Turn (angle)

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 Template:Math radians, 360 degrees or 400 gradians. As an angular unit, one turn also corresponds to one cycle (symbol cyc or c) or to one revolution (symbol rev or r). Common related units of frequency are cycles per second (cps) and revolutions per minute (rpm).Template:Efn 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 $2\pi$ radians, some have proposed [[#Tau proposals|representing Template:Math with a single letter]]. In 2010, Michael Hartl proposed using the Greek letter $\tau$ (tau), equal to the ratio of a circle's circumference to its radius ($2\pi$) and corresponding to one turn, for greater conceptual simplicity when stating angles in radians. This proposal did not initially gain widespread acceptance in the mathematical community, but the constant has become more widespread, 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. Template:Xref

Unit symbols

There are several unit symbols for the turn.

EU and Switzerland

The German standard DIN 1315 (March 1974) proposed the unit symbol "pla" (from Latin: Template:Lang 'full angle') for turns. Covered in Template:Ill (October 2010), the so-called Template:Lang ('full angle') is not an SI unit. However, it is a legal unit of measurement in the EU and Switzerland.

Calculators

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. An angular mode TURN was suggested for the WP 43S as well, but the calculator instead implements "MULTemplate:Pi" ([[multiples of π|multiples of Template:Pi]]) as mode and unit since 2019.

Divisions

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 Template:Val milliturns, with each milliturn corresponding to an angle of 0.36°, which can also be written as 21′ 36″. A protractor divided in centiturns is normally called a "percentage protractor". While percentage protractors have existed since 1922, the terms centiturns, milliturns and microturns were introduced much later by the British astronomer Fred Hoyle in 1962. Some measurement devices for artillery and satellite watching carry milliturn scales.

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 Template:Sfrac turn. 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 Template:Math equal parts for other values of Template:Mvar.

Proposals for a single letter to represent 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 is expressed here using the Greek letter tau (Template:Mvar).

Some special angles in radians, stated in terms of Template:Tau.

A comparison of angles expressed in degrees and radians.

The number Template:Math (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 $\pi$ was not originally fixed to the ratio of the circumference and the diameter. In 1697, David Gregory used Template:Math (pi over rho) to denote the perimeter of a circle (i.e., the circumference) divided by its radius. However, earlier in 1647, William Oughtred had used Template:Math (delta over pi) for the ratio of the diameter to perimeter. The first use of the symbol Template:Pi on its own with its present meaning (of perimeter divided by diameter) was in 1706 by the Welsh mathematician William Jones.

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 Template:Pi. Euler would later use the letter Template:Pi for the 3.14... constant in his 1736 Mechanica and 1748 Introductio in analysin infinitorum, 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 Mechanica, Euler instead used the letter Template:Pi for one-fourth of the circumference of a unit circle, or 1.57... . Usage of the letter Template:Pi, sometimes for 3.14... and other times for 6.28..., became widespread, with the definition varying as late as 1761; afterward, Template:Pi was standardized as being equal to 3.14... .

Several people have independently proposed using Template:Math, including:

• Joseph Lindenburg (Template:Circa 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 Template:Pi, 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 ($\pi \pi =2\pi$).

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: $\psi =2\pi$.

The same year, Thomas Colignatus proposed the uppercase Greek letter theta, Θ, to represent 2Template:Pi. 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. It has also been proposed to use the wheel symbol, teth, to represent the value 2Template:Pi, 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 2Template:Pi.

In 2010, Michael Hartl proposed to use the Greek letter tau to represent the circle constant: Template:Math. 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 Template:Sfrac turn would be represented as Template:Math rad instead of Template:Math rad. As for the choice of notation, he offered two reasons. First, Template:Mvar is the number of radians in one turn, and both Template:Mvar and turn begin with a Template:IPAc-en sound. Second, Template:Mvar visually resembles Template:Pi, whose association with the circle constant is unavoidable. Hartl's Tau ManifestoTemplate:Efn gives many examples of formulas that are asserted to be clearer where Template:Math is used instead of Template:Math. For example, Hartl asserts that replacing Euler's identity Template:Math by Template:Math (which Hartl also calls "Euler's identity") is more fundamental and meaningful. He also claims that the formula for circular area in terms of Template:Mvar, Template:Math, contains a natural factor of Template:Sfrac arising from integration.

Initially, this proposal did not receive significant acceptance by the mathematical and scientific communities. However, the use of Template:Math has become more widespread.

