Al-Kashi’s constant τ

Al-Kashi’s constant τ

Peter Harremoës

March 3, 2012
Here I collect some facts about al-Kashi’s constant that was appearently fist used by the Persian mathematician Jamshid al-Kashi. A previous version of this page was entitled Gregory’s constant, but since then I have learned that al-Kashi studied this number about 300 years before Gregory. On this page I will denote al-Kashi’s constant by τ (tau), but for those readers who do not like τ as notation for al-Kashi’s constant, I suggest to use ϖ (varpi).

1 Definition of Al-Kashi’s constant

Definition Al-Kashi’s constant τ is defined as the circumference of a unit circle.

As a consequence al-Kashi’s constant equals the circumference divided the radius for any circle. Since Archemedes’ constant π equals the circumference divided by the diameter we have that τ = 2π.

2 Decimal expansion

The first part of the digital expansion of al-Kashi’s constant is

τ =
        6.28318530717958647692528676655900576839433879875021
        16419498891846156328125724179972560696506842341359
        64296173026564613294187689219101164463450718816256

It is belived that τ is a normal number where any digit or string of digits asymptotically appear with the same frequency. The Feynman point of τ begins at digit 761 and consist of a sequence of seven 9s. The similar Feynman point of π has only six 9s.

3 Continued fraction

Like Archimedes’ constant π one can expand al-Kashi’s constant τ as a continued fraction. It is

 ----------------1---------------
        τ = 6+ 1 3 + ---------------1------------- 1+
        ------------------------- 1 + ----------1----------- 7+
        ---------1-------- ------1-------- 2 + 1 146+ ---------- 3 +
        --1--- 6+ ⋅⋅⋅

This leads to the following rational approximations of al-Kashi’s constant.





Approximation described by year Dec. exp.








6 The Bible 1st millenium BC 6




613

6.33




614 Babylonian math c. 1 600 BC 6.250




627 Archimedes c. 250 BC 6.2857




61553-

6.28302




613213 Zǔ Chongzhi 5th century AD 6.28318 584




6146685751

6.28318 53060




61449079366

6.28318 53072 37




683912514547

6.28318 53071 741





4 Angle measurements

When angles are measured in radians, τ corresponds to one turn (or 360). Similarly τ2 corresponds to a halfturn (straight angle, 180), and τ4 corresponds to a quaterturn (right angle, 90). For circular sectors we have that a semi-circle corresponds to τ2, a quadrant corresponds to τ4, a quintant correspomnds to τ5, a sextant corresponds to τ6, and a octant corresponds to τ8. Thus, when using τ there is no dichotomy between measuring angles in radians, measuring angles in turns, and the names of circular sectors.

A turn can be subdivided in centiturns (cτ) and milliturns (mτ). A centiturn corresponds to 3.6, which can also be written as 336. A milliturn corresponding to an angle of 0.36°, which can also be written as 21’36". Pie charts illustrate proportions of a whole as fractions of a turn. Each one percent is shown as an angle of one centiturn. Angles can be measured in centiturns by use of a centiturn protractor. Like degrees centiturns cannot be constructed by ruler and compass. A deciturn can be constructed because a pentagon is constructable and the angles can be divided in two.

Binary subdivisions of a turn are often used. Thus, 132 turn is called a point and has been popular in navigation. In programming one often uses 1256 = 2-8 turn (a binary radian) or 165336 = 2-16 turn as units. In this way an angle can be stored as integers in one or two bytes, and addition of angles is simplifyed because overflow does not create any problems. Binary subdivisions are constructable by ruler and compass.

5 Formulas

A lot of formulas simplify by using al-Kashi’s constant rather than Archimedes’ constant. A few formulas should simplify by changing to the circle constant η = τ4. Below you will find some important formulas. I have used a happy smiley if a formula has simplified and a sad smiley if a formula has become more complicated. No smiley means that there have not been any significant change in complexity. I have not included a list of integrals since all kinds of factors appear giving no clear preference to any special choice of circle constant.

