# Turn (angle)

Turn
Unit ofPlane angle
Symboltr, pla or τ
Conversions
1 tr in ...... is equal to ...
degrees   360°

A turn is a unit of plane angle measurement equal to 2π radians, 360 degrees or 400 gradians. A turn is also referred to as a cycle (abbreviated cyc. or cyl.), revolution (abbreviated rev.), complete rotation (abbreviated rot.) or full circle.

Subdivisions of a turn include half-turns, quarter-turns, centiturns, milliturns, points, etc.

## Subdivisions

A 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″. A protractor divided in centiturns is normally called a percentage protractor.

Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points. The binary degree, also known as the binary radian (or brad), is 1/256 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 2n equal parts for other values of n.

The notion of turn is commonly used for planar rotations.

## History

The word turn originates via Latin and French from the Greek word τόρνος (tórnos – a lathe).

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. 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. Euler adopted the symbol with that meaning in 1737, leading to its widespread use.

Percentage protractors have existed since 1922, but 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.

## Unit symbols

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

The standard ISO 80000-3:2006 mentions that the unit name revolution with symbol r is used with rotating machines, as well as using the term turn to mean a full rotation. The standard IEEE 260.1:2004 also uses the unit name rotation and symbol r.

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 MULπ (multiples of π) as mode and unit since 2019.

## Unit conversion

One turn is equal to 2π (≈ 6.283185307179586) radians.

Conversion of common angles
1/24 turn π/12 rad 15° 16+2/3g
1/16 turn π/8 rad 22.5° 25g
1/12 turn π/6 rad 30° 33+1/3g
1/10 turn π/5 rad 36° 40g
1/8 turn π/4 rad 45° 50g
1/2π turn 1 rad c. 57.3° c. 63.7g
1/6 turn π/3 rad 60° 66+2/3g
1/5 turn 2π/5 rad 72° 80g
1/4 turn π/2 rad 90° 100g
1/3 turn 2π/3 rad 120° 133+1/3g
2/5 turn 4π/5 rad 144° 160g
1/2 turn π rad 180° 200g
3/4 turn 3π/2 rad 270° 300g
1 turn 2π rad 360° 400g

## Proposals for a single letter to represent a full turn 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).

In 1746, Leonard Euler first used the Greek letter pi to represent the circumference divided by the radius (i.e. Pi is approx. 6.28...) of a circle. (see "How pi was almost 6.283185...", 3Blue1Brown).

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 ($\pi \!\;\!\!\!\pi =2\pi$ ).

In 2008 Thomas Colignatus proposed the uppercase Greek letter Theta, Θ or ϴ, to represent a full circle, i.e. Θ = 2π.

The Greek letter Theta derives from the Phoenician and Hebrew letter teth, 𐤈 or ט, and it has been observed that, especially the older version of the symbol which means wheel, resembles a wheel with four spokes, where it was also proposed to use the wheel symbol, teth, to represent the quantity 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.

In 2010, Michael Hartl proposed to use tau to represent a full circle: τ = 2π. He offered two reasons. First, τ is the number of radians in one turn, which 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. Second, τ visually resembles π, whose association with the circle constant is unavoidable. Hartl's Tau Manifesto gives many examples of formulas that are asserted to be clearer where τ is used instead of π.

Initially, neither of these proposals received widespread acceptance by the mathematical and scientific communities. However, the use of τ has become more widespread, for example:

• In 2012, the educational website Khan Academy began accepting answers expressed in terms of τ.
• In June 2017, for release 3.6, the Python programming language adopted the name tau to represent the number of radians in a turn.
• The τ-functionality is made available in the Google calculator and in several programming languages like Python, Raku, Processing, Nim, and Rust.
• It has also been used in at least one mathematical research article, authored by the τ-promoter Peter Harremoës.
• In 2020, for release 5.0, Tau was added to .NET Core (which is being rebranded as ".NET" for the 5.0 release).

The following table shows how various identities and inequalities appear if τ := 2π was used instead of π.

Formula Using π Using τ Notes
Circumference C of a circle of radius r C = 2πr C = τr
Area of a circle A = πr2 A = τr2/2 Recall that the area of a sector of angle θ (measured in radians) is A = θr2/2.
Area of a regular n-gon with unit circumradius A = n/2 sin /n A = n/2 sin τ/n
Volume of an n-ball $V_{n}(R)={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}R^{n}$ $V_{n}(R)={\frac {\tau ^{\left\lfloor {\frac {n}{2}}\right\rfloor }}{n!!}}(1+n\operatorname {mod} 2)R^{n}$ Surface area of an n-ball $S_{n}(R)={\frac {2\pi ^{\frac {n+1}{2}}}{\Gamma {\big (}{\frac {n+1}{2}}{\big )}}}R^{n}$ $S_{n}(R)={\frac {\tau ^{\left\lfloor {\frac {n+1}{2}}\right\rfloor }}{(n-1)!!}}(2-(n\operatorname {mod} 2))R^{n}$ Cauchy's integral formula $f(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$ $f(a)={\frac {1}{\tau i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$ Standard normal distribution $\varphi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}$ $\varphi (x)={\frac {1}{\sqrt {\tau }}}e^{-{\frac {x^{2}}{2}}}$ Stirling's approximation $n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}$ $n!\sim {\sqrt {\tau n}}\left({\frac {n}{e}}\right)^{n}$ Euler's identity 0      e = − 1
e + 1 = 0
0     e = 1
e − 1 = 0
nth roots of unity $e^{2\pi i{\frac {k}{n}}}=\cos {\frac {2k\pi }{n}}+i\sin {\frac {2k\pi }{n}}$ $e^{\tau i{\frac {k}{n}}}=\cos {\frac {k\tau }{n}}+i\sin {\frac {k\tau }{n}}$ Reduced Planck constant $\hbar ={\frac {h}{2\pi }}$ $\hbar ={\frac {h}{\tau }}$ h is the Planck constant.
Angular frequency $\omega ={{2\pi } \over T}={2\pi f}$ $\omega ={{\tau } \over T}={\tau f}$ Reactance of an inductor 2πfL τfL
Susceptance of a capacitor 2πfC τfC

## Examples of use

• As an angular unit, the turn or revolution is particularly useful for large angles, such as in connection with electromagnetic coils and rotating objects. See also winding number.
• The angular speed of rotating machinery, such as automobile engines, is commonly measured in revolutions per minute or RPM.
• Pie charts illustrate proportions of a whole as fractions of a turn. Each one percent is shown as an angle of one centiturn.

## Kinematics of turns

In kinematics, a turn is a rotation less than a full revolution. A turn may be represented in a mathematical model that uses expressions of complex numbers or quaternions. In the complex plane every non-zero number has a polar coordinate expression z = r cis a = r(cos a + i sin a) where r > 0 and a is in [0, 2π). A turn of the complex plane arises from multiplying z = x + iy by an element u = exp(bi) that lies on the unit circle:

$z\mapsto uz.$ Frank Morley consistently referred to elements of the unit circle as turns in the book Inversive Geometry, (1933) which he coauthored with his son Frank Vigor Morley.

The Latin term for turn is versor, which is a quaternion that can be visualized as an arc of a great circle. The product of two versors can be compared to a spherical triangle where two sides add to the third. For the kinematics of rotation in three dimensions, see quaternions and spatial rotation. This algebraic expression of rotation was initiated by William Rowan Hamilton in the 1840s (using the term versor), and is a recurrent theme in the works of Narasimhaiengar Mukunda as "Hamilton's theory of turns".