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Turn (angle)

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Turn
Unit ofPlane angle
Symboltr, pla
Conversions
1 tr in ...... is equal to ...
   radians   6.283185307179586... rad
   radians   2π rad
   degrees   360°
   gradians   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 degrees or 400 gradians. A turn is also referred to as a cycle (abbreviated cyc), revolution (abbreviated rev), complete rotation (abbreviated rot) or full circle.

Subdivisions of a turn include half turns, quarter turns, centiturns, milliturns, 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.[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.

History

The word turn originates via Latin and French from the Greek word τόρνος (tórnos – 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] In June 2017, for release 3.6, the Python programming language adopted the name tau to represent the number of radians in a turn.[12]

The standard ISO 80000-3:2006 mentions that the unit name revolution with symbol r is used with rotating machines, as well as using the term turn to mean a full rotation. The standard IEEE 260.1:2004 also uses the unit name rotation and symbol r.

Unit conversion

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

One turn is equal to 2π (≈ 6.283185307179586)[13] radians.

Conversion of common angles
Turns Radians Degrees Gradians
0 turn 0 rad 0° 0g
1/72 turn π/36 rad 5° 5+5/9g
1/24 turn π/12 rad 15° 16+2/3g
1/16 turn π/8 rad 22.5° 25g
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 or τ turn 1 rad approx. 57.3° approx. 63.7g
1/6 turn π/3 rad 60° 66+2/3g
1/5 turn 2π or τ/5 rad 72° 80g
1/4 turn π/2 rad 90° 100g
1/3 turn 2π or τ/3 rad 120° 133+1/3g
2/5 turn 4π or α/5 rad 144° 160g
1/2 turn π rad 180° 200g
3/4 turn 3π or ρ/2 rad 270° 300g
1 turn τ or 2π rad 360° 400g

Tau proposals

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 the Greek letter tau 𝜏 as a symbol for 1/2π, by analogy with the quarter period of elliptic functions, selecting the new symbol because π resembles two 𝜏 symbols conjoined (𝜏𝜏).[14]

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. His proposal used a "pi with three legs" symbol to denote the constant ( = 2π).[15]

In 2010, Michael Hartl proposed to use tau to represent Palais' circle constant (not Eagle's): 𝜏 = 2π. He offered 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.[16] Hartl's Tau Manifesto gives many examples of formulas that are asserted to be clearer where tau is used instead of pi.[17][18][19]

None of these proposals has been taken up by the mathematical and scientific communities.[20]

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 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:

z ↦ uz.

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.[21]

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

  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. ^ https://www.python.org/dev/peps/pep-0628/
  13. ^ Sequence OEIS: A019692
  14. ^ 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. pp. ix–x. {{cite book}}: Cite has empty unknown parameter: |lay-url= (help)
  15. ^ Palais, Robert (2001). "Pi is Wrong" (PDF). The Mathematical Intelligencer. 23 (3). New York, USA: Springer-Verlag: 7–8. doi:10.1007/bf03026846.
  16. ^ Hartl, Michael (2013-03-14). "The Tau Manifesto". Retrieved 2013-09-14.
  17. ^ 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.
  18. ^ Landau, Elizabeth (2011-03-14). "On Pi Day, is 'pi' under attack?". cnn.com.
  19. ^ "Why Tau Trumps Pi". Scientific American. 2014-06-25. Retrieved 2015-03-20.
  20. ^ "Life of pi in no danger – Experts cold-shoulder campaign to replace with tau". Telegraph India. 30 June 2011. Archived from the original on 13 July 2013. {{cite journal}}: Unknown parameter |deadurl= ignored (|url-status= suggested) (help)
  21. ^ 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

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