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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. 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 2 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 a point since these are identical for transformations in two-dimensions.

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

Depending on the application, one turn may be abbreviated as ${\displaystyle \tau }$, 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.
• With algebraic manipulation Euler's identity becomes: ${\displaystyle e^{i\tau }=1+0}$

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
Radians in terms of τ	0	${\displaystyle {\tfrac {1}{12}}\tau }$	${\displaystyle {\tfrac {1}{10}}\tau }$	${\displaystyle {\tfrac {1}{8}}\tau }$	${\displaystyle {\tfrac {1}{6}}\tau }$	${\displaystyle {\tfrac {1}{5}}\tau }$	${\displaystyle {\tfrac {1}{4}}\tau }$	${\displaystyle {\tfrac {1}{2}}\tau }$	${\displaystyle {\tfrac {3}{4}}\tau }$	${\displaystyle \tau }$
Radians in terms of π	0	${\displaystyle {\tfrac {1}{6}}\pi }$	${\displaystyle {\tfrac {1}{5}}\pi }$	${\displaystyle {\tfrac {1}{4}}\pi }$	${\displaystyle {\tfrac {1}{3}}\pi }$	${\displaystyle {\tfrac {2}{5}}\pi }$	${\displaystyle {\tfrac {1}{2}}\pi }$	${\displaystyle \pi \,}$	${\displaystyle {\tfrac {3}{2}}\pi }$	${\displaystyle 2\pi \,}$
Degrees	0°	30°	36°	45°	60°	72°	90°	180°	270°	360°
Grads	0	33⅓	40	50	66⅔	80	100	200	300	400

Since a turn ${\displaystyle \tau }$ is often identified with the circle constant ${\displaystyle 2\pi }$ there is essentially no difference between measuring angles in turns and in radians.

Mathematical constant

A full "turn" of a circle is represented by the greek letter ${\displaystyle \tau }$. ${\displaystyle \tau }$ is the circle constant and ${\displaystyle \tau =2\pi }$.

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

Robert Palais proposed in 2001 to use the turn as the fundamental circle constant instead of π, in order to make mathematics simpler and more intuitive, using a "pi with three legs" symbol to denote 1 turn (${\displaystyle \pi \!\;\!\!\!\pi =2\pi }$) In 2010, Michael Hartl proposed to use the Greek letter ${\displaystyle \tau }$ (or "tau," aka "pi with one leg in the denominator") instead (${\displaystyle \tau =2\pi }$).

History

The word turn originates via Latin and French from the Greek word ${\displaystyle \tau \mathrm {o} \rho \nu \mathrm {o} \sigma }$ (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 ${\displaystyle \pi /\rho }$ (pi/rho) to denote the perimeter of a circle (i.e. the circumference) divided by its radius, though ${\displaystyle \delta /\pi }$ (delta/pi) had been used by Oughtred in 1647 for the ratio of diameter to perimeter. The first use of ${\displaystyle \pi }$ on its own with its present meaning of perimeter/diameter was by William Jones in 1706. Euler adopted the symbol with that meaning in 1737, leading to its widespread use.

The idea of using centiturns and milliturns as units was introduced by the British astronomer and science writer Sir Fred Hoyle.

References

1. ooPIC Programmer's Guide www.oopic.com
2. Angles, integers, and modulo arithmetic Shawn Hargreaves blogs.msdn.com
3. Half Turn, Reflection in Point cut-the-knot.org
4. ^ Michael Hartl (June 28, 2010). "The Tau Manifesto". Retrieved January 12, 2011.
5. Palais, R. 2001: Pi is Wrong, The Mathematical Intelligencer. Springer-Verlag New York. Volume 23, Number 3, pp. 7-8
6. Aron, Jacob (8 January 2011), "Interview: Michael Hartl: It's time to kill off pi", New Scientist, 209 (2794), doi:10.1016/S0262-4079(11)60036-5
7. Landau, Elizabeth (14 March 2011), "On Pi Day, is 'pi' under attack?", cnn.com
8. Ashley, C. The Ashley Book of Knots, New York 1944. p. 604.
9. Harremoes, Peter (21 February 2011), Gregory's constant Tau
10. Beckmann, P., A History of Pi. Barnes & Noble Publishing, 1989.
11. Schwartzman, S., The Words of Mathematics. The Mathematical Association of America,1994. Page 165
12. Pi through the ages
13. Hoyle, F., Astronomy. London, 1962.

Further reading

• Aron, Jacob (8 January 2011), "Interview: Michael Hartl: It's time to kill off pi", New Scientist, 209 (2794), doi:10.1016/S0262-4079(11)60036-5
• Landau, Elizabeth (14 March 2011), "On Pi Day, is 'pi' under attack?", cnn.com

• Units of angle