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A turn is a unit of plane angle measurement equal to  
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s, 360  degrees or 400  gradians. Subdivisions of a turn include half-turns, quarter-turns, centiturns, milliturns, etc. The closely related terms ''cycle'' and ''revolution'' are not equivalent to a turn.


Subdivisions

A turn can be divided in 100 centiturns or milliturns, with each milliturn corresponding to an
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
of 0.36°, which can also be written as 21′ 36″. A
protractor A protractor is a measuring instrument, typically made of transparent plastic or glass, for measuring angles. Some protractors are simple half-discs or full circles. More advanced protractors, such as the bevel protractor, have one or two sw ...
divided in centiturns is normally called a "
percentage In mathematics, a percentage (from la, per centum, "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also use ...
protractor". 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 The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable uni ...
. Other measures of angle used in computing may be based on dividing one whole turn into equal parts for other values of . The notion of turn is commonly used for planar rotations.


History

The word turn originates via Latin and French from the Greek word ( – a
lathe A lathe () is a machine tool that rotates a workpiece about an axis of rotation to perform various operations such as cutting, sanding, knurling, drilling, deformation, facing, and turning, with tools that are applied to the workpiece ...
). In 1697, David Gregory used (pi over rho) to denote the perimeter of a circle (i.e., the
circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out ...
) divided by its radius. However, earlier in 1647,
William Oughtred William Oughtred ( ; 5 March 1574 – 30 June 1660), also Owtred, Uhtred, etc., was an Kingdom of England, English mathematician and Anglican ministry, Anglican clergyman.'Oughtred (William)', in P. Bayle, translated and revised by J.P. Bernar ...
had used (delta over pi) for the ratio of the diameter to perimeter. The first use of the symbol on its own with its present meaning (of perimeter divided by diameter) was in 1706 by the
Welsh Welsh may refer to: Related to Wales * Welsh, referring or related to Wales * Welsh language, a Brittonic Celtic language spoken in Wales * Welsh people People * Welsh (surname) * Sometimes used as a synonym for the ancient Britons (Celtic peopl ...
mathematician William Jones.
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
adopted the symbol with that meaning in 1737, leading to its widespread use. The Latin word for ''turn'' is versor, which represents a rotation about an arbitrary axis in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
. Versors form points in elliptic space and motivate the study of
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s, an algebra developed by W. R. Hamilton in the 1840s. Percentage protractors have existed since 1922, but the terms centiturns, milliturns and microturns were introduced much later by the British astronomer
Fred Hoyle Sir Fred Hoyle FRS (24 June 1915 – 20 August 2001) was an English astronomer who formulated the theory of stellar nucleosynthesis and was one of the authors of the influential B2FH paper. He also held controversial stances on other sci ...
in 1962. Some measurement devices for artillery and satellite watching carry milliturn scales.


Unit symbols

The German standard
DIN 1315 DIN or Din or din may refer to: People and language * Din (name), people with the name * Dīn, an Arabic word with three general senses: judgment, custom, and religion from which the name originates * Dinka language (ISO 639 code: din), spoken by ...
(March 1974) proposed the unit symbol "pla" (from Latin: 'full angle') for turns. Covered in (October 2010), the so-called ('full angle') is not an
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
. However, it is a legal unit of measurement in the EU and Switzerland. The scientific calculators
HP 39gII HP 39/40 series are graphing calculators from Hewlett-Packard, the successors of HP 38G. The series consists of six calculators, which all have algebraic entry modes, and can perform numeric analysis together with varying degrees of symbolic ...
and HP Prime support the unit symbol "tr" for turns since 2011 and 2013, respectively. Support for "tr" was also added to
newRPL RPL is a handheld calculator operating system and application programming language used on Hewlett-Packard's scientific graphing RPN (Reverse Polish Notation) calculators of the HP 28, 48, 49 and 50 series, but it is also usable on non-RPN c ...
for the HP 50g in 2016, and for the
hp 39g+ HP 39/40 series are graphing calculators from Hewlett-Packard, the successors of HP 38G. The series consists of six calculators, which all have algebraic entry modes, and can perform numeric analysis together with varying degrees of symboli ...
, HP 49g+,
HP 39gs HP 39/40 series are graphing calculators from Hewlett-Packard, the successors of HP 38G. The series consists of six calculators, which all have algebraic entry modes, and can perform numeric analysis together with varying degrees of symboli ...
, and
HP 40gs HP 39/40 series are graphing calculators from Hewlett-Packard, the successors of HP 38G. The series consists of six calculators, which all have algebraic entry modes, and can perform numeric analysis together with varying degrees of symboli ...
in 2017. An angular mode TURN was suggested for the
WP 43S The HP-42S RPN Scientific is a programmable RPN Scientific hand held calculator introduced by Hewlett Packard in 1988. It has advanced functions suitable for applications in mathematics, linear algebra, statistical analysis, computer scienc ...
as well, but the calculator instead implements "MUL" ('' multiples of '') as mode and unit since 2019.