5.1 Geometry

Circumference of a circle

τR.

Area of disc

1 2
        2 τR

become a special case of the formula for the area of a circular sector 12θR2.

Surface area of sphere

2τR2.

Volume of ball

2 3
        3τR .

Surface of a spherical segment

τhR.

Volume of spherical sector

τhR2
        --3--

Volume of spherical segment

τ-h2(3R- h)
        6

Surface area of torus

τ2R1R2.

Volume of torus

τ2 2
        -2 R1R 2.

Volume of d dimensional unit ball when d is even

 τd∕2
        2⋅4...d.

Volume of d dimensional unit ball when d is odd

 2τ⌊d∕2⌋
        1⋅3...d.

Gauss-Bonnet formula

∫ M KdA = τχ (M ) .

What is normally called the 2π theorem states: A Dehn filling of M with each filling slope greater than τ results in a 3-manifold with a complete metric of negative sectional curvature.

5.2 Complex numbers and complex analysis

Euler’s formula

τi
        e = 1.

The solutions to the equation zn = 1 are

 kτi∕n
        z = e ,k = 1,2,⋅⋅⋅,n.

Logarithms in the complex plane

lnz = ln r+ (θ+ nτ)i,n ∈ ℝ.

Cauchy’s integral formula

 ∮
        f (z0) = 1 f-(z)-dz, τi z - z0

 ∮
        f(n)(z) = n! --f-(z)--dz. 0 τi (z - z0)n+1

Residues

1 ∮ ∑n
        τi f (z)dz = Rz=easkf (z). k=1

Laurant series

 ∞∑ n
          f (z) = cn(z - a) , n=-∮∞ cn = 1- --f-(z)n+1dz. τi (z - z0)

5.3 Trigonometric function, oscilations, and harmonic analysis

Periodicity of trigonometric functions

 sin (x + τ) = sin (τ), cos(x +
        τ) = cos(x),
        tan(x + τ) = tan (x). 2

Derived functions

d ( τ )
          dx sin(x) = sin x+ 4- ,
          d ( τ)
          dx cos(x) = cos x+ 4-.
Harmonic oscillator
 τ-
        ω = T.

Kepler’s third law constant, relating the orbital period (T) and the semimajor axis (a) to the masses (M and m) of two co-orbiting bodies

(τ-)2 3 2 3 T a = ω a = G(M + m).

The circle group

T = ℝ∕τℤ.

Fourier series

f (x) = ∑ ∞ c ⋅einx,
        c = 1 ∫τnf=-(x∞)e-ninxdx.
        n τ 0

Fourier integrals

 1∫ ∞ iωt
        f(t) = τ∫ -∞∞F (ω)e dω, F(ω) = -∞f (t)e-iωtdt.

5.4 Probability and statistics

Density of Gaussian distribution

 1 ( x2)
        -1∕2 exp --- .
        τ 2

The error function

 ∫ ( )
        -1- x t2
        τ1∕2 0 exp -2 dt.

Density of inverse Gaussian distribution

 1 ( (x - μ )2)
        ----1∕2-exp -----2-- .
        (τ x3) 2μ x

Density of lognormal distribution

 ( )
        --1--- (lnx)2-
        x ⋅τ 1∕2 exp - 2 .

Cauchy distribution

2 1
        ---2---.
        τ x + 1

Density of von Mises distribution

 1
        τ-I-(κ) exp (κcos(x)) . 0

Buffon needle experiment gives probability

4
        τ.

Asymptotic minimax redundancy of a d-dimensional random variable with distribution Pθ

 ∫
        dln( n) +ln (detI(θ)1∕2dθ).
        2 τ

5.5 Physics

Planck’s constant

 h
        ℏ = τ.

Einstein’s field equation

 gikR- 4τG-
        Rik - 2 + Λgik = c4 Tik,

where the cosmological constant Λ is given by

Λ = 4τGρvac.

