| Turn | |
| Othernames: | Revolution, Cycles |
| Quantity: | Plane angle |
| Symbol: | tr |
| Symbol2: | pla |
| Symbol3: | rev |
| Symbol4: | cyc |
| Units1: | radians |
| Inunits1: | rad ≈ |
| Units3: | milliradians |
| Inunits3: | mrad ≈ |
| Units4: | degrees |
| Inunits4: | 360° |
| Units5: | gradians |
| Inunits5: | 400g |
The turn (symbol tr or pla) is a unit of plane angle measurement that is the angular measure subtended by a complete circle at its center. It is equal to radians, 360 degrees or 400 gradians. As an angular unit, one turn also corresponds to one cycle (symbol cyc or c) or to one revolution (symbol rev or r). Common related units of frequency are cycles per second (cps) and revolutions per minute (rpm). The angular unit of the turn is useful in connection with, among other things, electromagnetic coils (e.g., transformers), rotating objects, and the winding number of curves. Subdivisions of a turn include the half-turn and quarter-turn, spanning a straight angle and a right angle, respectively; metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc.
Because one turn is
2\pi
\tau
2\pi
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.
There are several unit symbols for the turn.
The German standard DIN 1315 (March 1974) proposed the unit symbol "pla" (from Latin: Latin: plenus angulus 'full angle') for turns. Covered in (October 2010), the so-called German: Vollwinkel ('full angle') is not an SI unit. However, it is a legal unit of measurement in the EU and Switzerland.
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. An angular mode TURN was suggested for the WP 43S as well, but the calculator instead implements "MUL" (multiples of ) as mode and unit since 2019.
A turn can be divided in 100 centiturns or 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".
While percentage protractors have existed since 1922, the terms centiturns, milliturns and microturns were introduced much later by the British astronomer Fred Hoyle in 1962. Some measurement devices for artillery and satellite watching carry milliturn scales.
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 turn. 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 equal parts for other values of .
The number (approximately 6.28) 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
The first known usage of a single letter to denote the 6.28... constant was in Leonhard Euler's 1727 Essay Explaining the Properties of Air, where it was denoted by the letter . Euler would later use the letter for the 3.14... constant in his 1736 Mechanica and 1748 Introductio in analysin infinitorum, though defined as half the circumference of a circle of radius 1—a unit circle—rather than the ratio of circumference to diameter. Elsewhere in Introductio in analysin infinitorum, Euler instead used the letter for one-fourth of the circumference of a unit circle, or 1.57... . Eventually, was standardized as being equal to 3.14..., and its usage became widespread.[1]
Several people have independently proposed using, including:[2]
In 2001, Robert Palais proposed using the number of radians in a turn as the fundamental circle constant instead of, 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
In 2008, Robert P. Crease proposed the idea of defining a constant as the ratio of circumference to radius, a proposal supported by John Horton Conway. Crease used the Greek letter psi:
\psi=2\pi
The same year, Thomas Colignatus proposed the uppercase Greek letter theta, Θ, to represent 2.The Greek letter theta derives from the Phoenician and Hebrew letter teth, or ט, and it has been observed that the older version of the symbol, which means wheel, resembles a wheel with four spokes. It has also been proposed to use the wheel symbol, teth, to represent the value 2, 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 2.
In 2010, Michael Hartl proposed to use the Greek letter tau to represent the circle constant: . He offered several reasons for the choice of constant, primarily that it allows fractions of a turn to be expressed more directly: for instance, a turn would be represented as rad instead of rad. As for the choice of notation, he offered two reasons. First, is the number of radians in one turn, and both and turn begin with a sound. Second, visually resembles, whose association with the circle constant is unavoidable. Hartl's Tau Manifesto gives many examples of formulas that are asserted to be clearer where is used instead of . For example, Hartl asserts that replacing Euler's identity by (which Hartl also calls "Euler's identity") is more fundamental and meaningful. He also claims that the formula for circular area in terms of,, contains a natural factor of arising from integration.
