User:Reddwarf2956/6.283185307179

circumference to radius ratio ... A fundamental mathematical constant that is the ratio of a circle's circumference to its radius has no formal symbolism. However, several individuals have recommended that it be symbolized by Greek letter τ tau and other characters of lower popularity τ_∠. Because of the lack of conformity, here we will use τ_∠ to represent the complete circle constant.

It has been observed that the value of τ is more mathematical fundamental than the constant π or pi. Because the functional use of the radius and radians, there is a dependency of the diameter to 2 times the radius which makes π := τ/2, and the number of radians in a complete circle is τ. A measure of an angle using a factional turn in units of τ is simple to use and understand because of its one to one correspondence with the number of radians. Most Scientific applications in mathematics and physics which use the constant π can be converted to measurements in τ using the equation π = τ/2.

A new mathematical constant tau (τ) has been proposed which equals two times π. Its supporters have argued that as the ratio of circle circumference to radius, this new constant is a more natural choice than π and would simplify many formulae.[1][2] While their proposals, which include celebrating 28 June as "Tau day", have been reported in the media, they have not been reflected in the scientific literature.[3][4] ^ Abbott, Stephen (April 2012). "My Conversion to Tauism". Math Horizons 19 (4): 34. doi:10.4169/mathhorizons.19.4.34. ^ Palais, Robert (2001). "π Is Wrong!". The Mathematical Intelligencer 23 (3): 7-8. doi:10.1007/BF03026846, also http://www.math.utah.edu/~palais/pi.pdf

The article has an incorrect citation. The citation for [2] indicates that Bob Palais suggested that tau represent 2pi is incorrect. Palais' article is found at his website here, http://www.math.utah.edu/~palais/pi.pdf and nowhere in the article is mention of tau. However, on Palais' website, http://www.math.utah.edu/~palais/pi, he mentions that he has be contacted by individuals wanting tau to represent 2pi.

^ Hartl, Michael. "The Tau Manifesto". Retrieved 28 April 2012. ^ Palmer, Jason (28 June 2011). "'Tau day' marked by opponents of maths constant pi". BBC News. Retrieved 28 April 2012.

TODO

• Gather more stuff from Harremoes and Palais' pages.

6.283185307179

A fundamental mathematical constant which has it [decimal expansion] approximately equal to 6.283185307179586476925286766559005768394338798750211641949889184615632... the ratio of a circle's circumference to its radius has no formal symbolism. However, several individuals have recommended that it be symbolized by Greek letter τ (/tau/) and other characters of lower popularity. _0τ "ought-tau"

It has been observed that the value of τ is more mathematical fundamental than the constant [π]. Because the functional use of the [radius] and [radians], the dependency of the diameter to 2 times the radius makes π := τ/2, and the number of radians in a complete circle τ. A measure of an angle using a factional [turn] in units of τ is simple to use and understand because of its [one to one correspondence] with the number of radians. Most Scientific applications in mathematics and physics which use the constant [π] or [/pi/] can be converted to measurements in τ using the equation π = τ/2.

It is at the OEIS as A058291, and A019692. A simple search on Google will find many, many more examples of the constant tau by other names and used as 6.2831853.... And, tau is mathematically consistent. Yet here on Wikipedia, there is a pious bias against tau or anything that looks like tau in value. Every where I look for the constant there is no references back to tau, two pi, or even 6.2831853. They all redirect back to pi or do not exist. The above commentators reference tau and know what it is and what its value is, yet you expect me to talk about it as if it was numerology, sorry if any number here is consider numerology that is pi because as a Wikipedia topic it has the lack of consistency. And, this Wikipedia topic "pi" does not reference any of these references above or any of many other places that I did not state. In fact there is the act of hijacking 2pi and saying that it, pi, in a lot of places http://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80 . But, that does not change the problem pi has as stated here earlier, it is inconsistent in that it does not use radius for consistency sake. And, no one wants to agree that tau is not the issue here, it is the unnecessary bias against tau because it is mathematically a better constant to deal with things like pi. Just look at how both pi and tau is used in spherical coordinates. On the x and y axes the use of the maximum value is tau. The maximum angle from the positive z axis is pi. Yet here you insist on C/d and ignore the facts. Just do a search for the number of times an even and odd multiples of pi are used in the period of functions, or just in functions here http://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80 . Now pi is not the first mathematically inconsistent thing. Just look at when 1 was called prime and 2 was called not prime.