The following table shows how various identities appear when Template:Math is used instead of Template:Pi. For a more complete list, see [[List of formulae involving π|List of formulae involving Template:Pi]].

Formula	Using Template:Pi	Using Template:Math	Notes
Angle subtended by Template:Sfrac of a circle	$\frac{\pi }{2}\text{ rad}$	$\frac{\tau }{4}\text{ rad}$	Template:Math
Circumference of a circle	$C=2\pi r$	$C=\tau r$	The length of an arc of angle Template:Math is Template:Math.
Area of a circle	$A=\pi r^2$	$A=\frac{1}{2}\tau r^2$	The area of a sector of angle Template:Math is Template:Math.
Area of a regular [[Polygon|Template:Mvar-gon]] with unit circumradius	$A=\frac{n}{2}\sin \frac{2\pi }{n}$	$A=\frac{n}{2}\sin \frac{\tau }{n}$
[[n-sphere#Volume and surface area|Template:Mvar-ball and Template:Math-sphere volume recurrence relation]]	$V_n(r)=\frac{r}{n}{S}_{n-1}(r)$ $S_n(r)=2\pi r{V}_{n-1}(r)$	$V_n(r)=\frac{r}{n}{S}_{n-1}(r)$ $S_n(r)=\tau r{V}_{n-1}(r)$	Template:MathTemplate:Br Template:Math
Cauchy's integral formula	$f(a)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f(z)}{z-a}dz$	$f(a)=\frac{1}{\tau i}{\oint }_{\gamma }\frac{f(z)}{z-a}dz$	$\gamma$ is the boundary of a disk containing $a$ in the complex plane.
Standard normal distribution	$\phi (x)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{x^2}{2}}$	$\phi (x)=\frac{1}{\sqrt{\tau }}{e}^{-\frac{x^2}{2}}$
Stirling's approximation	$n!\sim \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$	$n!\sim \sqrt{\tau n}{\left(\frac{n}{e}\right)}^n$
[[Root of unity|Template:Mvarth roots of unity]]	$e^{2\pi i\frac{k}{n}}=\cos \frac{2k\pi }{n}+i\sin \frac{2k\pi }{n}$	$e^{\tau i\frac{k}{n}}=\cos \frac{k\tau }{n}+i\sin \frac{k\tau }{n}$
Planck constant	$h=2\pi \hslash$	$h=\tau \hslash$	Template:Math is the reduced Planck constant.
Angular frequency	$\omega =2\pi f$	$\omega =\tau f$

In culture

Template:Tau has made numerous appearances in culture. It is celebrated annually on June 28, known as Tau Day. Template:Tau has been covered in videos by Vi Hart, Numberphile, SciShow, Steve Mould, Khan Academy, and 3Blue1Brown, and it has appeared in the comics xkcd, Saturday Morning Breakfast Cereal, and Sally Forth. The Massachusetts Institute of Technology usually announces admissions on March 14 at 6:28Template:Nbspp.m., which is on Pi Day at Tau Time.

Unit conversion

The circumference of the unit circle (whose radius is one) is Template:Math.

One turn is equal to Template:Math (≈ Template:Val) radians, 360 degrees, or 400 gradians.

Conversion of common angles

Turns	Radians		Degrees	Gradians
0 turn	0 rad		0°	0g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	5°	Template:Sfracg
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	15°	Template:Sfracg
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	22.5°	25g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	30°	Template:Sfracg
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	36°	40g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	45°	50g
Template:Sfrac turn	1 rad		Template:Circa 57.3°	Template:Circa 63.7g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	60°	Template:Sfracg
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	72°	80g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	90°	100g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	120°	Template:Sfracg
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	144°	160g
Template:Sfrac turn	Template:Sfrac rad	Template:Pi rad	180°	200g
Template:Sfrac turn	Template:Sfrac rad	Template:Sfrac rad	270°	300g
1 turn	Template:Tau rad	2Template:Pi rad	360°	400g

In the ISQ/SI

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

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:

$N=\frac{\phi }{2\pi \text{ rad}}$

where Template:Varphi 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), and adopted in the International System of Units (SI).

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:

$N=\frac{\phi }{\text{tr}}=\{\phi {\}}_{\text{tr}}$

where {Template:Varphi}_tr is the numerical value of the angle Template:Varphi in units of turns (see Template:Slink).

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

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

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).

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

In programming languages

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

See also

• 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 Template:Math steradians.
• Unit interval
• Divine Proportions: Rational Trigonometry to Universal Geometry
• Modulo operation
• Twist (mathematics)

External links

• The Tau Manifesto