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[[File:angle-fractions.png|250px|thumb|right|Counterclockwise [[rotation]]s about the center point where a complete rotation is equal to 1 turn]]
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A '''turn''' is a unit of [[angle]] measurement equal to 360[[degree (angle)|°]] or 2[[Pi|{{pi}}]] [[radian]]s. A turn is also referred to as a '''revolution''' or '''complete rotation''' or '''full circle''' or '''cycle''' or '''rev''' or '''rot'''.
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A turn can be subdivided in many different ways: into half turns, quarter turns, centiturns, milliturns, binary angles, points etc.
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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.
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[[Binary scaling#Binary angles|Binary fractions of a turn]] are also used. Sailors have traditionally divided a turn into 32 [[compass point]]s. The ''binary degree'', also known as the ''[[binary radian]]'' (or ''brad''), is 1/256 turn.<ref>[http://www.oopic.com/pgchap15.htm ooPIC Programmer's Guide] ''www.oopic.com''</ref> 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<sup>''n''</sup> equal parts for other values of ''n''.<ref>[http://blogs.msdn.com/shawnhar/archive/2010/01/04/angles-integers-and-modulo-arithmetic.aspx Angles, integers, and modulo arithmetic] Shawn Hargreaves ''blogs.msdn.com''</ref>
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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 (<math>\pi</math> radians),<ref>[http://www.cut-the-knot.org/Curriculum/Geometry/HalfTurn.shtml Half Turn, Reflection in Point] cut-the-knot.org</ref> 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.
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== History ==
  <li>[http://freshnhottrends.com/activity/p/144642/ http://freshnhottrends.com/activity/p/144642/]</li>
The word turn originates via Latin and French from the Greek word τόρνος (tornos – a [[lathe]]).
 
 
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In 1697, [[David Gregory (mathematician)|David Gregory]] used <math>\pi/\rho</math> (pi/rho) to denote the '''p'''erimeter of a circle (i.e., the [[circumference]]) divided by its '''r'''adius.<ref>Beckmann, P., A History of Pi. Barnes & Noble Publishing, 1989.</ref><ref>Schwartzman, S., The Words of Mathematics. The Mathematical Association of America,1994. Page 165</ref>  However, earlier in 1647, [[William Oughtred]] had used <math>\delta/\pi</math> (delta/pi) for the ratio of the '''d'''iameter to '''p'''erimeter. The first use of the symbol <math>\pi</math> on its own with its present meaning (of perimeter divided by diameter) was in 1706 by the [[Wales|Welsh]] mathematician [[William Jones (mathematician)|William Jones]].<ref>[http://www.veling.nl/anne/templars/Pi_through_the_ages.html Pi through the ages]</ref> [[Euler]] adopted the symbol with that meaning in 1737, leading to its widespread use.
 
 
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Percentage protractors have existed since 1922,<ref>Croxton, F. E. (1922), A Percentage Protractor Journal of the American Statistical Association, Vol. 18, pp. 108-109</ref> but the terms centiturns and milliturns were introduced much later by [[Sir Fred Hoyle]].<ref>Hoyle, F., Astronomy. London, 1962</ref>
 
 
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== Mathematical constants ==
 
One turn is equal to {{nowrap|1=2π}} (≈6.283185307179586)<ref>Sequence {{OEIS2C|A019692}}.</ref> [[radian]]s.
</ul>
 
[[File:2pi-unrolled.gif|400px|thumb|left|The [[circumference]] of the [[unit circle]] (whose [[radius]] is one) is 2{{pi}}.]]
{{clear}}
 
