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Tau (2π)

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A circle showing the center (magenta), circumference (black), diameter (cyan), and radius (red).
A circle with radius of 1 unit has a circumference of 𝜏, which equals 2π.
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 are expressed here using the Greek letter tau (τ).

The number 𝜏 is a proposed mathematical constant equal to the ratio of a circle's circumference, C, to its radius, r. Tau equals exactly twice the mathematical constant pi, conventionally written 2π. Tau has not been accepted by any significant portion of the mathematical community. And with few mathematicians aware of its existence and with most typically calling tau a fringe science subject, these mathematicians generally see the change as irrelevant or harmful due to the presence of π as an established notation. Using 𝜏 instead of π would be a very slight change in mathematics notation by effectively encapsulating a factor of 2. Adopting tau would be a major change in the habits of most scientists and engineers, who habitually use pi. Tau is approximated in decimal form as 6.28318530717958647692528676...OEIS: A019692. Proponents of tau, such as Bob Palais , Salman Khan , Michael Hartl and Vi Hart believe that tau is a more natural circle constant than pi, with Hartl calling pi a "confusing and unnatural choice." Others claim that the pedagogical benifits of tau are substantial, and outweigh the inconvenience of changing notation.

Notation

The modern conception of tau as 2 pi began with the article π is wrong! by Robert Palais. Palais initially used a three-legged pi symbol, π π {\displaystyle \pi \!\;\!\!\!\pi } , in place of 2π, and used the term "turns". Usage of 𝜏 was introduced independently by Michael Hartl, Peter Harremoës, and Joseph Lindenberg. The choice for 𝜏 was made for a number of reasons, including the fact that tau was an established glyph, the visual similarity to π, the tie-in with Palais' "turn", and the pun-potential of tau. Other symbols, including Θ (capital theta) , the registered copyright symbol ® , sampi (ϡ) and varpi (ϖ) have been suggested, but tau remains the dominant notation.

The argument for tau

Comparison of angles on a circle expressed in terms of tau and pi
Diagram showing graphs of functions
Sine and cosine functions repeat with period 𝜏 = 2π.

Proponents of tau claim that because the circle is defined as the locus of all points a fixed distance (the radius) from another point (the center) This implies that 𝜏 = C/r might be a more natural constant than π = C/d, with d being the diameter.

The use of radians would also be simplified by the use of tau. An angle of 1/N of a circle would be expressed as 2π/N radians or as 𝜏/N radians. For example, a quarter circle is π/2 radians or 𝜏/4 radians.

The table shows the conversion of some common angles.

Units Values
Turns   0 {\displaystyle 0} 1 12 {\displaystyle {\tfrac {1}{12}}} 1 8 {\displaystyle {\tfrac {1}{8}}} 1 6 {\displaystyle {\tfrac {1}{6}}} 1 4 {\displaystyle {\tfrac {1}{4}}} 1 2 {\displaystyle {\tfrac {1}{2}}} 3 4 {\displaystyle {\tfrac {3}{4}}} 1 {\displaystyle 1}
Radians in terms of 𝜏   0 {\displaystyle 0} 1 12 τ {\displaystyle {\tfrac {1}{12}}\tau } 1 8 τ {\displaystyle {\tfrac {1}{8}}\tau } 1 6 τ {\displaystyle {\tfrac {1}{6}}\tau } 1 4 τ {\displaystyle {\tfrac {1}{4}}\tau } 1 2 τ {\displaystyle {\tfrac {1}{2}}\tau } 3 4 τ {\displaystyle {\tfrac {3}{4}}\tau } 1 τ {\displaystyle 1\tau }
Radians in terms of π   0 {\displaystyle 0} 1 6 π {\displaystyle {\tfrac {1}{6}}\pi } 1 4 π {\displaystyle {\tfrac {1}{4}}\pi } 1 3 π {\displaystyle {\tfrac {1}{3}}\pi } 1 2 π {\displaystyle {\tfrac {1}{2}}\pi } π {\displaystyle \pi } 3 2 π {\displaystyle {\tfrac {3}{2}}\pi } 2 π {\displaystyle 2\pi }
Degrees   0° 30° 45° 60° 90° 180° 270° 360°


Common trigonometric functions, such as sine and cosine have a period of 𝜏.

