Turn (geometry)

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Unit of Plane angle
Symbol tr or pla 
Unit conversions
1 tr in ... ... is equal to ...
   radians    6.283185307179586... rad
   radians    2π rad
   degrees    360°
   gons    400g
Counterclockwise rotations about the center point where a complete rotation is equal to 1 turn

A turn is a unit of plane angle measurement equal to 2π radians, 360° or 400 gon. 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[edit]

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 1256 turn.[1] The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte. Other measures of angle used in computing may be based on dividing one whole turn into 2n 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 (π radians),[3] a rotation through 90° is referred to as a quarter-turn.


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

In 1697, David Gregory used π/ρ (pi over rho) to denote the perimeter of a circle (i.e., the circumference) divided by its radius.[4][5] However, earlier in 1647, William Oughtred had used δ/π (delta over pi) for the ratio of the diameter to perimeter. The first use of the symbol π 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]

The German standard DIN 1315 (1974-03) proposed the unit symbol pla (from Latin: plenus angulus "full angle") for turns.[9][10] Since 2011, the HP 39gII and HP Prime support the unit symbol tr for turns. In 2016, support for turns was also added to newRPL for the HP 50g.[11]

Unit conversion[edit]

The circumference of the unit circle (whose radius is one) is 2π.

One turn is equal to 2π (≈ 6.283185307179586)[12] radians.

Conversion of common angles
Turns Radians Degrees Gradians (Gons)
0 turn 0 rad 0g
1/24 turn π/12 rad 15° 16 2/3g
1/12 turn π/6 rad 30° 33 1/3g
1/10 turn π/5 rad 36° 40g
1/8 turn π/4 rad 45° 50g
1/2π turn 1 rad c. 57.3° c. 63.7g
1/6 turn π/3 rad 60° 66 2/3g
1/5 turn 2π/5 rad 72° 80g
1/4 turn π/2 rad 90° 100g
1/3 turn 2π/3 rad 120° 133 1/3g
2/5 turn 4π/5 rad 144° 160g
1/2 turn π rad 180° 200g
3/4 turn 3π/2 rad 270° 300g
1 turn 2π rad 360° 400g

Tau proposal[edit]

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 (τ).

In 1958, Albert Eagle proposed τ as a symbol for 1/2π, because π resembles two τ symbols conjoined (ττ).[13]

In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of π, 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 ( = 2π).[14]

In 2010, Michael Hartl proposed to use the Greek letter tau (τ) instead of pi (π) for two reasons. First, τ is the number of radians in one turn, which allows fractions of a turn to be expressed more directly: for instance, a 3/4 turn would be represented as 3/4τ rad instead of 3/2π rad. Second, τ visually resembles π, whose association with the circle constant is unavoidable.[15] Hartl's Tau Manifesto gives many examples of formulas that are simpler if tau is used instead of pi.[16][17][18]

Examples of use[edit]

  • 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[edit]

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 cis(a) = r cos(a) + ri sin(a) where r > 0 and a is in [0, 2π). A turn of the complex plane arises from multiplying z = x + iy by an element u = ebi that lies on the unit circle:


Frank Morley consistently referred to elements of the unit circle as turns in the book Inversive Geometry, (1933) which he coauthored with his son Frank Vigor Morley.[19]

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[edit]

Notes and references[edit]

  1. ^ "ooPIC Programmer's Guide". www.oopic.com. 
  2. ^ Hargreaves, Shawn. "Angles, integers, and modulo arithmetic". blogs.msdn.com. 
  3. ^ "Half Turn, Reflection in Point". cut-the-knot.org. 
  4. ^ Beckmann, Petr (1989). A History of Pi. Barnes & Noble Publishing. 
  5. ^ Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. The Mathematical Association of America. p. 165. 
  6. ^ "Pi through the ages". 
  7. ^ Croxton, Frederick E. (1922). "A Percentage Protractor". Journal of the American Statistical Association. 18: 108–109. doi:10.1080/01621459.1922.10502455. 
  8. ^ Hoyle, Fred (1962). Astronomy. London: Macdonald. 
  9. ^ German, Sigmar; Drath, Peter (2013-03-13) [1979]. Handbuch SI-Einheiten: Definition, Realisierung, Bewahrung und Weitergabe der SI-Einheiten, Grundlagen der Präzisionsmeßtechnik (in German) (1 ed.). Friedrich Vieweg & Sohn Verlagsgesellschaft mbH, reprint: Springer-Verlag. ISBN 3322836061. 978-3-528-08441-7, 9783322836069. Retrieved 2015-08-14. 
  10. ^ Kurzweil, Peter (2013-03-09) [1999]. Das Vieweg Einheiten-Lexikon: Formeln und Begriffe aus Physik, Chemie und Technik (in German) (1 ed.). Vieweg, reprint: Springer-Verlag. doi:10.1007/978-3-322-92920-4. ISBN 3322929205. 978-3-322-92921-1. Retrieved 2015-08-14. 
  11. ^ http://www.hpmuseum.org/forum/thread-4783-post-55836.html#pid55836
  12. ^ Sequence OEISA019692
  13. ^ Eagle, Albert (1958). The Elliptic Functions as They Should Be: An Account, with Applications, of the Functions in a New Canonical Form. Cambridge, England: Galloway and Porter. 
  14. ^ Palais, Robert (2001). "Pi is Wrong" (PDF). The Mathematical Intelligencer. New York, USA: Springer-Verlag. 23 (3): 7–8. doi:10.1007/bf03026846. 
  15. ^ Hartl, Michael (2013-03-14). "The Tau Manifesto". Retrieved 2013-09-14. 
  16. ^ Aron, Jacob (2011-01-08). "Interview: Michael Hartl: It's time to kill off pi". New Scientist. 209 (2794): 23. Bibcode:2011NewSc.209...23A. doi:10.1016/S0262-4079(11)60036-5. 
  17. ^ Landau, Elizabeth (2011-03-14). "On Pi Day, is 'pi' under attack?". cnn.com. 
  18. ^ "Why Tau Trumps Pi". Scientific American. 2014-06-25. Retrieved 2015-03-20. 
  19. ^ Morley, Frank; Morley, Frank Vigor (2014) [1933]. Inversive Geometry. Boston, USA; New York, USA: Ginn and Company, reprint: Courier Corporation, Dover Publications. ISBN 978-0-486-49339-8. 0-486-49339-3. Retrieved 2015-10-17. 

External links[edit]