Draft:Tau

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Tau

Tau (τ) is an irrational number equal to the circumference of a circle divided by it's radius. It is thus equal to 2 pi. It was popularized since 2010 by Michael Hartl in The Tau Manifesto, arguing that many mathematical formulals look nicer using tau instead of pi. Since then, tau has been adopted by many mathematicians, but most of them are still using pi.

History

In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of $\pi$, which amounts to the number of radians in half a turn, in order to make mathematics simpler and more intuitive. Unfortunately, his proposal used the ugly symbol "π with three legs" to denote the constant ($\pi \!\;\!\!\!\pi =2\pi$).. This symbol was never further used, and Palais himself later abandoned it.

In 2010, Michael Hartl proposed to use the Greek letter tau to represent the circle constant: τ = 2π . He offered several reasons for the choice of constant, primarily that it 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. As for the choice of notation, he offered two reasons. First, tau is the number of radians in one turn, and both tau and turn begin with a t, and etymologically it comes from Greek τόρνος, which begins with a τ. Second, tau visually resembles pi, whose association with the circle constant is unavoidable. Hartl's Tau Manifesto gives many examples of formulas that are asserted to be clearer where tau is used instead of pi..

Since then, tau became more and more popular, but most people still use pi.

Some formulas using pi vs tau

The following table shows how various identities appear when tau is used instead of pi.

Formula	Using $\pi$	Using τ	! Notes
Angle subtended by 1/4 of a circle	${\color {orangered}{\frac {\pi }{2}}}{\text{ rad}}$	${\color {orangered}{\frac {\tau }{4}}}{\text{ rad}}$	τ/4 rad = 1/4 turn
Circumference of a circle of radius r	$C={\color {orangered}2\pi }r$	$C={\color {orangered}\tau }r$
Area of a circle	$A={\color {orangered}\pi }r^{2}$	$A={\color {orangered}{\frac {1}{2}}\tau }r^{2}$	The area of a circular sector is A = 1/2θr2 .
Area of a regular n -gon with unit circumradius	$A={\frac {n}{2}}\sin {\frac {\color {orangered}2\pi }{n}}$	$A={\frac {n}{2}}\sin {\frac {\color {orangered}\tau }{n}}$
n -ball and n -sphere volume recurrence relation	$V_{n}(r)={\frac {r}{n}}S_{n-1}(r)$ $S_{n}(r)={\color {orangered}2\pi }rV_{n-1}(r)$	$V_{n}(r)={\frac {r}{n}}S_{n-1}(r)$ $S_{n}(r)={\color {orangered}\tau }rV_{n-1}(r)$	V0(r) = 1 S0(r) = 2
Cauchy's integral formula	$f(a)={\frac {1}{{\color {orangered}2\pi }i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$	$f(a)={\frac {1}{{\color {orangered}\tau }i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$	$\gamma$ is the boundary of a disk containing $a$ in the complex plane.
Standard normal distribution	$\varphi (x)={\frac {1}{\sqrt {\color {orangered}2\pi }}}e^{-{\frac {x^{2}}{2}}}$	$\varphi (x)={\frac {1}{\sqrt {\color {orangered}\tau }}}e^{-{\frac {x^{2}}{2}}}$
Stirling's approximation	$n!\sim {\sqrt {{\color {orangered}2\pi }n}}\left({\frac {n}{e}}\right)^{n}$	$n!\sim {\sqrt {{\color {orangered}\tau }n}}\left({\frac {n}{e}}\right)^{n}$
n -th roots of unity	$e^{{\color {orangered}2\pi }i{\frac {k}{n}}}=\cos {\frac {{\color {orangered}2}k{\color {orangered}\pi }}{n}}+i\sin {\frac {{\color {orangered}2}k{\color {orangered}\pi }}{n}}$	$e^{{\color {orangered}\tau }i{\frac {k}{n}}}=\cos {\frac {k{\color {orangered}\tau }}{n}}+i\sin {\frac {k{\color {orangered}\tau }}{n}}$
Planck constant	$h={\color {orangered}2\pi }\hbar$	$h={\color {orangered}\tau }\hbar$	The reduced Planck constant is ħ
Angular frequency	$\omega ={\color {orangered}2\pi }f$	$\omega ={\color {orangered}\tau }f$