Turn (angle)

It has been suggested that Draft:Tau (mathematical constant) be merged into this article. (Discuss) Proposed since July 2024.

Turn
Counterclockwise rotations about the center point starting from the right, where a complete rotation corresponds to an angle of rotation of 1 turn.
General information
Unit of	Plane angle
Symbol	tr, pla, rev, cyc
Conversions
1 tr in ...	... is equal to ...
radians	2π rad ≈ 6.283185307... rad
milliradians	2000π mrad ≈ 6283.185307... mrad
degrees	360°
gradians	400g

The turn (symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center. One turn is equal to 2π radians, 360 degrees or 400 gradians. As an angular unit, one turn also corresponds to one cycle (symbol cyc or c)^[1] or to one revolution (symbol rev or r).^[2] Common related units of frequency are cycles per second (cps) and revolutions per minute (rpm).^[a] The angular unit of the turn is useful in connection with, among other things, electromagnetic coils (e.g., transformers), rotating objects, and the winding number of curves. Divisions of a turn include the half-turn and quarter-turn, spanning a straight angle and a right angle, respectively; metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc.

Because one turn is ${\displaystyle 2\pi }$ radians, some have proposed representing 2π with a single letter. In 2010, Michael Hartl proposed using the Greek letter ${\displaystyle \tau }$ (tau), equal to the ratio of a circle's circumference to its radius (${\displaystyle 2\pi }$) and corresponding to one turn, for greater conceptual simplicity when stating angles in radians.^[3] This proposal did not initially gain widespread acceptance in the mathematical community,^[4] but the constant has become more widespread,^[5] having been added to several major programming languages and calculators.

In the ISQ, an arbitrary "number of turns" (also known as "number of revolutions" or "number of cycles") is formalized as a dimensionless quantity called rotation, defined as the ratio of a given angle and a full turn. It is represented by the symbol N. (See below for the formula.)

There are several unit symbols for the turn.

The German standard DIN 1315 (March 1974) proposed the unit symbol "pla" (from Latin: plenus angulus 'full angle') for turns.^[6][7] Covered in DIN 1301-1 [de] (October 2010), the so-called Vollwinkel ('full angle') is not an SI unit. However, it is a legal unit of measurement in the EU^[8][9] and Switzerland.^[10]

The scientific calculators HP 39gII and HP Prime support the unit symbol "tr" for turns since 2011 and 2013, respectively. Support for "tr" was also added to newRPL for the HP 50g in 2016, and for the hp 39g+, HP 49g+, HP 39gs, and HP 40gs in 2017.^[11][12] An angular mode TURN was suggested for the WP 43S as well,^[13] but the calculator instead implements "MULπ" (multiples of π) as mode and unit since 2019.^[14][15]

Many angle units are defined as a division of the turn. For example, the degree is defined such that one turn is 360 degrees.

Using metric prefixes, the turn can be divided in 100 centiturns or 1000 milliturns, with each milliturn corresponding to an angle of 0.36°, which can also be written as 21′ 36″.^[16][17] A protractor divided in centiturns is normally called a "percentage protractor". While percentage protractors have existed since 1922,^[18] the terms centiturns, milliturns and microturns were introduced much later by the British astronomer Fred Hoyle in 1962.^[16][17] Some measurement devices for artillery and satellite watching carry milliturn scales.^[19][20]

Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points, which implicitly have an angular separation of 1/32 turn. The binary degree, also known as the binary radian (or brad), is ⁠1/256⁠ turn.^[21] The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte. Other measures of angle used in computing may be based on dividing one whole turn into 2ⁿ equal parts for other values of n.^[22]

The number 2π (approximately 6.28) is the ratio of a circle's circumference to its radius, and the number of radians in one turn.

The meaning of the symbol ${\displaystyle \pi }$ was not originally fixed to the ratio of the circumference and the diameter. In 1697, David Gregory used ⁠π/ρ⁠ (pi over rho) to denote the perimeter of a circle (i.e., the circumference) divided by its radius.^[23][24] However, earlier in 1647, William Oughtred had used ⁠δ/π⁠ (delta over pi) for the ratio of the diameter to perimeter. The first use of the symbol π on its own with its present meaning (of perimeter divided by diameter) was in 1706 by the Welsh mathematician William Jones.^[25][26]

The first known usage of a single letter to denote the 6.28... constant was in Leonhard Euler's 1727 Essay Explaining the Properties of Air, where it was denoted by the letter π.^[27][28] Euler would later use the letter π for the 3.14... constant in his 1736 Mechanica^[29] and 1748 Introductio in analysin infinitorum,^[30] though defined as half the circumference of a circle of radius 1—a unit circle—rather than the ratio of circumference to diameter. Elsewhere in Introductio in analysin infinitorum, Euler instead used the letter π for one-fourth of the circumference of a unit circle, or 1.57... . 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;^[31] afterward, π was standardized as being equal to 3.14... .^[32][33]