== Conversion of some common angles ==
{|class = wikitable style="text-align:center;"
! Units !! colspan=13 | Values
|-
|style = "background:#f2f2f2; text-align:left;" | '''Turns'''&nbsp;&nbsp;
|style = "width:3em;" | 0
|style = "width:3em;" | <math>\tfrac{1}{24}</math>
|style = "width:3em;" | <math>\tfrac{1}{12}</math>
|style = "width:3em;" | <math>\tfrac{1}{10}</math>
|style = "width:3em;" | <math>\tfrac{1}{8}</math>
|style = "width:3em;" | <math>\tfrac{1}{6}</math>
|style = "width:3em;" | <math>\tfrac{1}{5}</math>
|style = "width:3em;" | <math>\tfrac{1}{4}</math>
|style = "width:3em;" | <math>\tfrac{1}{3}</math>
|style = "width:3em;" | <math>\tfrac{2}{5}</math>
|style = "width:3em;" | <math>\tfrac{1}{2}</math>
|style = "width:3em;" | <math>\tfrac{3}{4}</math>
|style = "width:3em;" | 1
|-
|style = "background:#f2f2f2; text-align:left;" | '''[[Radian]]s'''
| 0
| <math>\tfrac{\pi}{12}</math>
| <math>\tfrac{\pi}{6}</math>
| <math>\tfrac{\pi}{5}</math>
| <math>\tfrac{\pi}{4}</math>
| <math>\tfrac{\pi}{3}</math>
| <math>\tfrac{2\pi}{5}</math>
| <math>\tfrac{\pi}{2}</math>
| <math>\tfrac{2\pi}{3}</math>
| <math>\tfrac{4\pi}{5}</math>
| <math>\pi\,</math>
| <math>\tfrac{3\pi}{2}</math>
| <math>2\pi\,</math>
|-
|style = "background:#f2f2f2; text-align:left;" | '''[[Degree (angle)|Degrees]]'''&nbsp;&nbsp;
|style = "width:3em;" | 0°
|style = "width:3em;" | 15°
|style = "width:3em;" | 30°
|style = "width:3em;" | 36°
|style = "width:3em;" | 45°
|style = "width:3em;" | 60°
|style = "width:3em;" | 72°
|style = "width:3em;" | 90°
|style = "width:3em;" | 120°
|style = "width:3em;" | 144°
|style = "width:3em;" | 180°
|style = "width:3em;" | 270°
|style = "width:3em;" | 360°
|-
|style = "background:#f2f2f2; text-align:left;" | '''[[Grad (angle)|Grads]]'''
| 0<sup>g</sup>
| 16⅔<sup>g</sup>
| 33⅓<sup>g</sup>
| 40<sup>g</sup>
| 50<sup>g</sup>
| 66⅔<sup>g</sup>
| 80<sup>g</sup>
| 100<sup>g</sup>
| 133⅓<sup>g</sup>
| 160<sup>g</sup>
| 200<sup>g</sup>
| 300<sup>g</sup>
| 400<sup>g</sup>
|}
 
==Tau proposal==
[[File:Circle_radians_tau.gif|thumb|200px|right|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 ({{tau}}).]]
 
In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of <math>\pi</math>, 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 (<math>\pi\!\;\!\!\!\pi = 2 \pi</math>).<ref name=palais>[http://www.math.utah.edu/%7Epalais/pi.pdf Palais, R. 2001: Pi is Wrong, The Mathematical Intelligencer. Springer-Verlag New York. Volume 23, Number 3, pp. 7–8]</ref> In 2010, Michael Hartl proposed to use the Greek letter {{tau}} (tau) instead for two reasons. First, {{tau}} is the radian angle measure for one ''turn'' of a circle, which allows fractions of a turn to be expressed, such as <math>\tfrac{2}{5}\tau</math> for a <math>\tfrac{2}{5}</math> turn or <math>\tfrac{4}{5} \pi</math>. Second, {{tau}} visually resembles {{pi}}, whose association with the circle constant is unavoidable.<ref name="tauday">{{cite web |url=http://tauday.com/tau-manifesto |title=The Tau Manifesto |author=Michael Hartl |date=March 14, 2013  |accessdate=September 14, 2013}}</ref> Hartl's ''Tau Manifesto'' gives many examples of formulas that are simpler if tau is used instead of pi.<ref>{{Citation |first=Jacob |last=Aron |date=8 January 2011 |title=Interview: Michael Hartl: It's time to kill off pi |journal=[[New Scientist]] |volume=209 |issue=2794 |doi=10.1016/S0262-4079(11)60036-5|bibcode = 2011NewSc.209...23A }}</ref><ref>{{Citation |first=Elizabeth |last=Landau |date=14 March 2011 |title=On Pi Day, is 'pi' under attack? |work=cnn.com |url=http://edition.cnn.com/2011/TECH/innovation/03/14/pi.tau.math/index.html}}</ref>
 
== Examples of use ==
* As an angular unit, the turn or revolution is particularly useful for large angles, such as in connection with [[coil]]s and [[rotation|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 ray|external]] and internal angles. The sum of external angles of a polygon equals one turn. Angle [[doubling map]] is used.
* [[Pie chart]]s 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 number]]s or [[quaternion]]s. 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<sup>''b'' i</sup> that lies on the [[unit circle]]:
: <math>z \mapsto u z .</math>
[[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 (geometry)|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 ==
 
* [[Revolutions per minute]]
* [[Angle of rotation]]
 
== Notes and references ==
{{reflist|colwidth=30em}}
 
==External links==
* [http://www.math.utah.edu/%7Epalais/pi.pdf "π Is Wrong!" by Bob Palais]
 
{{DEFAULTSORT:Turn (Geometry)}}
[[Category:Units of angle]]

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