Turn (geometry)
Turn | |
---|---|
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° |
gradians | 400^{g} |
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.
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 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 2^{n} equal parts for other values of n.^{[2]}
The notion of turn is commonly used for planar rotations.
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.^{[3]}^{[4]} 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.^{[5]} Euler adopted the symbol with that meaning in 1737, leading to its widespread use.
Percentage protractors have existed since 1922,^{[6]} but the terms centiturns and milliturns were introduced much later by Fred Hoyle.^{[7]}
The German standard DIN 1315 (1974-03) proposed the unit symbol pla (from Latin: plenus angulus "full angle") for turns.^{[8]}^{[9]} 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.^{[10]} In June 2017, for release 3.6, the Python programming language adopted the name tau to represent the number of radians in a turn.^{[11]}
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
One turn is equal to 2π (≈ 6.283185307179586)^{[12]} radians.
Turns | Radians | Degrees | Gradians (Gons) |
---|---|---|---|
0 | 0 | 0° | 0^{g} |
1/24 | π/12 | 15° | 16 2/3^{g} |
1/12 | π/6 | 30° | 33 1/3^{g} |
1/10 | π/5 | 36° | 40^{g} |
1/8 | π/4 | 45° | 50^{g} |
1/2π | 1 | c. 57.3° | c. 63.7^{g} |
1/6 | π/3 | 60° | 66 2/3^{g} |
1/5 | 2π/5 | 72° | 80^{g} |
1/4 | π/2 | 90° | 100^{g} |
1/3 | 2π/3 | 120° | 133 1/3^{g} |
2/5 | 4π/5 | 144° | 160^{g} |
1/2 | π | 180° | 200^{g} |
3/4 | 3π/2 | 270° | 300^{g} |
1 | 2π | 360° | 400^{g} |
Tau proposals
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).^{[13]}
In 2010, Michael Hartl proposed to use tau to represent Palais' circle constant: 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.^{[14]} Hartl's Tau Manifesto^{[15]} gives many examples of formulas that are asserted to be clearer where tau is used instead of pi.^{[16]}^{[17]}^{[18]}
The proposal is implemented in the Google calculator and in several computer programs like Python^{[19]}, Perl^{[20]}, Processing^{[21]}, and Nim^{[22]}. It has also been used in at least one mathematical research article,^{[23]} authored by the τ-promoter P. Harremoës.^{[24]}
However, none of these proposals have received widespread acceptance by the mathematical and scientific communities.^{[25]}
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 = e^{bi} 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.^{[26]}
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
- Angle of rotation
- Revolutions per minute
- Repeating circle
- Spat (unit) — the 3D counterpart of the turn, equivalent to 4π steradians.
- Unit interval
- Turn (rational trigonometry)
- Spread (rational trigonometry)
- Modulo operation
Notes and references
- ^ "ooPIC Programmer's Guide". www.oopic.com. Archived from the original on 2008-06-28.
- ^ Hargreaves, Shawn. "Angles, integers, and modulo arithmetic". blogs.msdn.com.
- ^ Beckmann, Petr (1989). A History of Pi. Barnes & Noble Publishing.
- ^ Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. The Mathematical Association of America. p. 165.
- ^ "Pi through the ages".
- ^ Croxton, Frederick E. (1922). "A Percentage Protractor". Journal of the American Statistical Association. 18: 108–109. doi:10.1080/01621459.1922.10502455.
- ^ Hoyle, Fred (1962). Astronomy. London: Macdonald.
- ^ 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.
- ^ 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.
- ^ http://www.hpmuseum.org/forum/thread-4783-post-55836.html#pid55836
- ^ https://www.python.org/dev/peps/pep-0628/
- ^ Sequence A019692
- ^ Palais, Robert (2001). "Pi is Wrong" (PDF). The Mathematical Intelligencer. New York, USA: Springer-Verlag. 23 (3): 7–8. doi:10.1007/bf03026846.
- ^ Hartl, Michael (2013-03-14). "The Tau Manifesto". Retrieved 2013-09-14.
- ^ https://hexnet.org/files/documents/tau-manifesto.pdf
- ^ 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.
- ^ Landau, Elizabeth (2011-03-14). "On Pi Day, is 'pi' under attack?". cnn.com.
- ^ "Why Tau Trumps Pi". Scientific American. 2014-06-25. Retrieved 2015-03-20.
- ^ "Python 3.7.0 documentation".
- ^ "Perl 6".
- ^ "Processing".
- ^ "Nim".
- ^ Harremoës, Peter. "Bounds on tail probabilities for negative binomial distributions". Kybernetika. 52 (6): 943–966. doi:10.14736/kyb-2016-6-0943.
- ^ Harremoës, Peter. "Al-Kashi's constant τ" (PDF). Retrieved 20 September 2018.
- ^ "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.
- ^ 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
- Palais, Robert (2001). "Pi is Wrong" (PDF). The Mathematical Intelligencer. New York, USA: Springer-Verlag. 23 (3): 7–8. doi:10.1007/bf03026846.