Turn (angle)

A circle. A complete rotation about the center point is equal to 1 turn.

A turn is a unit of angle of rotation, equal to a full circle or 360° or 2π radians. 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".

Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 points or 128 quarter-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 efficiently represented in a single byte (albeit to limited precision). 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. Two special rotations have acquired appellations of their own: a rotation through 180° is commonly referred to as a half-turn, a rotation through 90° is referred to as a quarter-turn. A half-turn is often referred to as a reflection in point.

A turn is also named as revolution or complete rotation or full circle or cycle.

Depending on the application 1 turn may be abbreviated as $\tau$, rev or rot.

Examples of use

As an angular unit it is particularly useful for large angles, such as in connection with coils and rotating objects. See also winding number. Turn is used in complex dynamics for measure of external and internal angles. The sum of external angles of a polygon equals one turn.

Pie charts illustrate proportions of a whole as fractions of a turn. Each one percent is shown as an angle of one centiturn.

Conversion of some common angles

Units	Values
Turns	0	1/12	1/10	1/8	1/6	1/5	1/4	1/2	3/4	1
Degrees	0°	30°	36°	45°	60°	72°	90°	180°	270°	360°
Radians	0	${\tfrac {1}{6}}\pi$	${\tfrac {1}{5}}\pi$	${\tfrac {1}{4}}\pi$	${\tfrac {1}{3}}\pi$	${\tfrac {2}{5}}\pi$	${\tfrac {1}{2}}\pi$	$\pi \,$	${\tfrac {3}{2}}\pi$	$2\pi \,$
Grads	0g	33⅓g	40g	50g	66⅔g	80g	100g	200g	300g	400g

Mathematical constant

A full "turn" of a circle is represented by the greek letter tau. Tau is the circle constant and is equal to 2*pi.

Half a turn is often identified with the mathematical constant $\pi$ because half a turn is $\pi$ radians. Similarly 1 turn can be identified with $2\pi \approx 6.283185307$.

History

The word turn originates via Latin and French from the Greek word $\tau \mathrm {o} \rho \nu \mathrm {o} \sigma$ (tornos - a lathe).

The geometric notion of a turn has its origin in the sailors terminology of knots where a turn means one round of rope on a pin or cleat, or one round of a coil. For knots the English terms of single turn, round turn and double round turn do not translate directly into the geometric notion of turn, but in German the correspondence is exact.

In 1697 David Gregory used $\pi /\rho$ to denote the circumference of a circle divided by its radius, though $\delta /\pi$ had been used by Oughtred in 1647 for the ratio of diameter to circumference. The first use of $\pi$ on its own with its present meaning was by William Jones in 1706., and Euler adopted the symbol in 1737.

The idea of using centiturns and milliturns as units was introduced by the British astronomer and science writer Sir Fred Hoyle. Robert Palais proposed to use a "pi with three legs" to denote 1 turn , while physicist Michael Hartl proposed to use the Greek letter $\tau$ to refer to the constant $2\pi$.