# Turn (geometry)

A **turn** is a unit of angle measurement equal to 360° or 2π radians. A turn is also referred to as a **revolution** or **complete rotation** or **full circle** or **cycle** or **rev** or **rot**.

A turn can be subdivided in many different ways: into half turns, quarter turns, centiturns, milliturns, binary angles, points etc.

## Contents

## 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". 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.^{1} 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^{n} equal parts for other values of *n*.^{2}

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 ( radians),^{3} 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.

## History

The word turn originates via Latin and French from the Greek word τόρνος (tornos – a lathe).

In 1697, David Gregory used (pi/rho) to denote the **p**erimeter of a circle (i.e., the circumference) divided by its **r**adius.^{4}^{5} However, earlier in 1647, William Oughtred had used (delta/pi) for the ratio of the **d**iameter to **p**erimeter. 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.^{6} Euler adopted the symbol with that meaning in 1737, leading to its widespread use.

Percentage protractors have existed since 1922,^{7} but the terms centiturns and milliturns were introduced much later by Sir Fred Hoyle.^{8}

## Mathematical constants

One turn is equal to 2π (≈6.283185307179586)^{9} radians.

## Conversion of some common angles

Units | Values | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Turns |
0 | 1 | |||||||||||

Radians |
0 | ||||||||||||

Degrees |
0° | 15° | 30° | 36° | 45° | 60° | 72° | 90° | 120° | 144° | 180° | 270° | 360° |

Grads |
0^{g} |
16⅔^{g} |
33⅓^{g} |
40^{g} |
50^{g} |
66⅔^{g} |
80^{g} |
100^{g} |
133⅓^{g} |
160^{g} |
200^{g} |
300^{g} |
400^{g} |

## Tau proposal

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, using a "pi with three legs" symbol to denote the constant ().^{10} In 2010, Michael Hartl proposed to use the Greek letter τ (tau) instead for two reasons. First, τ is the radian angle measure for one *turn* of a circle, which allows fractions of a turn to be expressed, such as for a turn or . Second, τ visually resembles π, whose association with the circle constant is unavoidable.^{11} Hartl's *Tau Manifesto* gives many examples of formulas that are simpler if tau is used instead of pi.^{12}^{13}

## 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 *z* = *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* = e^{b i} that lies on the unit circle:

Frank Morley consistently referred to elements of the unit circle as *turns* in the book *Inversive Geometry* (1933) that 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.

## See also

## Notes and references

**^**ooPIC Programmer's Guide*www.oopic.com***^**Angles, integers, and modulo arithmetic Shawn Hargreaves*blogs.msdn.com***^**Half Turn, Reflection in Point cut-the-knot.org**^**Beckmann, P., A History of Pi. Barnes & Noble Publishing, 1989.**^**Schwartzman, S., The Words of Mathematics. The Mathematical Association of America,1994. Page 165**^**Pi through the ages**^**Croxton, F. E. (1922), A Percentage Protractor Journal of the American Statistical Association, Vol. 18, pp. 108-109**^**Hoyle, F., Astronomy. London, 1962**^**Sequence A019692.**^**Palais, R. 2001: Pi is Wrong, The Mathematical Intelligencer. Springer-Verlag New York. Volume 23, Number 3, pp. 7–8**^**Michael Hartl (March 14, 2013). "The Tau Manifesto". Retrieved September 14, 2013.**^**Aron, Jacob (8 January 2011), "Interview: Michael Hartl: It's time to kill off pi",*New Scientist***209**(2794), Bibcode:2011NewSc.209...23A, doi:10.1016/S0262-4079(11)60036-5**^**Landau, Elizabeth (14 March 2011), "On Pi Day, is 'pi' under attack?",*cnn.com*

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