Initially, neither of these proposals received widespread acceptance by the mathematical and scientific communities. However, the use of has become more widespread. For example:
The following table shows how various identities appear when is used instead of . For a more complete list, see List of formulae involving .
| Formula | Using | Using | Notes | ||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Angle subtended by of a circle |
|
| |||||||||||||||||||||
| Circumference of a circle of radius | C={\color{orangered}2\pi}r | C={\color{orangered}\tau}r | |||||||||||||||||||||
| Area of a circle | A={\color{orangered}\pi}r2 | A={\color{orangered}
\tau}r2 | The area of a sector of angle is . | ||||||||||||||||||||
| Area of a regular -gon with unit circumradius | A=
\sin
\pi}}{n} | A=
\sin
| |||||||||||||||||||||
| -ball and -sphere volume recurrence relation | Vn(r)=
Sn-1(r) Sn(r)={\color{orangered}2\pi}rVn-1(r) | Vn(r)=
Sn-1(r) Sn(r)={\color{orangered}\tau}rVn-1(r) | |||||||||||||||||||||
| Cauchy's integral formula | f(a)=
2\pi}i}\oint\gamma
dz | f(a)=
\tau}i}\oint\gamma
dz | \gamma a | ||||||||||||||||||||
| Standard normal distribution | \varphi(x)=
| \varphi(x)=
| |||||||||||||||||||||
| Stirling's approximation | n | \sim \sqrt\left(\frac\right)^n | n | \sim \sqrt\left(\frac\right)^n | |||||||||||||||||||
| th roots of unity | e{\color{orangered2\pi}i
| e{\color{orangered\tau}i
| |||||||||||||||||||||
| Planck constant | h={\color{orangered}2\pi}\hbar | h={\color{orangered}\tau}\hbar | is the reduced Planck constant. | ||||||||||||||||||||
| Angular frequency | \omega={\color{orangered}2\pi}f | \omega={\color{orangered}\tau}f | |||||||||||||||||||||
One turn is equal to (≈ ) radians, 360 degrees, or 400 gradians.
| Turns | Radians | Degrees | Gradians | ||
|---|---|---|---|---|---|
| 0 turn | 0 rad | 0° | 0g | ||
| turn | rad | rad | 5° | g | |
| turn | rad | rad | 15° | g | |
| turn | rad | rad | 22.5° | 25g | |
| turn | rad | rad | 30° | g | |
| turn | rad | rad | 36° | 40g | |
| turn | rad | rad | 45° | 50g | |
| turn | 1 rad | 57.3° | 63.7g | ||
| turn | rad | rad | 60° | g | |
| turn | rad | rad | 72° | 80g | |
| turn | rad | rad | 90° | 100g | |
| turn | rad | rad | 120° | g | |
| turn | rad | rad | 144° | 160g | |
| turn | rad | rad | 180° | 200g | |
| turn | rad | rad | 270° | 300g | |
| 1 turn | rad | 2 rad | 360° | 400g | |
| Rotation | |
| Othernames: | number of revolutions, number of cycles, number of turns, number of rotations |
| Unit: | Unitless |
| Symbols: | N |
| Dimension: | 1 |
In the International System of Quantities (ISQ), rotation (symbol N) is a physical quantity defined as number of revolutions:
N is the number (not necessarily an integer) of revolutions, for example, of a rotating body about a given axis. Its value is given by:where denotes the measure of rotational displacement.N=
\varphi 2\pirad
The above definition is part of the ISQ, formalized in the international standard ISO 80000-3 (Space and time), and adopted in the International System of Units (SI).
Rotation count or number of revolutions is a quantity of dimension one, resulting from a ratio of angular displacement.It can be negative and also greater than 1 in modulus.The relationship between quantity rotation, N, and unit turns, tr, can be expressed as:
N=
| \varphi | |
| tr |
=\{\varphi\}tr
In the ISQ/SI, rotation is used to derive rotational frequency (the rate of change of rotation with respect to time), denoted by :
n=
| dN | |
| dt |
The SI unit of rotational frequency is the reciprocal second (s−1). Common related units of frequency are hertz (Hz), cycles per second (cps), and revolutions per minute (rpm).
| Revolution | |
| Othernames: | Cycle |
| Quantity: | Rotation |
| Symbol: | rev |
| Symbol2: | r |
| Symbol3: | cyc |
| Symbol4: | c |
| Units1: | Base units |
| Inunits1: | 1 |
The superseded version ISO 80000-3:2006 defined "revolution" as a special name for the dimensionless unit "one",which also received other special names, such as the radian.Despite their dimensional homogeneity, these two specially named dimensionless units are applicable for non-comparable kinds of quantity: rotation and angle, respectively.[4] "Cycle" is also mentioned in ISO 80000-3, in the definition of period.