The circumference of a circle relates to one of the most fundamental and important mathematical constant in all of mathematics. This constant pi, is represented by the Greek letter π. π has the numerical value of 3.14159 26535 89793 ..., and is defined by two straight line linear correlations. The first correlation is the ratio of a circle's circumference to its diameter and equals π. While the second is the ratio of the diameter and two times the radius, and is used as to convert the diameter to radius in the same ratio as the first correlation, π. Both linear correlations combine in respect with circumference c, diameter d, and radius r to become:

${c}=\pi \cdot {d}={2}\cdot \pi \cdot {r}.\!$

The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science. While the mathematical constant ratio of the circumference to radius, ${c}/{r}=2\cdot \pi$, also has many uses in mathematics, engineering, and science, it is not formally named. These uses include but are not limited to radians, computer programming, and physical constants.

Some special angles in terms of τ

The ratio of any circle's circumference to its radius is a mathematical constant equal to two times the number pi (2π). This number has a value of approximately 6.2831853 and appears in many common formulas, often because it is the period of some very common functions — sine, cosine, e^ix, and others that involve trips around the unit circle. Some individuals have proposed giving this number its own symbol and using that instead of π in mathematics notation. This proposition has been relayed in several news articles, but has not been echoed in scientific publications nor by any scientific authority.

decimal expansion and Continued fraction

The decimal expansion of τ = c/r = 2π is approximately equal to 6.283185307179586476925286766559005768394338798750211641949889184615632...OEIS: A019692. Continued fraction is found in OEIS: A058291 and is

$\tau =6+\textstyle {\frac {1}{3+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{7+\textstyle {\frac {1}{2+\textstyle {\frac {1}{146+\textstyle {\frac {1}{3+\textstyle {\frac {1}{6+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\ddots }}}}}}}}}}}}}}}}}}}}$

Version numbering

The major/minor version numbers of Pugs converges to 2π (being reminiscent of TeX and METAFONT, which use a similar scheme); each significant digit in the minor version represents a successfully completed milestone. The third digit is incremented for each release. The current milestones are:

• 6.0: Initial release.
• 6.2: Basic IO and control flow elements; mutable variables; assignment.
• 6.28: Classes and traits.
• 6.283: Rules and Grammars.
• 6.2831: Type system and linking.
• 6.28318: Macros.
• 6.283185: Port Pugs to Perl 6, if needed.

Advocacy

In an opinion column in The Mathematical Intelligencer, Robert Palais argued that π is "wrong" as a circle measure, and that a better value would be 2π, being the measure of the circle's circumference and the period of the sine, cosine, and complex exponential functions. He suggested a symbol like π but with three legs be used in place of 2π, demonstrating how it simplifies many mathematical formulas.

In popular culture

In 2010, Michael Hartl posted an essay called The Tau Manifesto on his personal website. In it, he proposed using the Greek letter tau (τ) to represent that number instead. Hartl argued that an existing symbol like τ would face fewer barriers to adoption than a new symbol like the "three-legged pi" proposed in the Intelligencer. A number of news outlets reported on "Tau Day", a holiday proposed in The Tau Manifesto' for June 28 to honour 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 [the] 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". Thus 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.

The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the Greek letter π. That letter (and therefore the number π itself) can be denoted by the Latin word pi. In English, π is pronounced as "pie" ( /paɪ/, /ˈpaɪ/). The lower-case letter π (or π in sans-serif font) is not to be confused with the capital letter Π, which denotes a product of a sequence.

The first mathematician to use the Greek letter π to represent the ratio of a circle's circumference to its diameter was William Jones, who used it in his work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics, of 1706.. A facsimile of Jones' text is in Jones' first use of the Greek letter was in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius one. He may have chosen π because it was the first letter in the Greek spelling of the word periphery.

After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler used it in 1736. Before then, mathematicians sometimes used letters such as c or p instead. Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly. In 1748, Euler used π in his widely read work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1") and the practice was universally adopted thereafter in the Western world.