Unit conversion

One turn is equal to (≈ )
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s, 360 degrees, or 400 gradians.


Proposals for a single letter to represent 2

In 1746, Leonard Euler first used the Greek letter pi to represent the circumference divided by the radius of a circle (i.e., = 6.28...). 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, Thomas Colignatus proposed the uppercase Greek letter
theta Theta (, ; uppercase: Θ or ; lowercase: θ or ; grc, ''thē̂ta'' ; Modern: ''thī́ta'' ) is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth . In the system of Greek numerals, it has a value of 9. G ...
, Θ, to represent 2. The Greek letter theta derives from the Phoenician and Hebrew letter
teth Teth, also written as or Tet, is the ninth letter of the Semitic abjads, including Phoenician Ṭēt , Hebrew Tēt , Aramaic Ṭēth , Syriac Ṭēṯ ܛ, and Arabic . It is the 16th letter of the modern Arabic alphabet. The Persian ṭa ...
, 𐤈 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 quantity 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 two reasons. First, is the number of radians in ''one turn'', which allows fractions of a turn to be expressed more directly: for instance, a  turn would be represented as  rad instead of  rad. 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 . Initially, neither of these proposals received widespread acceptance by the mathematical and scientific communities. However, the use of has become more widespread, for example: * In 2012, the educational website
Khan Academy Khan Academy is an American non-profit educational organization created in 2008 by Sal Khan. Its goal is creating a set of online tools that help educate students. The organization produces short lessons in the form of videos. Its website also i ...
began accepting answers expressed in terms of . * The constant is made available in the Google calculator and in several programming languages such as Python, Raku,
Processing Processing is a free graphical library and integrated development environment (IDE) built for the electronic arts, new media art, and visual design communities with the purpose of teaching non-programmers the fundamentals of computer programming ...
, Nim,
Rust Rust is an iron oxide, a usually reddish-brown oxide formed by the reaction of iron and oxygen in the catalytic presence of water or air moisture. Rust consists of hydrous iron(III) oxides (Fe2O3·nH2O) and iron(III) oxide-hydroxide (FeO( ...
, Java, .NET, and Haskell. * It has also been used in at least one mathematical research article, authored by the -promoter Peter Harremoës. The following table shows how various identities appear if was used instead of . For a more complete list, see '' List of formulae involving ''.


Examples of use

* As an angular unit, the turn is particularly useful in many applications, such as in connection with
electromagnetic coil An electromagnetic coil is an electrical conductor such as a wire in the shape of a coil (spiral or helix). Electromagnetic coils are used in electrical engineering, in applications where electric currents interact with magnetic fields, in d ...
s and rotating objects. See also ''
Winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of t ...
''. *
Pie chart A pie chart (or a circle chart) is a circular statistical graphic, which is divided into slices to illustrate numerical proportion. In a pie chart, the arc length of each slice (and consequently its central angle and area) is proportional t ...
s illustrate proportions of a whole as fractions of a turn. Each one percent is shown as an angle of one centiturn.


See also

*
Ampere-turn The ampere-turn (A⋅t) is the MKS (metre–kilogram–second) unit of magnetomotive force (MMF), represented by a direct current of one ampere flowing in a single-turn loop in a vacuum. " Turns" refers to the winding number of an electrical ...
*
Hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
(modern) or
Cycle per second The cycle per second is a once-common English name for the unit of frequency now known as the hertz (Hz). The plural form was typically used, often written cycles per second, cycles/second, c.p.s., c/s, or, ambiguously, just cycles (Cy./Cyc.). The ...
(older) *
Angle of rotation In mathematics, the angle of rotation is a measurement of the amount, of namely angle, that a figure is rotated about a fixed point, often the center of a circle. A clockwise rotation is considered a negative rotation, so that, for instanc ...
*
Revolutions per minute Revolutions per minute (abbreviated rpm, RPM, rev/min, r/min, or with the notation min−1) is a unit of rotational speed or rotational frequency for rotating machines. Standards ISO 80000-3:2019 defines a unit of rotation as the dimensio ...
*
Repeating circle Developed from the reflecting circle, the repeating circle is an instrument for geodetic surveying, invented by Etienne Lenoir in 1784, while an assistant of Jean-Charles de Borda, who later improved the instrument. It was notable as being the ...
* Spat (unit) – the
solid angle In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The poi ...
counterpart of the turn, equivalent to   steradians. *
Unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
* Divine Proportions: Rational Trigonometry to Universal Geometry *
Modulo operation In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
* Twist (mathematics)


References


External links


Tau manifesto
{{DEFAULTSORT:Turn (Geometry) Units of plane angle Mathematical concepts 1 (number)