In the ISQ, an arbitrary "number of turns" (also known as "number of revolutions" or "number of cycles") is formalized as a dimensionless quantity called rotation, defined as the ratio of a given angle and a full turn. It is represented by the symbol N. (See below for the formula.) Subdivisions of a turn include half-turns and quarter-turns, spanning a semicircle and a right angle, respectively; metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc.
Another common unit for representing angles is radians, which are usually stated in terms of $\pi$ (pi). The symbol $\pi$, as representing one half-turn, was developed by William Jones in 1706 and then popularized by Leonhard Euler.^{[3]}^{[4]} In 2010, Michael Hartl proposed instead using the symbol $\tau$ (tau), equal to $2\pi$ and corresponding to one turn, for greater conceptual simplicity.^{[5]} This proposal did not initially gain widespread acceptance in the mathematical community,^{[6]} but the constant has become more widespread,^{[7]} having been added to several major programming languages and calculators.
Unit symbols
There are several unit symbols for the turn and for related concepts.
SI units
Rotation
Other names
number of revolutions, number of cycles, number of turns, number of rotations
The above definition is part of the ISQ, formalized in the international standard ISO 80000-3 (Space and time),^{[8]} and adopted in the International System of Units (SI).^{[9]}^{[10]} In the ISQ/SI, rotation is used to derive rotational frequency (the rate of change of rotation with respect to time), denoted by n:
The superseded version ISO 80000-3:2006 defined "revolution" as a special name for the dimensionless unit "one",^{[b]}
which also received other special names, such as the radian.^{[c]}
Despite their dimensional homogeneity, these two specially named dimensionless units are applicable for non-comparable kinds of quantity: rotation and angle, respectively.^{[12]}
"Cycle" is also mentioned in ISO 80000-3, in the definition of period.^{[d]}
EU and Switzerland
The German standard DIN 1315 (March 1974) proposed the unit symbol "pla" (from Latin: plenus angulus 'full angle') for turns.^{[13]}^{[14]} Covered in DIN 1301-1 [de] (October 2010), the so-called Vollwinkel ('full angle') is not an SI unit. However, it is a legal unit of measurement in the EU^{[15]}^{[16]} and Switzerland.^{[17]}
Calculators
The scientific calculators HP 39gII and HP Prime support the unit symbol "tr" for turns since 2011 and 2013, respectively. Support for "tr" was also added to newRPL for the HP 50g in 2016, and for the hp 39g+, HP 49g+, HP 39gs, and HP 40gs in 2017.^{[18]}^{[19]} An angular mode TURN was suggested for the WP 43S as well,^{[20]} but the calculator instead implements "MULπ" (multiples of π) as mode and unit since 2019.^{[21]}^{[22]}
Subdivisions
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″.^{[23]}^{[24]} A protractor divided in centiturns is normally called a "percentage protractor".
While percentage protractors have existed since 1922,^{[25]} the terms centiturns, milliturns and microturns were introduced much later by the British astronomer Fred Hoyle in 1962.^{[23]}^{[24]} Some measurement devices for artillery and satellite watching carry milliturn scales.^{[26]}^{[27]}
Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points, which implicitly have an angular separation of 1/32 turn. The binary degree, also known as the binary radian (or brad), is 1/256 turn.^{[28]} 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.^{[29]}
The number 2π is the ratio of a circle's circumference to its radius, and the number of radians in one turn.
