Tau (mathematics)

Constant equal to twice pi

This article is about the proposed mathematical constant. For other uses, see Tau § Mathematics.

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). A comparison of angles expressed in degrees and radians.

The number 𝜏 (/ˈtaʊ, ˈtɔː, ˈtɒ/ ⓘ; spelled out as tau) is a mathematical constant that is the ratio of a circle's circumference to its radius. It is approximately equal to 6.28 and exactly equal to 2π.

𝜏 and π are both circle constants relating the circumference of a circle to its linear dimension: the radius in the case of 𝜏; the diameter in the case of π.

While π is used almost exclusively in mainstream mathematical education and practice, it has been proposed, most notably by Michael Hartl in 2010, that 𝜏 should be used instead. Hartl and other proponents argue that 𝜏 is the more natural circle constant and its use leads to conceptually simpler and more intuitive mathematical notation.[1]

Critics have responded that the benefits of using 𝜏 over π are trivial and that given the ubiquity and historical significance of π a change is unlikely to occur.[2]

The proposal did not initially gain widespread acceptance in the mathematical community, but awareness of 𝜏 has become more widespread,[3] having been added to several major programming languages and calculators.

## Fundamentals

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### Definition

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𝜏 is commonly defined as the ratio of a circle's circumference C {\textstyle {C}} to its radius  r {\textstyle {r}} : τ = C r {\displaystyle \tau ={\frac {C}{r}}} A circle is defined as a closed curve formed by the set of all points in a plane that are a given distance from a fixed point, where the given distance is called the radius.

The distance around the circle is the circumference, and the ratio  C r {\textstyle {\frac {C}{r}}} is constant regardless of the circle's size. Thus, 𝜏 denotes the fixed relationship between the circumference of any circle and the fundamental defining property of that circle, the radius.

### Units of angle

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Some special angles in radians, stated in terms of 𝜏.

When radians are used as the unit of angular measure there are 𝜏 radians in one full turn of a circle, and the radian angle is aligned with the proportion of a full turn around the circle:  τ 8 {\textstyle {\frac {\tau }{8}}} rad is an eighth of a turn;  3 τ 4 {\textstyle {\frac {3\tau }{4}}} rad is three-quarters of a turn.

### Relationship to π

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As 𝜏 is exactly equal to 2π it shares many of the properties of π including being both an irrational and transcendental number.

## History

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The proposal to use the Greek letter 𝜏 as a circle constant representing 2π dates to Michael Hartl's 2010 publication, The Tau Manifesto,[a] although the symbol had been independently suggested earlier by Joseph Lindenburg (c.1990), John Fisher (2004) and Peter Harremoës (2010).[5]

Hartl offered two reasons for the choice of notation. First, τ is the number of radians in one turn, and both τ and turn begin with a /t/ sound. Second, τ visually resembles π, whose association with the circle constant is unavoidable.

### Earlier proposals

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There had been a number of earlier proposals for a new circle constant equal to 2π, together with varying suggestions for its name and symbol.

In 2001, Robert Palais of the University of Utah proposed that π was "wrong" as the fundamental circle constant arguing instead that 2π was the proper value.[6] His proposal used a "π with three legs" symbol to denote the constant ( π π = 2 π {\displaystyle \pi \\!\;\\!\\!\\!\pi =2\pi } ), and referred to angles as fractions of a "turn" ( 1 4 π π = 1 4 t u r n {\displaystyle {\tfrac {1}{4}}\pi \\!\;\\!\\!\\!\pi ={\tfrac {1}{4}}\,\mathrm {turn} } ). Palais stated that the word "turn" served as both the name of the new constant and a reference to the ordinary language meaning of turn.[7]

In 2008, Robert P. Crease proposed defining a constant as the ratio of circumference to radius, an idea supported by John Horton Conway. Crease used the Greek letter psi:  ψ = 2 π {\displaystyle \psi =2\pi } .[8]

The same year, Thomas Colignatus proposed the uppercase Greek letter theta, Θ, to represent 2π due to its visual resemblance of a circle.[9] For a similar reason another proposal suggested the Phoenician and Hebrew letter teth, 𐤈 or ט, (from which the letter theta was derived), due to its connection with wheels and circles in ancient cultures.[10][11]

### Use of the symbol π to represent 6.28

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See also: Pi § Adoption of the symbol π

The meaning of the symbol  π {\displaystyle \pi } was not originally defined as the ratio of circumference to diameter, and at times was used in representations of the 6.28...constant.