Historical usage of 2π as a constant

• Islamic mathematicians like Jamshid al-Kashi (c. 1380–1429) focused on the circle constant 6.283... although they were fully aware of the work of Archimedes focusing on the circle constant that is nowadays called π.
• William Oughtred used ⁠π/δ⁠ to represent ⁠perimeter/diameter⁠.
• David Gregory used ⁠π/ρ⁠ to represent ⁠perimeter/radius⁠.
• William Jones first used π as it is used today to represent ⁠perimeter/diameter⁠.
• Leonhard Euler adopted this use of π and popularized it.
• Paul Matthieu Hermann Laurent, though never explaining why, treated 2π as if it were a single symbol in Traité D'Algebra by consistently not simplifying expressions like ⁠2π/4⁠ to ⁠π/2⁠.
• Fred Hoyle, in Astronomy, A history of man's investigation of the universe, proposed using centiturns (hundredths of a turn) and milliturns (thousandths of a turn) as units for angles.

Mathematical publications about τ

• Palais, Robert (2001). "π Is Wrong!" (PDF). The Mathematical Intelligencer. 23 (3): 7–8. doi:10.1007/BF03026846. S2CID 120965049.
• Abbott, Stephen (April 2012). "My Conversion to Tauism" (PDF). Math Horizons. 19 (4): 34. doi:10.4169/mathhorizons.19.4.34. S2CID 126179022.

Notable endorsements

• Salman Khan of Khan Academy: Tau versus Pi (2011) and Happy Tau Day! (2012)
• Vi Hart: Pi Is (still) Wrong (2011) and A Song About A Circle Constant (2012)
• Robert Dixon: Pi ain't all that
• Michael Cavers of Spiked Math (and The Pi Manifesto): Math Fact Tuesday: Tau
• MITadmissions.org: I have SMASHING news!
• Eric S. Raymond: Tau versus Pi

Established mathematicians

• Stanley Max: Radian Measurement: What It Is, and How to Calculate It More Easily Using τ Instead of π
• Kevin Houston: Pi is wrong! Here comes Tau Day

Celebration of 2π day before Hartl's manifesto (2010)

• 1998: K.C. Cole (12 March 1998). "Easy as Pi". LA Times. some people even celebrate 2 pi day, June 28 (6/28)
• 2009: Lance Fortnow and William Gasarch (1 July 2009). "2pi-day? Other holiday possibilities!". Computational Complexity. Retrieved 2011-07-24. {{cite web}}: External link in |author= (help)
• 2009: Mathematics (28 June 2009). "2pi Day". Facebook. Retrieved 2011-07-24. {{cite web}}: |author= has generic name (help); External link in |author= (help)CS1 maint: numeric names: authors list (link)
• 2009: Gerald Thurman (6 July 2009). Eating Pie in Pie Town on Two Pi Day (video on YouTube).

Support in programming languages

Programming language	Name	Value
Fortran	TWOPI
OGRE	TWO_PI
OpenGUI	TWO_PI
Java	TWOPI	6.28318 53071 79586
Pascal	TwoPI	6.28318 53071 79586
Processing	TWO_PI	6.28318 53071 79586 47693
Wiring	TWO_PI	6.28318 53071 79586 47693
Extreme Optimization Libraries	TwoPi	6.28318 53071 79586 47692 52867 66558
Haskell (module)
Python (PEP, bug)
Ruby (bug)

The constant tau is used in programming languages as names like twopi, two_pi, TWOPI, TWO_PI, ... for example MATHEMATICA: 2Pi, FORTRAN: http://naif.jpl.nasa.gov/pub/naif/toolkit_docs/FORTRAN/spicelib/twopi.html . Or, as here a way numbering software versions: https://en.wikipedia.org/wiki/Pugs#Version_numbering .

Textbooks

• T. Colignatus. Trigonometry reconsidered. Measuring angles in unit meter around and using the unit radius functions Xur and Yur. T. Colignatus, 2008.
• T. Colignatus. Conquest of the Plane. Using The Economics Pack Applications of Mathematica for a didactic primer on Analytic Geometry and Calculus. Consultancy & Econometrics, March 2011. ISBN 978-90-804774-6-9.