The reactance of an inductor is

τfL.

The susceptance of a capacitor is

τfC.

Coulomb’s law for the electric force, describing the force between two electric charges, q1 and q2, separated by the distance r (with ϵ0 representing the vacuum permittivity of free space)

 -|q1q2|-
        F = 2τε0 ⋅r2.

Magnetic permeability of free space relates the production of a magnetic field in a vacuum by an electric current in units of Newtons (N) and Amperes (A):

μ0 = 2τ ⋅10-7N∕A2.

5.6 Miscellaneous

Stirling’s approximation

n! ≈ (nτe)1∕2nne-n.

The Riemann function evaluated for a positive even integer n

 τn-
        ζ(n) = 2n!Bn∕2.

Functional equation for the Riemann function

 ( )
        ζ(1- s) =-2 sin τ (1--s) Γ (s)ζ(s). τs 4

Euler’s reflection formula

Γ (z)Γ (1- z) =---τ∕2---.
        sin(τz∕2)

Jacobi’s identities for the theta function involves in its normal formulation both π and τ, that in this case denotes the half period ratio.

6 Programming languages

For many applications in computer programming it is convenient to use al-Kashi’s constant rather than Archimedes’ constant. Therefore several programming languages have a name for al-Kashi’s constant.




Program 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




In the Haskell programming language there is now a module that defines the constant τ.

7 The symbol τ

The symbol “τ” is the 19th letter in the Greek alphabet and denotes ’t’-sound. It is not an ASCII character, but is available in most modern text processing systems.



Typesetting system code




LATEX \tau


Unicode U+03C4 or U+F074


HTML entity τ


HTML decimal τ


HTML Hex τ



In MSWord and OpenOffice “τ” can be inserted by choosing the font “Symbol” and typing “t”.

8 Other meanings of τ

Like all other letters in the Latin and Greek alphabet the letter τ is used in different ways in different parts of mathematics and physics.

9 Alternative symbols

Several symbols have been proposed to denote al-Kashi’s constant.

In German speaking languages there have been some attempts to introduce “pla” (from Latin plenus angulus) as an abbreviation for a turn but at present the word “Vollwinkel” is used without abbreviation and without SI prefixes.

In 1994 Dhananjay Ostawal proposed to use omega as symbol for al-Kashi’s constant. This proposal was presented at the 63rd annual conference of the Indian Mathematical Society in 1997

Robert Palais wrote an article in 2001 entitled “Pi is wrong!” [Pal01] where he proposed to have a symbol and suggested to use a ’three-legged pi’ with the LATEX code \pi\mskip-7.8mu\pi . This symbol did not get any popularity because it is only possible to write it using LATEX. Therefore several other symbols have been proposed.

The symbols ϖ (varpi) and sampi have been proposed. Both these symbols are variations of π and are seldomly used so they will not create any notational conflicts. Varpi is available in most programs that can write Greek letters. Sampi can be written in LATEX using the code \sampi when the babel package and the teubner package are used.

The symbol for registrered trademark ®; has been proposed because it contains both a circle and a R that may refer to the radius, and similarly has been proposed.

The Greek letter τ was proposed independently by several people. In 2010 Michael Hartl launced a Tau Manifesto where he advocated for using τ and declared June 28th (6.28) as tau-day [Har10].

Thomas Cool has published two textbooks on geometry that used al-Kashi’s constant [Col08Col11]. As symbol for al-Kashi’s constant he used Θ.

10 History of circle constants

During the history of mathematics various circle constants have been used.

Babylonian constant The old babylonians used the circumference of a regular hexagon divided by the circumference of the circumscribed circle. The approximate value they used was 5760 + 36602 .