Early works in circle geometry used the letter π to designate the perimeter (i.e., circumference) in different fractional representations of circle constants and in 1697 David Gregory used ⁠π/ρ⁠ (pi over rho) to denote the perimeter divided by the radius (6.28...).[12][13]

Subsequently π came to be used as a single symbol to represent the ratios in whole. Leonhard Euler initially used the single letter π was to denote the constant 6.28... in his 1727 Essay Explaining the Properties of Air.[14][15] Euler would later use the letter π for 3.14... in his 1736 Mechanica[16] and 1748 Introductio in analysin infinitorum,[17] though defined as half the circumference of a circle of radius 1 rather than the ratio of circumference to diameter. Elsewhere in Mechanica, Euler instead used the letter π for one-fourth of the circumference of a unit circle, or 1.57... .[18][19] Usage of the letter π, sometimes for 3.14... and other times for 6.28..., became widespread, with the definition varying as late as 1761;[20] afterward, π was standardized as being equal to 3.14... .[21][22]

## Notion using 𝜏

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Proponents argue that while use of 𝜏 in place of 2π does not change any of the underlying mathematics, it does lead to simpler and more intuitive notation in many areas. Michael Hartl's Tau Manifesto[b] gives many examples of formulas that are asserted to be clearer where τ is used instead of π.[23][24][25]

### Units of angle

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Hartl and Robert Palais[7] have argued that 𝜏 allows radian angles to be expressed more directly and in a way that makes clear the link between the radian measure and rotation around the unit circle. For instance, ⁠3τ/4⁠ rad can be easily interpreted as ⁠3/4⁠⁠ of a turn around the unit circle in contrast with the numerically equal ⁠⁠3π/2⁠⁠ rad, where the meaning could be obscured, particularly for children and students of mathematics.

Critics have responded that a full rotation is not necessarily the correct or fundamental reference measure for angles and two other possibilities, the right angle and straight angle, each have historical precedent. Euclid used the right angle as the basic unit of angle, and David Butler has suggested that ⁠τ/4⁠ = ⁠π/2⁠ ≈ 1.57, which he denotes with the Greek letter η (eta), should be seen as the fundamental circle constant.[26][27]

### Trigonometric Functions

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Hartl has argued that the periodic trigonometric functions are simplified using 𝜏 as it aligns the function argument (radians) with the function period: sin θ repeats with period T = τ rad, reaches a maximum at ⁠T/4⁠=⁠τ/4⁠ rad and a minimum at ⁠3T/4⁠=⁠3τ/4⁠ rad.

### Area of a circle

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Critics have argued that the formula for the area of a circle is more complicated when restated as A = ⁠1/2⁠𝜏r2. Hartl and others respond that the ⁠1/2⁠ factor is meaningful, arising from either integration or geometric proofs for the area of a circle as half the circumference times the radius.

### Euler's identity

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A common criticism of τ is that Euler's identity, eiπ \+ 1 = 0, sometimes claimed to be "the most beautiful theorem in mathematics"[28] is made less elegant rendered as eiτ/2  \+ 1 = 0.[29] Hartl has asserted that eiτ = 1 (which he also called "Euler's identity") is more fundamental and meaningful. John Conway noted[8] that Euler's identity is a specific case of the general formula of the nth roots of unity, n√1 = eiτk/n (k = 1,2,..,n), which he maintained is preferable and more economical than Euler's.