The meaning of the symbol $\pi$ was not originally fixed to the ratio of the circumference and the diameter. 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.^{[30]}^{[31]} 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.^{[32]}^{[33]}Euler then adopted the symbol with that meaning, leading to its widespread use.^{[3]}
In 2001, Robert Palais proposed instead 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 "π with three legs" symbol to denote the constant ($\pi \!\;\!\!\!\pi =2\pi$).^{[34]}
In 2008, Thomas Colignatus proposed the uppercase Greek letter theta, Θ, to represent 2π.^{[35]}
The Greek letter theta derives from the Phoenician and Hebrew letter teth, 𐤈 or ט, and it has been observed that the older version of the symbol, which means wheel, resembles a wheel with four spokes.^{[36]} It has also been proposed to use the wheel symbol, teth, to represent the value 2π, and more recently a connection has been made among other ancient cultures on the existence of a wheel, sun, circle, or disk symbol—i.e. other variations of teth—as representation for 2π.^{[37]}
In 2010, Michael Hartl proposed to use the Greek letter tau to represent the 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 3τ/4 rad instead of 3π/2 rad. Second, τ visually resembles π, whose association with the circle constant is unavoidable.^{[5]} Hartl's Tau Manifesto^{[38]} gives many examples of formulas that are asserted to be clearer where τ is used instead of π.^{[39]}^{[40]}^{[41]} For example, Hartl asserts that replacing Euler's identitye^{iπ} = −1 by e^{iτ} = 1 (which Hartl also calls "Euler's identity") is more fundamental and meaningful.^{[38]}
Initially, neither of these proposals received widespread acceptance by the mathematical and scientific communities.^{[6]} However, the use of τ has become more widespread,^{[7]} for example:
In 2012, the educational website Khan Academy began accepting answers expressed in terms of τ.^{[42]}
It has also been used in at least one mathematical research article,^{[56]} authored by the τ-promoter Peter Harremoës.^{[57]}
The following table shows how various identities appear when τ = 2π is used instead of π.^{[58]}^{[34]} For a more complete list, see List of formulae involving π.
^Euler, Leonhard (1746). Nova theoria lucis et colorum. Opuscula varii argumenti (in Latin). sumtibus Ambr. Haude & Jo. Carol. Speneri, bibliop. p. 200. unde constat punctum B per datum tantum spatium de loco fuo naturali depelli, ad quam maximam distantiam pertinget, elapso tempore t=π/m denotante π angulum 180°, quo fit cos(mt)=- 1 & B b=2α. [from which it is clear that the point B is pushed by a given distance from its natural position, and it will reach the maximum distance after the elapsed time t=π/m, π denoting an angle of 180°, which becomes cos(mt)=- 1 & B b=2α.]
^Paul, Matthias R. (2016-01-12) [2016-01-11]. "RE: WP-32S in 2016?". HP Museum. Archived from the original on 2019-08-05. Retrieved 2019-08-05. […] I'd like to see a TURN mode being implemented as well. TURN mode works exactly like DEG, RAD and GRAD (including having a full set of angle unit conversion functions like on the WP 34S), except for that a full circle doesn't equal 360 degree, 6.2831... rad or 400 gon, but 1 turn. (I […] found it to be really convenient in engineering/programming, where you often have to convert to/from other unit representations […] But I think it can also be useful for educational purposes. […]) Having the angle of a full circle normalized to 1 allows for easier conversions to/from a whole bunch of other angle units […]
^Hayes, Eugene Nelson (1975) [1968]. Trackers of the Skies. History of the Smithsonian Satellite-tracking Program. Cambridge, Massachusetts, USA: Academic Press / Howard A. Doyle Publishing Company.
^Jones, William (1706). Synopsis Palmariorum Matheseos. London: J. Wale. pp. 243, 263. p. 263: There are various other ways of finding the Lengths, or Areas of particular Curve Lines or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to Circumference as 1 to ${\overline ((\tfrac {16}{5))-{\tfrac {4}{239))))-{\tfrac {1}{3)){\overline ((\tfrac {16}{5^{3))}-{\tfrac {4}{239^{3))))}+{\tfrac {1}{5)){\overline ((\tfrac {16}{5^{5))}-{\tfrac {4}{239^{5))))}-,\,\&c.=$ 3.14159, &c. = π. This Series (among others for the same purpose, and drawn from the same Principle) I receiv'd from the Excellent Analyst, and my much Esteem'd Friend Mr. John Machin; and by means thereof, Van Ceulen's Number, or that in Art. 64.38. may be Examin'd with all desireable Ease and Dispatch. Reprinted in Smith, David Eugene (1929). "William Jones: The First Use of π for the Circle Ratio". A Source Book in Mathematics. McGraw–Hill. pp. 346–347.
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