# 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.

## 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. 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 ($\pi$ radians), 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 $\pi /\rho$ (pi/rho) to denote the perimeter of a circle (i.e., the circumference) divided by its radius. However, earlier in 1647, William Oughtred had used $\delta /\pi$ (delta/pi) for the ratio of the diameter to perimeter. The first use of the symbol $\pi$ on its own with its present meaning (of perimeter divided by diameter) was in 1706 by the Welsh mathematician William Jones. Euler adopted the symbol with that meaning in 1737, leading to its widespread use.

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

## Mathematical constants

One turn is equal to (≈6.283185307179586) radians.

## Conversion of some common angles

Units Values
Turns   0 {{ safesubst:#invoke:Unsubst $B=1/24}} {{ safesubst:#invoke:Unsubst$B=1/12}} {{ safesubst:#invoke:Unsubst $B=1/10}} {{ safesubst:#invoke:Unsubst$B=1/8}} {{ safesubst:#invoke:Unsubst $B=1/6}} {{ safesubst:#invoke:Unsubst$B=1/5}} {{ safesubst:#invoke:Unsubst $B=1/4}} {{ safesubst:#invoke:Unsubst$B=1/3}} {{ safesubst:#invoke:Unsubst $B=2/5}} {{ safesubst:#invoke:Unsubst$B=1/2}} {{ safesubst:#invoke:Unsubst $B=3/4}} 1 Radians 0 {{ safesubst:#invoke:Unsubst$B=π/12}} {{ safesubst:#invoke:Unsubst $B=π/6}} {{ safesubst:#invoke:Unsubst$B=π/5}} {{ safesubst:#invoke:Unsubst $B=π/4}} {{ safesubst:#invoke:Unsubst$B=π/3}} {{ safesubst:#invoke:Unsubst $B=2π/5}} {{ safesubst:#invoke:Unsubst$B=π/2}} {{ safesubst:#invoke:Unsubst $B=2π/3}} {{ safesubst:#invoke:Unsubst$B=4π/5}} π {{ safesubst:#invoke:Unsubst $B=3π/2}} 2π Degrees 15° 30° 36° 45° 60° 72° 90° 120° 144° 180° 270° 360° Grads 0g {{ safesubst:#invoke:Unsubst$B=16 2/3}}g {{ safesubst:#invoke:Unsubst $B=33 1/3}}g 40g 50g {{ safesubst:#invoke:Unsubst$B=66 2/3}}g 80g 100g {{ safesubst:#invoke:Unsubst \$B=133 1/3}}g 160g 200g 300g 400g

## Tau proposal An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to a full turn, or approximately 6.28 radians, which are expressed here using the Greek letter tau (Template:Tau).

In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of $\pi$ , 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 ($\pi \!\;\!\!\!\pi =2\pi$ ). In 2010, Michael Hartl proposed to use the Greek letter Template:Tau (tau) instead for two reasons. First, Template:Tau is the radian angle measure for one turn of a circle, which allows fractions of a turn to be expressed, such as ${\tfrac {2}{5}}\tau$ for a ${\tfrac {2}{5}}$ turn or ${\tfrac {4}{5}}\pi$ . Second, Template:Tau visually resembles π, whose association with the circle constant is unavoidable. Hartl's Tau Manifesto gives many examples of formulas that are simpler if tau is used instead of pi.

## 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 = eb i that lies on the unit circle:

$z\mapsto uz.$ 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.

## Notes and 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. Beckmann, P., A History of Pi. Barnes & Noble Publishing, 1989.
5. Schwartzman, S., The Words of Mathematics. The Mathematical Association of America,1994. Page 165
6. Pi through the ages
7. Croxton, F. E. (1922), A Percentage Protractor Journal of the American Statistical Association, Vol. 18, pp. 108-109
8. Hoyle, F., Astronomy. London, 1962
9. Palais, R. 2001: Pi is Wrong, The Mathematical Intelligencer. Springer-Verlag New York. Volume 23, Number 3, pp. 7–8
10. Template:Cite web
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