User:Danstronger/Draft:Tau (proposed mathematical constant)

An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to a full turn, or approximately 6.28 radians, which is expressed here using the Greek letter tau (τ).

The number 𝜏 (/taʊ/; spelled out as "tau") is a proposed mathematical constant, approximately equal to 6.2831853. It is defined as the ratio of a circle's circumference to its radius. The value of 𝜏 is equal to 2 times π, and it is proposed as a replacement for π, in the sense that in any context where "2π" appears, such as in Cauchy's integral formula, it could be replaced with 𝜏, and anywhere π appears by itself, such as in the circle area formula, it could be replaced with 𝜏/2. Advocates for the proposal argue that these replacements make formulas simpler on average, and make many mathematical facts easier to teach and understand. Arguments against the proposal include claims that the distinction between π and 𝜏 is a triviality, and that the symbol 𝜏 is already in widespread use (such as for torque).

The use of the Greek letter 𝜏 to represent 2π was first proposed by Michael Hartl in his 2010 essay The Tau Manifesto. The proposal has been mostly ignored by academics, but it has some enthusiastic proponents. In recent years, it has become supported in some programming languages.

History

In 1746, Leonard Euler first used the Greek letter pi to represent the circumference divided by the radius (i.e. Pi is approx. 6.28...) of a circle.

In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of π, which amounts to the number of radians in half a turn, in order to make mathematics simpler and more intuitive. His proposal used a "π with three legs" symbol to denote the constant ($\pi \!\;\!\!\!\pi =2\pi$).

In 2008, Thomas Colignatus proposed the uppercase Greek letter theta, Θ, to represent 2π.

The Greek letter theta derives from the Phoenician and Hebrew letter teth, 𐤈 or ט, and it has been observed that the older version of the symbol, which means wheel, resembles a wheel with four spokes. It has also been proposed to use the wheel symbol, teth, to represent the quantity 2π, and more recently a connection has been made among other ancient cultures on the existence of a wheel, sun, circle, or disk symbol—i.e. other variations of teth—as representation for 2π.

In 2010, Michael Hartl proposed to use the Greek letter tau to represent the circle constant: τ = 2π. 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 3τ/4 rad instead of 3π/2 rad. Second, τ visually resembles π, whose association with the circle constant is unavoidable. Hartl's Tau Manifesto gives many examples of formulas that are asserted to be clearer where τ is used instead of π.

Reception

Initially, this proposal did not receive widespread acceptance by the mathematical and scientific communities. However, the use of τ has become more widespread, for example:

• In 2012, the educational website Khan Academy began accepting answers expressed in terms of τ.
• In June 2017, for release 3.6, the Python programming language adopted the name tau to represent the number of radians in a turn.
• The τ-functionality is made available in the Google calculator and in several programming languages like Python, Raku, Processing, Nim, and Rust.
• It has also been used in at least one mathematical research article, authored by the τ-promoter Peter Harremoës.
• In 2020, for release 5.0, Tau was added to .NET Core (which is being rebranded as ".NET" for the 5.0 release).

Analysis

The following table shows how various identities and inequalities appear if τ := 2π was used instead of π.

Formula	Using π	Using τ	Notes
1/4 of a circle	π/2 rad	τ/4 rad	τ/4 rad is a quarter of a circle and a quarter of τ
Circumference C of a circle of radius r	C = 2πr	C = τr
Area of a circle	A = πr2	A = τr2/2	Recall that the area of a sector of angle θ (measured in radians) is A = θr2/2.
Area of a regular n-gon with unit circumradius	A = n/2 sin 2π/n	A = n/2 sin τ/n
Volume of an n-ball	$V_{n}(R)={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}R^{n}$	$V_{n}(R)={\frac {\tau ^{\left\lfloor {\frac {n}{2}}\right\rfloor }}{n!!}}(1+n\operatorname {mod} 2)R^{n}$
Surface area of an n-ball	$S_{n}(R)={\frac {2\pi ^{\frac {n+1}{2}}}{\Gamma {\big (}{\frac {n+1}{2}}{\big )}}}R^{n}$	$S_{n}(R)={\frac {\tau ^{\left\lfloor {\frac {n+1}{2}}\right\rfloor }}{(n-1)!!}}(2-(n\operatorname {mod} 2))R^{n}$
Cauchy's integral formula	$f(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$	$f(a)={\frac {1}{\tau i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$
Standard normal distribution	$\varphi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}$	$\varphi (x)={\frac {1}{\sqrt {\tau }}}e^{-{\frac {x^{2}}{2}}}$
Stirling's approximation	$n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}$	$n!\sim {\sqrt {\tau n}}\left({\frac {n}{e}}\right)^{n}$
Euler's identity	0       eiπ = − 1 eiπ + 1 = 0	0     eiτ = 1 eiτ − 1 = 0
nth roots of unity	$e^{2\pi i{\frac {k}{n}}}=\cos {\frac {2k\pi }{n}}+i\sin {\frac {2k\pi }{n}}$	$e^{\tau i{\frac {k}{n}}}=\cos {\frac {k\tau }{n}}+i\sin {\frac {k\tau }{n}}$
Reduced Planck constant	$\hbar ={\frac {h}{2\pi }}$	$\hbar ={\frac {h}{\tau }}$	h is the Planck constant.
Angular frequency	$\omega ={{2\pi } \over T}={2\pi f}$	$\omega ={{\tau } \over T}={\tau f}$
Reactance of an inductor	$2\pi fL$	$\tau fL$
Susceptance of a capacitor	$2\pi fC$	$\tau fC$