### Comparison of identities

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The following table shows how various identities appear when τ = 2π is used instead of π.[30][6] 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 {\displaystyle {\color {orangered}{\frac {\pi }{2}}}{\text{ rad}}} |  τ 4  rad {\displaystyle {\color {orangered}{\frac {\tau }{4}}}{\text{ rad}}} | ⁠τ/4⁠ rad = ⁠1/4⁠ turn
Circumference of a circle  |  C = 2 π r {\displaystyle C={\color {orangered}2\pi }r} |  C = τ r {\displaystyle C={\color {orangered}\tau }r} | The length of an arc of angle θ is L = θr.
Area of a circle |  A = π r 2 {\displaystyle A={\color {orangered}\pi }r^{2}} |  A = 1 2 τ r 2 {\displaystyle A={\color {orangered}{\frac {1}{2}}\tau }r^{2}} | The area of a sector of angle θ is A = ⁠1/2⁠θr2.
Area of a regular n-gon with unit circumradius  |  A = n 2 sin ⁡ 2 π n {\displaystyle A={\frac {n}{2}}\sin {\frac {\color {orangered}2\pi }{n}}} |  A = n 2 sin ⁡ τ n {\displaystyle A={\frac {n}{2}}\sin {\frac {\color {orangered}\tau }{n}}} |
n-ball and n-sphere volume recurrence relation |  V n ( r ) = r n S n − 1 ( r ) {\displaystyle V_{n}(r)={\frac {r}{n}}S_{n-1}(r)}

S n ( r ) = 2 π r V n − 1 ( r ) {\displaystyle S_{n}(r)={\color {orangered}2\pi }rV_{n-1}(r)}

|  V n ( r ) = r n S n − 1 ( r ) {\displaystyle V_{n}(r)={\frac {r}{n}}S_{n-1}(r)}

S n ( r ) = τ r V n − 1 ( r ) {\displaystyle S_{n}(r)={\color {orangered}\tau }rV_{n-1}(r)}

| V0(r) = 1
S0(r) = 2
Cauchy's integral formula |  f ( a ) = 1 2 π i ∮ γ f ( z ) z − a d z {\displaystyle f(a)={\frac {1}{{\color {orangered}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}{{\color {orangered}\tau }i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz} |  γ {\displaystyle \gamma } is the boundary of a disk containing  a {\displaystyle a} in the complex plane.
Standard normal distribution |  φ ( x ) = 1 2 π e − x 2 2 {\displaystyle \varphi (x)={\frac {1}{\sqrt {\color {orangered}2\pi }}}e^{-{\frac {x^{2}}{2}}}} |  φ ( x ) = 1 τ e − x 2 2 {\displaystyle \varphi (x)={\frac {1}{\sqrt {\color {orangered}\tau }}}e^{-{\frac {x^{2}}{2}}}} |
Stirling's approximation |  n ! ∼ 2 π n ( n e ) n {\displaystyle n!\sim {\sqrt {{\color {orangered}2\pi }n}}\left({\frac {n}{e}}\right)^{n}} |  n ! ∼ τ n ( n e ) n {\displaystyle n!\sim {\sqrt {{\color {orangered}\tau }n}}\left({\frac {n}{e}}\right)^{n}} |
nth roots of unity |  e 2 π i k n = cos ⁡ 2 k π n + i sin ⁡ 2 k π n {\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}}} |  e τ i k n = cos ⁡ k τ n + i sin ⁡ k τ n {\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 |  h = 2 π ℏ {\displaystyle h={\color {orangered}2\pi }\hbar } |  h = τ ℏ {\displaystyle h={\color {orangered}\tau }\hbar } | ħ is the reduced Planck constant.
Angular frequency |  ω = 2 π f {\displaystyle \omega ={\color {orangered}2\pi }f} |  ω = τ f {\displaystyle \omega ={\color {orangered}\tau }f} |
Riemann's functional equation |  ζ ( s ) = 2 s π s − 1   sin ⁡ ( s π 2 )   Γ ( 1 − s )   ζ ( 1 − s ) {\displaystyle \zeta (s)={\color {orangered}2^{s}\pi ^{s-1}}\ \sin \left(s{\color {orangered}{\frac {\pi }{2}}}\right)\ \Gamma (1-s)\ \zeta (1-s)} |  ζ ( s ) = 2 τ s − 1   sin ⁡ ( s τ 4 )   Γ ( 1 − s )   ζ ( 1 − s ) {\displaystyle \zeta (s)={\color {orangered}2\tau ^{s-1}}\ \sin \left(s{\color {orangered}{\frac {\tau }{4}}}\right)\ \Gamma (1-s)\ \zeta (1-s)} |  2 s ( τ 2 ) s − 1 {\displaystyle 2^{s}({\frac {\tau }{2}})^{s-1}} reduces to  2 τ s − 1 {\displaystyle 2\tau ^{s-1}}