Archimedes’ constant This constant is defined as the circumference of a circle divided by its diameter. Since ancient times round objects have been characterized by their diameter. At an early time it was realized that the circumference could be calculated by multiplying the diamter by a certain number. Archimedes proved that it is the same constant one need to use for calculating the area of a circle. Before that sometimes different constants were used to calculate the area and circumference of a circle. The first use of π on its own with its present meaning was by William Jones in 1706. Jones introduces π as

1Periphery(π)
        2

and used the following formulas for circumference and area of a circle

c = πd 1 2 2
          α = 4πd = c ÷ 4π
where α denoted the area.

Oughtred’s constant William Oughtred used for the diameter of a circle divided by its circumference as circle constant in 1647. It was denoted δπ and he used this ’fraction’ as one symbol rather than a numerator divided by a denomination.

Al-Kashi’s constant Ramshid Al-Kashi used the circumference of a circle divided by its radius as circle constant in Treatise on the Circumference published 1424. The idea of using this constant rather than Archimedes’ constant may have been used earlier among Islamic mathematicians than al-Kashi, but al-Kashi calculated it with higher precision than any previous mathematician. In Europe D. Gregory seems to be the first to use al-Kashi’s constant.

Eagle’s constant Albert Eagle published a book on elliptic functions that introduced a lot of non-standard notation [Eag58]. For instance he used “τ” as notation for π2, but it should be noted that π still appear in many of his formulas. Eagle’s notational proposals have never been adapted. Recently, David Butler has proposed to use the symbol η to denote Eagle’s constant. In geometry the idea of using a right angle as unit dates back to Euclid. In Germany and Schwitzerland the symbol has been used to denote a right angle and it has been officially recognized as a unit for angle measurements in the period 1970-1996.

11 History

The Persian mathematician Jamshid al-Kashi seems to have been the first to use the circumference divided by the radius as circle constant rather than Archimedes’ constant. In Treatise on the Circumference published 1424 he calculated the circumference of a unit circle to 9 sexagesimal places, converted that into 16 decimal places. It took about 200 years before a more precise circle constant was calculated by Ludolph van Ceulen. In 1697 David Gregory used πρ to denote the circumference of a circle divided by its radius, and he used this ’fraction’ as one symbol rather than a numerator divided by a denomination. The first use of π on its own with its present meaning was by William Jones in 1706. It took almost 100 years before the notation π became standard notation. For instance M. Nicole [Nic47] did not use any special symbol for the circle constant but made tables of the circumference of inscribes and circumscribed polygons of the unit circle. This gave 6.2831853070319616 and 6.2831853072678912 as lower and upper bounds on the circumference. Leonard Euler adopted the symbol π in 1737 [Eul37, Thm. 3, p. 165]. Because of a very influential book on analysis by Euler [Eul48, Chapter VII] and his prestige in general, mathematicians have followed Euler in the use of π. For instance T. Bugge in 1797 [Bug95, p. 237-239] describes the idea of finding the value of the circumference divided by the radius by inscribing and circumscribing a regular polygons leading to the value 6.283185307. As a consequence, he writes, the circumference divided by the diameter is equal to 3.141592653. Bugge then explains that this number was studied in more detail by L. Euler in 1737 and is denoted as π. After π had become standard notation, some mathematicians have used 2π as if it was one symbol. For instance H. Laurant always wrote 2π4 rather than π2 [Lau89].

The idea of using centiturns and milliturns as units was introduced by the British astronomer and science writer Sir Fred Hoyle [Hoy62].

The idea of using τ as symbol for al-Kashi’s constant was first discussed in an unpublished manuscript by Joseph Lindenberg in 1992 [Lin11]. Dhananjay Ostawal from Pune, India claims that he sought a copyright for ’omega’ to denote al-Kashi’s constant in 1997 and the idea was presented in a paper at the annual conference of Indian Mathematical Society in 1997. Robert Palais wrote an article in 2001 entitled “Pi is wrong!” [Pal01] where he proposed to have a symbol and suggested to use a ’three-legged pi’ with the LATEX code \pi\mskip-7.8mu\pi . This symbol did not get any popularity because it is only possible to write it using LATEX. After the paper of Palais a number of people have proposed to replace the three legged pi by another more convenient symbols as for instance ϖ, or τ that has been proposed independently by several people including the author of this page [Fre07wee10]. In 2010 Michael Hartl launced a Tau Manifesto where he advocated for using τ [Har10] and declared June 28th to be Tauday. In 2008 Thomas Cool published the first textbook on geometry using al-Kashi’s constant [Col08Col11].

