# Turn (angle)

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**), **revolution** (abbreviated **rev**), **complete rotation** (abbreviated **rot**) or **full circle**.

Turn | |
---|---|

Unit of | Plane angle |

Symbol | tr or pla |

Conversions | |

1 tr in ... | ... is equal to ... |

radians | 6.283185307179586... rad |

radians | 2π rad |

degrees | 360° |

gradians | 400^{g} |

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

## Subdivision of turns

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″.[1][2] 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.[3] 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 2^{n} equal parts for other values of *n*.[4]

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.[5][6] 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.[7] Euler adopted the symbol with that meaning in 1737, leading to its widespread use.

Percentage protractors have existed since 1922,[8] but the terms centiturns, milliturns and microturns were introduced much later by the British astronomer Fred Hoyle in 1962.[1][2] Some measurement devices for artillery and satellite watching carry milliturn scales.[9][10]

## Unit symbols

The German standard DIN 1315 (March 1974) proposed the unit symbol *pla* (from Latin: *plenus angulus* "full angle") for turns.[11][12] 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[13][14] and in Switzerland.[15]

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.[16][17] An angular mode `TURN`

was suggested for the WP 43S as well,[18] but the calculator instead implements `MULπ`

(*multiples of π*) as mode and unit since 2019.[19][20]

## Unit conversion

One turn is equal to 2π (≈ 6.283185307179586)[21] radians.

Turns | Radians | Degrees | Gradians, or gons |
---|---|---|---|

0 | 0 | 0° | 0^{g} |

1/24 | π/12 | 15° | 16+2/3^{g} |

1/12 | π/6 | 30° | 33+1/3^{g} |

1/10 | π/5 | 36° | 40^{g} |

1/8 | π/4 | 45° | 50^{g} |

1/2π | 1 | c. 57.3° | c. 63.7^{g} |

1/6 | π/3 | 60° | 66+2/3^{g} |

1/5 | 2π/5 | 72° | 80^{g} |

1/4 | π/2 | 90° | 100^{g} |

1/3 | 2π/3 | 120° | 133+1/3^{g} |

2/5 | 4π/5 | 144° | 160^{g} |

1/2 | π | 180° | 200^{g} |

3/4 | 3π/2 | 270° | 300^{g} |

1 | 2π | 360° | 400^{g} |

## Tau proposals

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 "pi with three legs" symbol to denote the constant ( = 2π).[22]

In 2010, Michael Hartl proposed to use tau to represent Palais' circle constant: *τ* = 2*π*. 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 3/4*τ* rad instead of 3/2*π* rad. Second, τ visually resembles π, whose association with the circle constant is unavoidable.[23] Hartl's *Tau Manifesto*[24] gives many examples of formulas that are asserted to be clearer where tau is used instead of pi.[25][26][27]

In June 2017, for release 3.6, the Python programming language adopted the name *tau* to represent the number of radians in a turn.[28]

The τ-functionality is made available in the Google calculator and in several programming languages like Python,[29] Perl 6,[30] Processing,[31] and Nim.[32] It has also been used in at least one mathematical research article,[33] authored by the τ-promoter Peter Harremoës.[34]

However, none of these proposals have received widespread acceptance by the mathematical and scientific communities.[35]

## 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.[8]

## 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*) + *r*i sin(*a*) where *r* > 0 and *a* is in [0, 2π).
A turn of the complex plane arises from multiplying *z* = *x* + i*y* by an element *u* = exp(*b* i) 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.[36]

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".

## See also

- Angle of rotation
- Revolutions per minute
- Repeating circle
- Spat (unit) — the 3D counterpart of the turn, equivalent to 4π steradians.
- Unit interval
- Turn (rational trigonometry)
- Spread (rational trigonometry)
- Modulo operation

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