 Turn (geometry)
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Turn Geometry

Turn
Unit ofPlane angle
Symboltr or pla
Conversions
degrees360°

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

Subdivisions of a turn include half-turns, quarter-turns, centi-turns, milli-turns, points, etc.

## Subdivision of turns

A turn can be divided in 100 centiturns or 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 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 (tórnos - a lathe).

In 1697, David Gregory used (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 (delta over pi) for the ratio of the diameter to perimeter. The first use of the symbol ? 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 no SI unit, but a legal unit of measurement in the EU and in 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 MUL? (multiples of ?) as mode and unit since 2019.

## Unit conversion

One turn is equal to 2? (? )radians.

Conversion of common angles
0 0 0g
15° g
30° g
36° 40g
45° 50g
1 c. 57.3° c. 63.7g
60° g
72° 80g
90° 100g
120° g
144° 160g
? 180° 200g
270° 300g
1 2? 360° 400g

## Tau proposals 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 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of ?, which amounts to the number of radians in half a turn, in order to make mathematics simpler and more intuitive. His proposal used a "? with three legs" symbol to denote the constant ($\pi \!\;\!\!\!\pi =2\pi$ ).

In 2010, Michael Hartl proposed to use tau to represent Palais' circle constant: 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  turn would be represented as τ rad instead of π rad. Second, τ visually resembles ?, 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 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 ? := 2 ? was used instead of ?.

Using ? := 2 ? Using ? Formula Notes
? ? of a circle (as an angle in radians)
C = ? r C = 2 ? r Circumference C of a circle of radius r
0ei ? = 1
ei ? - 1 = 0
0ei ? = - 1
ei ? + 1 = 0
Euler's formula
A = ? r2 A = ? r2 Area of a circle
• Recall the area of a sector of angle ? (measured in radians) is A = ? r2.
• The more clearly expresses that area is the integral of circumference.
• Compare to Kinetic energy m v2 and spring energy k x2.
$\hbar ={\frac {h}{\tau }}$ $\hbar ={\frac {h}{2\pi }}$ Reduced Planck constant
$\omega ={{\tau } \over T}={\tau f}$ $\omega ={{2\pi } \over T}={2\pi f}$ Angular frequency
A = sin A = n sin cos Area of a regular n-gon with unit circumradius
$V_{n}(R)={\frac {\tau ^{\left\lfloor {\frac {n}{2}}\right\rfloor }R^{n}}{n!!}}\cdot (1+n\operatorname {mod} 2)$ $V_{n}(R)={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}R^{n}$ Volume of an n-ball
$S_{n}(R)={\frac {\tau ^{\left\lfloor {\frac {n+1}{2}}\right\rfloor }R^{n}}{(n-1)!!}}\cdot (2-(n\operatorname {mod} 2))$ $S_{n}(R)={\frac {2\pi ^{\frac {n+1}{2}}}{\Gamma {\big (}{\frac {n+1}{2}}{\big )}}}R^{n}$ Surface area of an n-ball
$f(a)={\frac {1}{\tau i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$ $f(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz$ Cauchy's integral formula
$\varphi (x)={\frac {1}{\sqrt {\tau }}}e^{-{\frac {1}{2}}x^{2}}$ $\varphi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}$ Standard normal distribution
$\left\{e^{i\tau k/n}\right\}$ $\left\{e^{i2\pi k/n}\right\}$ n-th roots of unity
$e^{\tau i{\frac {k}{n}}}=\cos {\frac {k\tau }{n}}+i\sin {\frac {k\tau }{n}}$ $e^{2\pi i{\frac {k}{n}}}=\cos {\frac {2k\pi }{n}}+i\sin {\frac {2k\pi }{n}}$ Roots of unity
$n!\sim {\sqrt {\tau n}}\left({\frac {n}{e}}\right)^{n}$ $n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}$ Stirling's approximation
? f L 2 ? f L Reactance of an inductor
? f C 2 ? f C Susceptance of a capacitor

## 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.
• Turn is used in complex dynamics for measure of external and internal angles. The sum of external angles of a polygon equals one turn. Angle doubling map is used.
• 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 where and a is in . A turn of the complex plane arises from multiplying by an element that lies on the unit circle:

z ? 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".