# Turn (angle)

A **turn** is a unit of plane angle measurement equal to 2*π* 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, centi-turns, milli-turns, points, etc.

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 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]}

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 not an SI unit. However, it is a legal unit of measurement in the EU^{[13]}^{[14]} and 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]}

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*τ*/4 rad instead of
3*π*/2 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 *τ* is used instead of *π*.^{[25]}^{[26]}^{[27]}

Initially, neither of these proposals received widespread acceptance by the mathematical and scientific communities.^{[28]} However, the use of *τ* has become more widespread,^{[29]} for example:

The following table shows how various identities and inequalities appear if *τ* := 2*π* was used instead of π.^{[40]}^{[41]}

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, 2*π*).
A turn of the complex plane arises from multiplying *z* = *x* + *iy* by an element *u* = exp(*bi*) that lies on the unit circle:

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.^{[42]}

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".