The Greek character τ comes from a similar character in the Phoenician alphabet and is derived from a cross. The Greeks aslo took over the name “tau” from the Phoenicians and at that time the original meaning of the word was already forgotten.

The word turn originates from Old English tyrnan and turnian. It comes from Medieval Latin tornare, from Latin, to turn on a lathe, from Greek τóρνoσ a ’lathe’. The word was influenced by Anglo-French turner, tourner to turn, from Medieval Latin tornare, akin to Latin tenere to rub. The geometric notion of a turn has its origin in the sailors terminology of knots where a turn means one round of rope on a pin or cleat, or one round of a coil. For knots the English terms of single turn, round turn and double round turn do not translate directly into the geometric notion of turn, but in German the correspondence is exact.

References

[Bug95]   T. Bugge. De første Grunde til Regning, Geometrie, Plan-Trigonometrie og Landmaaling. S. Poulsens Forlag, Copenhagen, 1795.

[Col08]    T. Colignatus. Trigonometry reconsidered. Measuring angles in unit meter around and using the unit radius functions Xur and Yur. T. Colignatus, 2008.

[Col11]    T. Colignatus. Conquest of the Plane. Consultancy & Econometrics, 2011.

[Eag58]   A. Eagle. Elliptic Functions as They Should Be: An Account, with Applications, of the Functions in a New Canonical Form. Galloway & Porter, Cambridge, 1958.

[Eul37]    L. Euler. Variae observationes circa series infinitas. Commentarii academiae scientiarum Petropolitanae, 9:160–188, 1737.

[Eul48]    L. Euler. Introducio in Analysia Infinetarium, volume 1. Lausanne, 1748.

[Fre07]    Freiddy. pi & 2 pi. Posted on the discussion forum of MathKB, Jan. 2007.

[Har10]   M. Hartl. The tau manifesto. Website, June 28 2010.

[Hoy62]   F. Hoyle. Astronomy. A history of man’s investigation of the universe. Crescent Books, Inc, London, 1962.

[Lau89]   H. Laurant. Traité D’Algebra. 4th. edition edition, 1889.

[Lin11]    J. Lindenberg. Tau before it was cool, July 2011. The site contains an unpublished manuscript from 1992.

[Nic47]     M. Nicole. Dans lequel on détermine en quantités incommensurables & en parties décimales, les valeurs des cótes & des espaces, de la suite en progression double, des polygones régulieres, inscrits & circonscrits au circle. Histoire de l’Académie royale des sciences avec les mémoires de mathématique et de physique tirés des registres de cette Académie., pages 437–448, 1747. Published 1751.

[Pal01]    R. Palais. π is wrong. Mathematical Intelligencer, 23(3):78, 2001.

[wee10]    william e emba. π really is wrong! Posted at the discussion forum of Good Math, Bad Math, Dec. 2010.

12 Link collection

Below are a number of links related to the use of the circle constant τ.

12.1 Webpages

Pi is wrong! Page edited by Robert Palais.

Tau Manifesto Page edited by Michael Hartl.

Tau before it was Cool Joseph Lindenberg describes that he proposed to use τ to denote al-Kashi’s constant allready in 1991.

Tauism, Pi and the fundamental particles Page edited by Michael Taylor.

http://www.math.utah.edu/~palais/cose.html

The Last Stoic

Tau and dozenal notation

The way of the tau at hexnet.

Square CircleZ The author of this page proposes to use the symbol for registered trademark ®; as symbol for al-Kashi’s constant.

Happy Tau (6.28) day by Alex Masterley.