Several people have independently proposed using 𝜏 = 2π, including:^[34]

• Joseph Lindenburg (c. 1990)
• John Fisher (2004)
• Peter Harremoës (2010)
• Michael Hartl (2010)

In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of π, which amounts to the number of radians in half a turn, in order to make mathematics simpler and more intuitive. His proposal used a "π with three legs" symbol to denote the constant (${\displaystyle \pi \!\;\!\!\!\pi =2\pi }$).^[35]

In 2008, Robert P. Crease proposed the idea of defining a constant as the ratio of circumference to radius, a proposal supported by John Horton Conway. Crease used the Greek letter psi: ${\displaystyle \psi =2\pi }$.^[36]

The same year, Thomas Colignatus proposed the uppercase Greek letter theta, Θ, to represent 2π.^[37] The Greek letter theta derives from the Phoenician and Hebrew letter teth, 𐤈 or ט, and it has been observed that the older version of the symbol, which means wheel, resembles a wheel with four spokes.^[38] It has also been proposed to use the wheel symbol, teth, to represent the value 2π, and more recently a connection has been made among other ancient cultures on the existence of a wheel, sun, circle, or disk symbol—i.e. other variations of teth—as representation for 2π.^[39]

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, 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. Hartl's Tau Manifesto^[b] gives many examples of formulas that are asserted to be clearer where τ is used instead of π.^[41][42][43] For example, Hartl asserts that replacing Euler's identity e^iπ = −1 by e^iτ = 1 (which Hartl also calls "Euler's identity") is more fundamental and meaningful. He also claims that the formula for circular area in terms of τ, A = ⁠1/2⁠𝜏r², contains a natural factor of ⁠1/2⁠ arising from integration.

Initially, this proposal did not receive significant acceptance by the mathematical and scientific communities.^[4] However, the use of τ has become more widespread.^[5] For example:

• In 2012, the educational website Khan Academy began accepting answers expressed in terms of τ.^[44]
• The constant τ is made available in the Google calculator, Desmos graphing calculator^[45] and in several programming languages such as Python,^[46][47] Raku,^[48] Processing,^[49] Nim,^[50] Rust,^[51] GDScript,^[52] UE Blueprints,^[53] Java,^[54][55] .NET,^[56][57] Odin,^[58] and Julia.^[59] Support is upcoming in Liberty Eiffel.^[60]
• It has been used in at least one mathematical research article,^[61] authored by the τ promoter Peter Harremoës.^[62]
• The iPhone's Convert Angle option expresses the turn as τ.^[63]

The following table shows how various identities appear when τ = 2π is used instead of π.^[64][35] 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	${\displaystyle {\color {orangered}{\frac {\pi }{2}}}{\text{ rad}}}$	${\displaystyle {\color {orangered}{\frac {\tau }{4}}}{\text{ rad}}}$	⁠τ/4⁠ rad = ⁠1/4⁠ turn
Circumference of a circle	${\displaystyle C={\color {orangered}2\pi }r}$	${\displaystyle C={\color {orangered}\tau }r}$	The length of an arc of angle θ is L = θr.
Area of a circle	${\displaystyle A={\color {orangered}\pi }r^{2}}$	${\displaystyle A={\color {orangered}{\frac {1}{2}}\tau }r^{2}}$	The area of a sector of angle θ is A = ⁠1/2⁠θr2.
Area of a regular n-gon with unit circumradius	${\displaystyle A={\frac {n}{2}}\sin {\frac {\color {orangered}2\pi }{n}}}$	${\displaystyle A={\frac {n}{2}}\sin {\frac {\color {orangered}\tau }{n}}}$
n-ball and n-sphere volume recurrence relation	${\displaystyle V_{n}(r)={\frac {r}{n}}S_{n-1}(r)}$ ${\displaystyle S_{n}(r)={\color {orangered}2\pi }rV_{n-1}(r)}$	${\displaystyle V_{n}(r)={\frac {r}{n}}S_{n-1}(r)}$ ${\displaystyle S_{n}(r)={\color {orangered}\tau }rV_{n-1}(r)}$	V0(r) = 1 S0(r) = 2
Cauchy's integral formula	${\displaystyle f(a)={\frac {1}{{\color {orangered}2\pi }i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz}$	${\displaystyle f(a)={\frac {1}{{\color {orangered}\tau }i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz}$	${\displaystyle \gamma }$ is the boundary of a disk containing ${\displaystyle a}$ in the complex plane.
Standard normal distribution	${\displaystyle \varphi (x)={\frac {1}{\sqrt {\color {orangered}2\pi }}}e^{-{\frac {x^{2}}{2}}}}$	${\displaystyle \varphi (x)={\frac {1}{\sqrt {\color {orangered}\tau }}}e^{-{\frac {x^{2}}{2}}}}$
Stirling's approximation	${\displaystyle n!\sim {\sqrt {{\color {orangered}2\pi }n}}\left({\frac {n}{e}}\right)^{n}}$	${\displaystyle n!\sim {\sqrt {{\color {orangered}\tau }n}}\left({\frac {n}{e}}\right)^{n}}$
nth roots of unity	${\displaystyle e^{{\color {orangered}2\pi }i{\frac {k}{n}}}=\cos {\frac {{\color {orangered}2}k{\color {orangered}\pi }}{n}}+i\sin {\frac {{\color {orangered}2}k{\color {orangered}\pi }}{n}}}$	${\displaystyle e^{{\color {orangered}\tau }i{\frac {k}{n}}}=\cos {\frac {k{\color {orangered}\tau }}{n}}+i\sin {\frac {k{\color {orangered}\tau }}{n}}}$
Planck constant	${\displaystyle h={\color {orangered}2\pi }\hbar }$	${\displaystyle h={\color {orangered}\tau }\hbar }$	ħ is the reduced Planck constant.
Angular frequency	${\displaystyle \omega ={\color {orangered}2\pi }f}$	${\displaystyle \omega ={\color {orangered}\tau }f}$