## In culture

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𝜏 has made numerous appearances in culture. It is celebrated annually on June 28, known as Tau Day.[31] Supporters of 𝜏 are called tauists.[25] 𝜏 has been covered in videos by Vi Hart,[32][33][34] Numberphile,[35][36][37] SciShow,[38] Steve Mould,[39][40][41] Khan Academy,[42] and 3Blue1Brown,[43][44] and it has appeared in the comics xkcd,[45][46] Saturday Morning Breakfast Cereal,[47][48][49] and Sally Forth.[50] The Massachusetts Institute of Technology usually announces admissions on March 14 at 6:28 p.m., which is on Pi Day at Tau Time.[51] Peter Harremoës has used τ in a mathematical research article which was granted Editor's award of the year.[52]

## In programming languages and calculators

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The following table documents various programming languages that have implemented the circle constant for converting between turns and radians. All of the languages below support the name "Tau" in some casing, but Processing also supports "TWO_PI" and Raku also supports the symbol "τ" for accessing the same value.

Language | Identifiers | First Version | Year Released

|
|
|

C# / .NET | System.Math.Tau and System.MathF.Tau | 5.0 | 2020
Crystal | TAU | 0.36.0 | 2021
Eiffel | math_constants.Tau | Curtiss | Not yet released
GDScript | TAU | Godot 3.0 | 2018
Java | Math.TAU | 19 | 2022
Nim | TAU | 0.14.0 | 2016
Processing | TAU and TWO_PI | 2.0 | 2013
Python | math.tau | 3.6 | 2016
Raku | tau and τ |  |
Rust | std::f64::consts::TAU | 1.47.0 | 2020
Zig | std.math.tau | 0.6.0 | 2019

The constant τ is made available in the Google calculator, Desmos graphing calculator,[53] and the iPhone's Convert Angle option expresses the turn as τ.[54]

## Notes

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1. ^ Original version,[4] current version[1]
2. ^ Original version,[4] current version[1]

