Draft: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[edit]
In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi\!\;\!\!\!\pi = 2\pi} ).[1]. 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.[2].
Since then, tau became more and more popular, but most people still use pi.
Some formulas using pi vs tau[edit]
The following table shows how various identities appear when tau is used instead of pi.[3]
Formula | Using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} | Using τ | ! Notes |
---|---|---|---|
Angle subtended by 1/4 of a circle | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\color{orangered}\frac{\pi}{2}} \text{ rad}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\color{orangered}\frac{\tau}{4}} \text{ rad}} | τ/4 rad = 1/4 turn |
Circumference of a circle of radius r | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = {\color{orangered}2 \pi} r} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = {\color{orangered}\tau} r} | |
Area of a circle | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = {\color{orangered}\pi}r^2} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \frac{n}{2} \sin \frac{{\color{orangered}2 \pi}}{n}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \frac{n}{2} \sin \frac{{\color{orangered}\tau}}{n}} | |
n
-ball and n -sphere volume recurrence relation |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_n(r) = \frac{r}{n} S_{n-1}(r)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n(r) = {\color{orangered} 2 \pi} r V_{n-1}(r)} |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_n(r) = \frac{r}{n} S_{n-1}(r)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n(r) = {\color{orangered}\tau}rV_{n-1}(r)} |
V0(r) = 1
S0(r) = 2 |
Cauchy's integral formula | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a) = \frac{1}{{\color{orangered}2\pi} i} \oint_\gamma \frac{f(z)}{z-a}\, dz} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a) = \frac{1}{{\color{orangered}\tau} i} \oint_\gamma \frac{f(z)}{z-a}\, dz} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} is the boundary of a disk containing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} in the complex plane. |
Standard normal distribution | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi(x) = \frac{1}{\sqrt{{\color{orangered}2\pi}}}e^{-\frac{x^2}{2}}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi(x) = \frac{1}{\sqrt{{\color{orangered}\tau}}}e^{-\frac{x^2}{2}}} | |
Stirling's approximation | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n! \sim \sqrt{{\color{orangered}2 \pi} n}\left(\frac{n}{e}\right)^n} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n! \sim \sqrt{{\color{orangered}\tau} n}\left(\frac{n}{e}\right)^n} | |
n
-th roots of unity |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h = {\color{orangered}2 \pi} \hbar} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h = {\color{orangered}\tau} \hbar} | The reduced Planck constant is ħ |
Angular frequency | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = {\color{orangered}2 \pi} f} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = {\color{orangered}\tau} f} |
Arguments for tau[edit]
Arguments against tau[edit]
References[edit]
- ↑ Palais, Robert (2001). "Pi is Wrong" (PDF). The Mathematical Intelligencer. 23 (3). New York, USA: Springer-Verlag: 7–8. doi:10.1007/bf03026846. S2CID 120965049. Archived (PDF) from the original on 2019-07-18. Retrieved 2019-08-05.
- ↑ Hartl, Michael (2010-03-14). "The Tau Manifesto" (PDF). Archived (PDF) from the original on 2019-07-18. Retrieved 2019-08-05.
- ↑ Abbott, Stephen (April 2012). "My Conversion to Tauism" (PDF). Math Horizons. 19 (4): 34. doi:10.4169/mathhorizons.19.4.34. S2CID 126179022. Archived (PDF) from the original on 2013-09-28.