph_2011 was invoked but never defined (see the help page).
^ Cite error: The named reference McMillan_2020 was invoked but never defined (see the help page).
^ "Happy Tau Day!". 28 June 2012.
^ Cite error: The named reference Desmos was invoked but never defined (see the help page).
^ Cite error: The named reference Python_2017 was invoked but never defined (see the help page).
^ Cite error: The named reference Python_370 was invoked but never defined (see the help page).
^ Cite error: The named reference Perl6 was invoked but never defined (see the help page).
^ Cite error: The named reference Processing was invoked but never defined (see the help page).
^ Cite error: The named reference Nim was invoked but never defined (see the help page).
^ Cite error: The named reference Rust was invoked but never defined (see the help page).
^ Cite error: The named reference Java was invoked but never defined (see the help page).
^ Cite error: The named reference Java-docs was invoked but never defined (see the help page).
^ Cite error: The named reference dotnet was invoked but never defined (see the help page).
^ Cite error: The named reference dotnet-docs was invoked but never defined (see the help page).
^ Cite error: The named reference Harremoes_2017 was invoked but never defined (see the help page).
^ Cite error: The named reference Harremoes_2018 was invoked but never defined (see the help page).
^ Cite error: The named reference Abbott_2012 was invoked but never defined (see the help page).
^ Cite error: The named reference Palais_2001 was invoked but never defined (see the help page).

[Hartl_2019-1] Cite error: The named reference Hartl_2019 was invoked but never defined (see the help page).

[Hartl_2010-2] Cite error: The named reference Hartl_2010 was invoked but never defined (see the help page).

[Aron_2011-3] Cite error: The named reference Aron_2011 was invoked but never defined (see the help page).

[Landau_2011-4] Cite error: The named reference Landau_2011 was invoked but never defined (see the help page).

[Bartholomew_2014-5] Cite error: The named reference Bartholomew_2014 was invoked but never defined (see the help page).

[Telegraph_2011-6] Cite error: The named reference Telegraph_2011 was invoked but never defined (see the help page).

[McMillan_2020-7] Cite error: The named reference McMillan_2020 was invoked but never defined (see the help page).

[8] "Happy Tau Day!". 28 June 2012.

[Desmos-9] Cite error: The named reference Desmos was invoked but never defined (see the help page).

[Python_2017-10] Cite error: The named reference Python_2017 was invoked but never defined (see the help page).

[Python_370-11] Cite error: The named reference Python_370 was invoked but never defined (see the help page).

[Perl6-12] Cite error: The named reference Perl6 was invoked but never defined (see the help page).

[Processing-13] Cite error: The named reference Processing was invoked but never defined (see the help page).

[Nim-14] Cite error: The named reference Nim was invoked but never defined (see the help page).

[Rust-15] Cite error: The named reference Rust was invoked but never defined (see the help page).

[Java-16] Cite error: The named reference Java was invoked but never defined (see the help page).

[Java-docs-17] Cite error: The named reference Java-docs was invoked but never defined (see the help page).

[dotnet-18] Cite error: The named reference dotnet was invoked but never defined (see the help page).

[dotnet-docs-19] Cite error: The named reference dotnet-docs was invoked but never defined (see the help page).

[Harremoes_2017-20] Cite error: The named reference Harremoes_2017 was invoked but never defined (see the help page).

[Harremoes_2018-21] Cite error: The named reference Harremoes_2018 was invoked but never defined (see the help page).

[Abbott_2012-22] Cite error: The named reference Abbott_2012 was invoked but never defined (see the help page).

[Palais_2001-23] Cite error: The named reference Palais_2001 was invoked but never defined (see the help page).

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