The difference of a pair of unequal coterminal angles is integer multiple of 𝜏.

Euler's identity, e + 1 = 0, appears at first to be a clear argument in favor of π. However, proponents of 𝜏 note that this comes from Euler's formula, e = cos x + i sin x, with π substituted for x. Substituting 𝜏 into the formula instead, the formula e = 1 is obtained.


Proponents of tau also note a number of important formulae in which the number 2π appears. Selected examples given by Hartl in The Tau Manifesto and/or Palais in Pi is Wrong! include:

Name Formula with π Formula with 𝜏
Normal Distribution f ( x ) = 1 σ 2 π e − 1 2 ( x − μ σ ) 2 {\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}} f ( x ) = 1 σ τ e − 1 2 ( x − μ σ ) 2 {\displaystyle f(x)={\frac {1}{\sigma {\sqrt {\tau }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}
Fourier Transform f ^ ( ξ ) = ∫ − ∞ ∞ f ( x )   e − 2 π i x ξ d x {\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }\,dx} f ^ ( ξ ) = ∫ − ∞ ∞ f ( x )   e − i τ x ξ d x {\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i\tau x\xi }\,dx}
Nth roots of unity e 2 π i k n {\displaystyle e^{2\pi i{\frac {k}{n}}}\;} e τ i k n {\displaystyle e^{\tau i{\frac {k}{n}}}\;}
Cauchy's integral formula f ( a ) = 1 2 π i ∮ γ f ( z ) z − a d z {\displaystyle f(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz} f ( a ) = 1 τ i ∮ γ f ( z ) z − a d z {\displaystyle f(a)={\frac {1}{\tau i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz}
Angular Frequency ω = 2 π f {\displaystyle \scriptstyle \omega \;=\;2\pi f} ω = τ f {\displaystyle \scriptstyle \omega \;=\;\tau f}
Wavenumber k = 2 π λ {\displaystyle \scriptstyle k\;=\;{\frac {2\pi }{\lambda }}} k = τ λ {\displaystyle \scriptstyle k\;=\;{\frac {\tau }{\lambda }}}

Hartl claims that these examples are not cherry-picked. Proponents of pi dispute this.

The argument for pi

Areas of segments of the unit circle expressed in terms of 𝜏 and π.

Proponents of π argue that π is already very well-established, and that there is no mathematical reason to make a switch. While the use of 𝜏 may make certain formulae simpler, the formulae can always be correctly expressed in terms of π by simply substituting π = 𝜏/2. Proponents of π also often dislike the choice of 𝜏 as a symbol, citing other common usages, such as shear stress, torque, and proper time.

Proponents of π also claim that π is a more convenient constant in certain applications. For example, they claim that the area of a unit circle is more conveniently expressed using pi. In addition, the diameter of a circle is generally much easier to physically measure than the radius, making the ratio π = C/d a more convenient constant than 𝜏 = C/r in many contexts. Proponents of pi also state that cases can be made for alternate circle constants, including π/2, π/4, and 2πi. They believe that this means that the choice is somewhat arbitrary, and therefore π, the established constant, should be used.

The common trigonometric function tangent has a period of π.

The period of the tangent function is also equal to pi.

Proponents of pi also point to several common formulae more naturally expressed with pi. Selected examples are shown below.

Name Formula with π Formula with 𝜏
Error function erf ⁡ ( x ) = 2 π ∫ 0 x e − t 2 d t . {\displaystyle \operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,\mathrm {d} t.} erf ⁡ ( x ) = 2 2 τ ∫ 0 x e − t 2 d t . {\displaystyle \operatorname {erf} (x)={\frac {2{\sqrt {2}}}{\sqrt {\tau }}}\int _{0}^{x}e^{-t^{2}}\,\mathrm {d} t.}
Sum of interior angles of an n-gon π ( n − 2 ) {\displaystyle \pi (n-2)} 1 2 τ ( n − 2 ) {\displaystyle {\frac {1}{2}}\tau (n-2)}
Area of an ellipse π a b {\displaystyle \pi ab} 1 2 τ a b {\displaystyle {\frac {1}{2}}\tau ab}
Gaussian integral ∫ − ∞ ∞ e − x 2 d x = π . {\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}.} ∫ − ∞ ∞ e − x 2 d x = 2 τ 2 . {\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\frac {\sqrt {2\tau }}{2}}.}
Area of a circle π r 2 {\displaystyle \pi {r^{2}}} 1 2 τ r 2 {\displaystyle {\frac {1}{2}}\tau r^{2}}
Basel problem ∑ n = 1 ∞ 1 n 2 = π 2 6 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}} ∑ n = 1 ∞ 1 n 2 = τ 2 24 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\tau ^{2}}{24}}}