## References

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1. ^ a b c Hartl, Michael (2019-03-14) [2010-03-14]. "The Tau Manifesto". Archived from the original on 2019-06-28. Retrieved 2013-09-14.
2. ^ "Life of pi in no danger – Experts cold-shoulder campaign to replace with tau". Telegraph India. 2011-06-30. Archived from the original on 2013-07-13. Retrieved 2019-08-05.
3. ^ McMillan, Robert (2020-03-13). "For Math Fans, Nothing Can Spoil Pi Day—Except Maybe Tau Day". Wall Street Journal. ISSN 0099-9660. Retrieved 2020-05-21.
4. ^ a b Hartl, Michael (2010-03-14). "The Tau Manifesto" (PDF). Archived (PDF) from the original on 2019-07-18. Retrieved 2019-08-05.
5. ^ sudgylacmoe; Hartl, Michael (28 June 2023). The Tau Manifesto - With Michael Hartl (YouTube video). Information shown at 18:35. Retrieved 24 July 2024.
6. ^ a b 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.
7. ^ a b "Pi Is Wrong!". www.math.utah.edu. Retrieved 2025-04-26.
8. ^ a b Crease, Robert (2008-02-01). "Constant failure". Physics World. Institute of Physics. Retrieved 2024-08-03.
9. ^ Cool, Thomas "Colignatus" (2008-07-18) [2008-04-08, 2008-05-06]. "Trig rerigged. Trigonometry reconsidered. Measuring angles in 'unit meter around' and using the unit radius functions Xur and Yur" (PDF). Archived from the original (PDF) on 2023-07-18. Retrieved 2023-07-18. (18 pages)
10. ^ Mann, Steve; Janzen, Ryan E.; Ali, Mir Adnan; Scourboutakos, Pete; Guleria, Nitin (22–24 October 2014). "Integral Kinematics (Time-Integrals of Distance, Energy, etc.) and Integral Kinesiology". Proceedings of the 2014 IEEE GEM. Toronto, Ontario, Canada: 627–629\. S2CID 6462220. Retrieved 2023-07-18.
11. ^ Mann, Steve; Chen, Hongyu; Aylward, Graeme; Jorritsma, Megan; Mann, Christina; Defaz Poveda, Diego David; Pierce, Cayden; Lam, Derek; Stairs, Jeremy; Hermandez, Jesse; Li, Qiushi; Xiang, Yi Xin; Kanaan, Georges (June 2019). "Keynote - Eye Itself as a Camera: Sensors, Integrity, and Trust". The 5th ACM Workshop on Wearable Systems and Applications. pp. 1–2\. doi:10.1145/3325424.3330210. ISBN 978-1-4503-6775-2. S2CID 189926593. Retrieved 2023-07-18.
12. ^ Beckmann, Petr (1989) [1970]. A History of Pi. Barnes & Noble Publishing.
13. ^ Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. The Mathematical Association of America. p. 165. ISBN 978-0-88385511-9.
14. ^ Euler, Leonhard (1727). "Tentamen explicationis phaenomenorum aeris" (PDF). Commentarii Academiae Scientiarum Imperialis Petropolitana (in Latin). 2: 351. E007. Archived (PDF) from the original on 1 April 2016. Retrieved 15 October 2017. "Sumatur pro ratione radii ad peripheriem, I : π" English translation by Ian Bruce Archived 10 June 2016 at the Wayback Machine: "π is taken for the ratio of the radius to the periphery [note that in this work, Euler's π is double our π.]"
15. ^ Euler, Leonhard (1747). Henry, Charles (ed.). Lettres inédites d'Euler à d'Alembert. Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche (in French). Vol. 19 (published 1886). p. 139\. E858. "Car, soit π la circonference d'un cercle, dout le rayon est = 1" English translation in Cajori, Florian (1913). "History of the Exponential and Logarithmic Concepts". The American Mathematical Monthly. 20 (3): 75–84\. doi:10.2307/2973441. JSTOR 2973441. "Letting π be the circumference (!) of a circle of unit radius"
16. ^ Euler, Leonhard (1736). "Ch. 3 Prop. 34 Cor. 1". Mechanica sive motus scientia analytice exposita. (cum tabulis) (in Latin). Vol. 1\. Academiae scientiarum Petropoli. p. 113\. E015. "Denotet 1 : π rationem diametri ad peripheriam" English translation by Ian Bruce Archived 10 June 2016 at the Wayback Machine : "Let 1 : π denote the ratio of the diameter to the circumference"
17. ^ Euler, Leonhard (1922). Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus / ediderunt Adolf Krazer et Ferdinand Rudio (in Latin). Lipsae: B.G. Teubneri. pp. 133–134\. E101. Archived from the original on 16 October 2017. Retrieved 15 October 2017.
18. ^ Euler, Leonhard (1736). Mechanica sive motus scientia analytice exposita. p. 185. Retrieved 2025-02-12.
19. ^ Sanderson, Grant (2018-03-14). How pi was almost 6.283185... Event occurs at 2:29. Retrieved 2025-02-11.
20. ^ Segner, Johann Andreas von (1761). Cursus Mathematicus: Elementorum Analyseos Infinitorum Elementorum Analyseos Infinitorvm (in Latin). Renger. p. 374\. "Si autem π notet peripheriam circuli, cuius diameter eſt 2"