𝜏 has made numerous appearances in culture. It is celebrated annually on June 28, known as Tau Day.^[65] 𝜏 has been covered in videos by Vi Hart,^[66][67][68] Numberphile,^[69][70][71] SciShow,^[72] Steve Mould,^[73][74][75] Khan Academy,^[76] and 3Blue1Brown,^[77][78] and it has appeared in the comics xkcd,^[79][80] Saturday Morning Breakfast Cereal,^[81][82][83] and Sally Forth.^[84] The Massachusetts Institute of Technology usually announces admissions on March 14 at 6:28 p.m., which is on Pi Day at Tau Time.^[85]

One turn is equal to 2π (≈ 6.283185307179586)^[86] radians, 360 degrees, or 400 gradians.

In the International System of Quantities (ISQ), rotation (symbol N) is a physical quantity defined as number of revolutions:^[87]

N is the number (not necessarily an integer) of revolutions, for example, of a rotating body about a given axis. Its value is given by:

${\displaystyle N={\frac {\varphi }{2\pi {\text{ rad}}}}}$

where 𝜑 denotes the measure of rotational displacement.

The above definition is part of the ISQ, formalized in the international standard ISO 80000-3 (Space and time),^[87] and adopted in the International System of Units (SI).^[88][89]

Rotation count or number of revolutions is a quantity of dimension one, resulting from a ratio of angular displacement. It can be negative and also greater than 1 in modulus. The relationship between quantity rotation, N, and unit turns, tr, can be expressed as:

${\displaystyle N={\frac {\varphi }{\text{tr}}}=\{\varphi \}_{\text{tr}}}$

where {𝜑}_tr is the numerical value of the angle 𝜑 in units of turns (see Physical quantity § Components).

In the ISQ/SI, rotation is used to derive rotational frequency (the rate of change of rotation with respect to time), denoted by n:

${\displaystyle n={\frac {\mathrm {d} N}{\mathrm {d} t}}}$

The SI unit of rotational frequency is the reciprocal second (s⁻¹). Common related units of frequency are hertz (Hz), cycles per second (cps), and revolutions per minute (rpm).

The superseded version ISO 80000-3:2006 defined "revolution" as a special name for the dimensionless unit "one",^[c] which also received other special names, such as the radian.^[d] Despite their dimensional homogeneity, these two specially named dimensionless units are applicable for non-comparable kinds of quantity: rotation and angle, respectively.^[91] "Cycle" is also mentioned in ISO 80000-3, in the definition of period.^[e]

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.

• Ampere-turn
• Hertz (modern) or Cycle per second (older)
• Angle of rotation
• Revolutions per minute
• Repeating circle
• Spat (angular unit) – the solid angle counterpart of the turn, equivalent to 4π steradians.
• Unit interval
• Divine Proportions: Rational Trigonometry to Universal Geometry
• Modulo operation
• Twist (mathematics)

• The Tau Manifesto