Response

Many mathematicians consider the difference between tau and pi to be irrelevant. Because the constant for a circle has been historically named pi, and the choice whether to use one or the other ratio (related by a factor of two) as the preferred named constant is arbitrary, causing no substantive change to mathematics, there has been relatively little impact in the mathematical community due to tau. Tau has gained a small but significant amount of attention in the mainstream media, with articles by The Sunday Times , Fox News , and TheStar.com among others, and has been used in at least one serious paper not on the subject of tau.

In popular culture

Tauists have declared June 28 (6.28) to be "tau day". This is in direct response to Pi Day, a popular celebration of pi on March 14 (3.14). A number of news outlets reported on "Tau Day", a holiday proposed in The Tau Manifesto' for June 28 to honor the number 2π. The Royal Institution of Australia's Tau Day celebration in 2011 featured the performance of a musical work based on tau.

According to the Massachusetts Institute of Technology's Dean of Admissions Stuart Schmill, "over past year or so, there has been a bit of a debate in the math universe over which is a better number to use, whether it is Pi or Tau". The school chose to inform 2012 applicants whether or not they were accepted on Pi Day at what MIT called Tau Time, 6:28 pm.

References

  1. ^ Cavers, Michael. "The Pi Manifesto". Retrieved 26 January 2013.
  2. Palais, Robert (2001). "π Is Wrong!" (PDF). The Mathematical Intelligencer. 23 (3): 7–8. {{cite journal}}: Cite has empty unknown parameter: |coauthors= (help)
  3. Khan, Salman. "Tau vs. Pi: Why Tau might be a better number to look at than Pi". Retrieved 24 January 2013.
  4. Wilkins, Alasdair. "Why we have to get rid of pi for the sake of good math". Retrieved 23 January 2013.
  5. ^ "On National Tau Day, Pi Under Attack". Fox News Channel. NewsCore. June 28, 2011. Retrieved 2011-07-03.
  6. Abbott, Stephen. "My Conversion to Tauism" (PDF). Retrieved 3 February 2013.
  7. Wolchover, Natalie. "Mathematicians Want to Say Goodbye to Pi". Retrieved 27 January 2013.
  8. ^ Hartl, Michael. "The Tau Manifesto". Retrieved 26 January 2013.
  9. Cool, Thomas. Conquest of the Plane Using The Economics Pack Applications of Mathematica for a didactic primer on Analytic Geometry and Calculus (PDF). Cool, T. (Consultancy & Econometrics) Rotterdamsestraat 69, NL-2586 GH Scheveningen, The Netherlands. ISBN 978-90-804774-6-9. {{cite book}}: line feed character in |publisher= at position 38 (help)
  10. Bourne, Murray. "Let's drop pi". Retrieved 2 February 2013.
  11. "Life of pi in no danger". Retrieved 27 January 2013.
  12. "Bye to Pi". Retrieved 27 January 2013.
  13. ^ Black, Debra. "Down with ugly pi, long live elegant Tau, physicist urges". Retrieved 27 January 2013.
  14. Thibaut, Pugin (2011), "An Algebraic Circle Method", http://www.math.columbia.edu/~thaddeus/theses/2011/pugin.pdf {{citation}}: Missing or empty |title= (help)
  15. Palmer, Jason (28 June 2011). "'Tau day' marked by opponents of maths constant pi". BBC News. Retrieved 2011-07-03.
  16. "Tau: Is two pi better than one?". Today. BBC Radio 4. 28 June 2011. Retrieved 2012-03-27.
  17. Young, Colin A. "In nod to Pi Day, MIT releasing admissions decisions Wednesday evening". Boston Globe. Retrieved 17 March 2012.

See also