Tau Day 6.28 Page where it is proposed to use the symbol sampi to denote al-Kashi’s constant.

The Pi Manifesto Page edited by Michael Cavers, where he argues in favor of π.

Pi is wrong! Long live Tau! Page edited by Dimitri Brant.

Mathematical constant Archimedes Θ = 2π = 6.2831853 Blog by Thomas Cool.

12.2 Videos and animations

Fun video by Vi Hart describing the problem of cutting a pie when using pi.

Pi is wrong! Here comes Tau day. Video by Kevin Houston.

Animation of cosine labelled by τ.

Animation of sine labelled by τ.

Pi may be wrong, but so is Tau! Video by David Butler, where he argues that it is more reasonable to use η = τ∕4 (Eagle’s constant) as circle constant.

Tau versus Pi by Khan Academy.

Pi ain’t all that by Robert Dixon.

12.3 News groups

The idea of using τ to denote al-Kashi’s constant is discussed at numerous math oriented news groups and only afew are listed here.

Is Pi wrong? Is Tau the correct circle constant? Stated as a question at Quora.com with a lot of people stating their opinion.

Spiked Math Forums

TeamLiquid Discussion forum where julianto proposes to use the symbol .

12.4 Articles in news medias

The use of τ has got a lot of attention in news medias.

Pi’s nemesis: Mathematics is better with tau Interview with M. Hartl in New Scientist.

Why we have to get rid of pi for the sake of good math Interview with M. Hartl at i09 webpage.

On Pi-Day, ’pi’ is under attack Article by CNN.

Mathematics Upstarts Look to Replace Pi With New Circle Constant Article at Daily Tech.

Life of pi over? ’Tau’ may set calculations aright Article in the Times of India.

Tau Day: An Even More Fundamental Holiday Than Pi Day by Alessondra Springmann.

’Tau day’ marked by opponents of maths constant pi BBC News.

Forget Pi, Here Comes Tau Using a new constant would simplify things, say experts. Article by Evann Gastaldo from the Newser Staff.

Happy Tau Day, everybody! Article at CBSNews.

Happy Tau Day! Article in International Business Times.

Tau Day today: Mathematicians show their work Article in Oregon Live.

On National Tau Day, Pi Under Attack Fox News.

Push to roll Pi Discussion at ABC in Australia.

Down with ugly pi, long live elegant Tau, physicist urgesArticle in the Star.

Pi’s 4,000-yr reign faces Tau challenge Article in Deccan Chronicle.

Mathematicians Want to Say Goodbye to Pi Yahoo News.

Second Annual Tau Day: Interview and Ideas!

What is Tau Day?

Your number’s up: Why mathematicians are campaigning for pi to be replaced with alternate value tau

Tau Day Generates Controversy Among Math Scholars

Math wars: Debate sparks anti-pi day

Bye Bye Pi: Mathematic Scholars Want To Replace The Circle Constant

Moves to replace Pi with Tau

Pi Is Wrong! Mathematicians Declare Today ’Tau Day’

Life of pi in no danger Experts cold-shoulder campaign to replace with tau

Life of pi over? ’Tau’ may set calculations aright

Pune researcher claims he derived ’tau’ constant first Researcher Dhananjay Ostawal from Pune, India said that he sought a copyright for ’omega’ as a constant derivative for circle in 1997 and that he has presented a paper at the annual conference of Indian Mathematical Society in 1997.

Pi is wrong McGill Daily.

Why we have to get rid of pi for the sake of good math

12.5 Various links

http://breadpig.com/tees/tau/ Here you can buy a τ-shirt.

http://www.zazzle.com/tau_day_shirt-235815786260277849Here you can buy a τ-day shirt.

Yet another tau-shirt page

PI should be 6.283185... A Facebook group.

Tau (The True Circle Constant) Yet another Facebook group.

Tau Day - June 28 A facebook profile of Tau Day.

Half-tau Day The day where τ2 is celebrated.

6.283 A song on how to use τ.