Tau conversion hubs

• The Tau Manifesto by Michael Hartl (since 28 June 2010)
• Al-Kashi’s constant τ by Peter Harremöes (since... 2010? confirm)
• Pi is Wrong! by Robert Palais (since September 2001)
• Tau Before It Was Cool by Joseph Lindenberg (since... 2010? confirm)

Neat stuff

• Circle is formally defined as all points at same distance —radius, not diameter— of a center point.
  • Main conventions regarding circles are based in radius: unit circle, radians, standard circle formulas.
• Tau day is a perfect day, because 6 and 28 are the two first perfect numbers.
  • 6:28 is a more convenient time to start celebrating than 3:15 (besides being after the actual start of the day rather than midnight)
• Feynman point better in τ: starts earlier (761 digits after the radix mark rather than 762 in π), is longer (7 nines rather than 6 nines in π), and thus more improbable (0.008% vs. 0.08% in π)
• Decimal expansion of 2*Pi and related links at the On-Line Encyclopedia of Integer Sequences
• "You can't eat pie on Tau Day!"
  1. First of all, the pun is not that strong of an argument: it only works because π is mispronounced "pie" in English, rather than "pea" as in the original Greek and most other languages. Even if people decided to eat peas instead, the pun would still only work for English speakers, which doesn't play well with the universality of a mathematical constant.
  2. Second, pi radians is half a circle, not a full circle as most pies are, which weakens the association. If this inconvenience is ignored, then this ends up actually backfiring into favoring Tau, since on Tau day you can eat two pies!

See also

• Turn (geometry)

• OEIS: A019692 Decimal expansion of 2*Pi and related links at the On-Line Encyclopedia of Integer Sequences
• OEIS: A058291 Continued fraction for 2 Pi.and related links at the On-Line Encyclopedia of Integer Sequences

http://en.wikipedia.org/w/index.php?title=Tau_(2%CF%80)&oldid=482586630 Category:Pi

References

• Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. English translation by Catriona and David Lischka.
• Ayers, Frank (1964). Calculus. McGraw-Hill. ISBN 978-0-070-02653-7.
• Berggren, Lennart; Borwein, Jonathan; Borwein, Peter (1997). Pi: a Source Book. Springer-Verlag. ISBN 978-0-387-20571-7.
• Beckmann, Peter (1989) [1974]. History of Pi. St. Martin's Press. ISBN 978-0-88029-418-8.
• Borwein, Jonathan; Borwein, Peter (1987). Pi and the AGM: a Study in Analytic Number Theory and Computational Complexity. Wiley. ISBN 978-0-471-31515-5.
• Boyer, Carl B.; Merzbach, Uta C. (1991). A History of Mathematics (2 ed.). Wiley. ISBN 978-0-471-54397-8.
• Bronshteĭn, Ilia; Semendiaev, K. A. (1971). A Guide Book to Mathematics. H. Deutsch. ISBN 978-3-871-44095-3.
• Eymard, Pierre; Lafon, Jean Pierre (1999). The Number Pi. American Mathematical Society. ISBN 978-0-8218-3246-2., English translation by Stephen Wilson.
• Joseph, George Gheverghese (1991). The Crest of the Peacock: Non-European Roots of Mathematics. Princeton University Press. ISBN 978-0-691-13526-7.
• Posamentier, Alfred S.; Lehmann, Ingmar (2004). Pi: A Biography of the World's Most Mysterious Number. Prometheus Books. ISBN 978-1-59102-200-8.
• Reitwiesner, George (1950). "An ENIAC Determination of pi and e to 2000 Decimal Places". Mathematical Tables and Other Aids to Computation. 4 (29): 11–15. doi:10.2307/2002695. JSTOR 2002695.
• Roy, Ranjan (1990). "The Discovery of the Series Formula for pi by Leibniz, Gregory, and Nilakantha". Mathematics Magazine. 63 (5): 291–306. doi:10.2307/2690896. JSTOR 2690896.
• Schepler, H. C. (1950). "The Chronology of Pi". Mathematics Magazine. 23 (3). Mathematical Association of America: 165–170 (Jan/Feb), 216–228 (Mar/Apr), and 279–283 (May/Jun). doi:10.2307/3029284. JSTOR 3029284.. issue 3 Jan/Feb, issue 4 Mar/Apr, issue 5 May/Jun

News (not published around pi day or tau day, or otherwise significant)