21. ^ "Pi". Encyclopaedia Brittanica. 2024-03-14. Retrieved 2024-03-26.
22. ^ Euler, Leonhard (1746). Nova theoria lucis et colorum. Opuscula varii argumenti (in Latin). sumtibus Ambr. Haude & Jo. Carol. Speneri, bibliop. p. 200\. "unde constat punctum B per datum tantum spatium de loco fuo naturali depelli, ad quam maximam distantiam pertinget, elapso tempore t=π/m denotante π angulum 180°, quo fit cos(mt)=- 1 & B b=2α." [from which it is clear that the point B is pushed by a given distance from its natural position, and it will reach the maximum distance after the elapsed time t=π/m, π denoting an angle of 180°, which becomes cos(mt)=- 1 & B b=2α.]
23. ^ Aron, Jacob (2011-01-08). "Michael Hartl: It's time to kill off pi". New Scientist. Interview. 209 (2794): 23. Bibcode:2011NewSc.209...23A. doi:10.1016/S0262-4079(11)60036-5.
24. ^ Landau, Elizabeth (2011-03-14). "On Pi Day, is 'pi' under attack?". cnn.com. CNN. Archived from the original on 2018-12-19. Retrieved 2019-08-05.
25. ^ a b Bartholomew, Randyn Charles (2014-06-25). "Let's Use Tau--It's Easier Than Pi - A growing movement argues that killing pi would make mathematics simpler, easier and even more beautiful". Scientific American. Archived from the original on 2019-06-18. Retrieved 2015-03-20.
26. ^ Butler, David. "Pi, Tau and Eta".
27. ^ David Butler (2011-06-28). Pi may be wrong, but so is Tau!. Retrieved 2025-05-01 – via YouTube.
28. ^ Peshin, Akash (2017-12-24). "Euler's Identity: 'The Most Beautiful Theorem In Mathematics'". ScienceABC. Retrieved 2025-04-27.
29. ^ Colin (2011-06-28). "Tau versus Pi". Proving the Obviously Untrue. Retrieved 2025-04-27.
30. ^ 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.
31. ^ Hartl, Michael. "Tau Day". Retrieved 1 November 2024.
32. ^ Hart, Vi (14 March 2011). "Pi is (still) Wrong". YouTube. Retrieved 1 November 2024.
33. ^ Hart, Vi (28 June 2012). "A Song About A Circle Constant". YouTube. Retrieved 1 November 2024.
34. ^ Hart, Vi (28 June 2015). "360 Video for Tau Day". YouTube. Retrieved 1 November 2024.
35. ^ Haran, Brady; Moriarty, Phil (9 November 2012). "Tau replaces Pi - Numberphile". YouTube. Retrieved 1 November 2024.
36. ^ Haran, Brady; Moriarty, Phil (19 November 2012). "Tau of Phi - Numberphile". YouTube. Retrieved 1 November 2024.
37. ^ Haran, Brady; Mould, Steve; Parker, Matthew (14 December 2012). "Tau vs Pi Smackdown - Numberphile". YouTube. Retrieved 1 November 2024.
38. ^ Hofmeister, Caitlin (26 June 2015). "Happy Tau Day!". YouTube. Retrieved 1 November 2024.
39. ^ Mould, Steve (2018-11-06). Stand-up comedy routine about bad science. Event occurs at 10:31. Retrieved 2024-11-17.
40. ^ Mould, Steve (2023-11-06). A cast saw on human skin. Event occurs at 7:22. Retrieved 2024-11-13.
41. ^ Mould, Steve (2024-03-14). world record calculation of tau by hand. Retrieved 2024-11-13.
42. ^ Khan, Sal (2011-07-11). Tau versus pi | Graphs of trig functions | Trigonometry | Khan Academy. Retrieved 2024-11-24.
43. ^ Sanderson, Grant (2018-03-14). How pi was almost 6.283185... Retrieved 2024-11-24.
44. ^ Sanderson, Grant (2019-07-07). e^(iπ) in 3.14 minutes, using dynamics | DE5. Event occurs at 3:08. Retrieved 2024-11-24.
45. ^ Munroe, Randall. "Pi vs. Tau". xkcd. Retrieved 1 November 2024.
46. ^ Munroe, Randall. "Symbols". xkcd. Retrieved 1 November 2024.
47. ^ Weinersmith, Zachary. "Fresh". Saturday Morning Breakfast Cereal. Retrieved 2 November 2024.
48. ^ Weinersmith, Zachary. "Better than Pi". Saturday Morning Breakfast Cereal. Retrieved 2 November 2024.
49. ^ Weinersmith, Zachary. "Social". Saturday Morning Breakfast Cereal. Retrieved 2 November 2024.
50. ^ Marciuliano, Francesco. "Sally Forth Comic Strip 2018-10-13". Comics Kingdom. Retrieved 13 November 2024.
51. ^ "Fun & Culture – MIT Facts". Massachusetts Institute of Technology. Retrieved 2 November 2024.
52. ^ Harremoës, Peter (2017). "Bounds on tail probabilities for negative binomial distributions". Kybernetika. 52 (6): 943–966\. arXiv:1601.05179. doi:10.14736/kyb-2016-6-0943. S2CID 119126029.
53. ^ "Supported Functions". help.desmos.com. Archived from the original on 2023-03-26. Retrieved 2023-03-21.
54. ^ Naumovski, Jovana (2022-08-05). "iOS 16 Has a Hidden Unit Converter for Temperatures, Time Zones, Distance, and Other Measurements". Gadget Hacks. Retrieved 21 October 2024.

## External links

[edit]

* The Tau Manifesto

*[c.]: circa