• Geoffrey Fattah (4 July 2011). "University of Utah math professor starts international math movement". Deseret News. (they actually contacted Palais, and took a picture of him wearing a Tau Day T-shirt)
• Debra Black (28 June 2011). "Down with ugly pi, long live elegant Tau, physicist urges". Toronto Star. (they seem to have actually interviewed Hartl)
• Alasdair Wilkins (2 February 2011). "Why we have to get rid of pi for the sake of good math". io9. (interviewed Hartl, and wasn't published around Tau day)

External links

• Robert Dixon (Author) (18 June 2011). Pi ain't all that (flv) (YouTube).
• Vi Hart (Author) (14 March 2011). Pi Is (still) Wrong (flv) (YouTube).
• Salman Khan (11 July 2011). Tau versus Pi (flv) (YouTube). Khan Academy.

Amira, Dan (14 March 2011). "Pi Is Very Slowly and Nerdily Going Out of Style". New York. Retrieved 2012-04-03.

Anthony, Sebastian (28 June 2011). "Down with pi! Today is Tau Day". ExtremeTech. Retrieved 2012-04-03.

Black, Debra (28 June 2011). "Down with ugly pi, long live elegant Tau, physicist urges". Toronto Star. Retrieved 2012-04-03.

Chant, Ian (28 June 2011). "Happy Tau Day!". Science Friday. Retrieved 2012-04-03.

Connelly, Claire (28 June 2011). "What does Tau sound like? Pi, except better". The Advertiser (Adelaide). Retrieved 2012-04-03.

Fattah, Geoffrey (4 July 2011). "University of Utah math professor starts international math movement". Deseret News. Retrieved 2012-04-03.

Haught, Nancy (28 June 2011). "Tau Day today: Mathematicians show their work". The Oregonian. Retrieved 2012-04-03.

Henderson, Mark (28 June 2011). "Maths mutineers say number's up for pi". The Australian. Retrieved 2012-04-03.

Hudson, John (28 June 2011). "Down with Pi! The Math Nerds Behind the Tau Movement". The Atlantic Wire. Retrieved 2012-04-03.

Landau, Elizabeth (28 June 2011). "In case Pi Day wasn't enough, it's now 'Tau Day' on the Internet". CNN. Retrieved 2012-04-03.

Palmer, Jason (28 June 2011). "'Tau day' marked by opponents of maths constant pi". BBC News. Retrieved 2011-04-03.

Peddie, Clare (28 June 2011). "Moves to replace Pi with Tau". The Advertiser (Adelaide). Retrieved 2012-04-03.

Shen, Anqi (29 June 2011). "Math wars: Debate sparks anti-pi day". PhysOrg. Retrieved 2012-04-03.

Springmann, Alessondra (June 28, 2011). "Tau Day: An Even More Fundamental Holiday Than Pi Day". PC World. Retrieved 2011-04-03.

Tagg, Cori (28 June 2011). "Happy National Tau Day, brainiacs". Deseret News. Retrieved 2012-04-03.

Tovrov, Daniel (28 June 2011). "Happy Tau Day!". International Business Times. Retrieved 2012-04-03.

Usigan, Ysolt (June 28, 2011). "Happy Tau Day, everybody!". Tech Talk. CBS News. Retrieved 2012-04-03.

Wolchover, Natalie (June 29, 2011). "Mathematicians want to say goodbye to pi". Life's Little Mysteries. MSNBC. Retrieved 2012-04-03.

"Life of pi over? 'Tau' may set calculations aright". The Times of India. June 29, 2011. Retrieved 2012-04-03.

"On National Tau Day, Pi Under Attack". Fox News Channel. NewsCore. June 28, 2011. Retrieved 2011-04-03.

"Tau: Is two pi better than one?". Today (BBC Radio 4). 28 June 2011. Retrieved 2012-04-03.

"Your number's up: Why mathematicians are campaigning for pi to be replaced with alternate value tau". Daily Mail. June 29, 2011. Retrieved 2012-04-03.

A new mathematical constant tau (τ) has been proposed which equals two times π. Its supporters have argued that as the ratio of circle circumference to radius, this new constant is a more natural choice than π and would simplify many formulae. While their proposals, which include celebrating 28 June as "Tau day", have been reported in the media, they have not been reflected in the scientific literature.