![angle-fractions](https://upload.wikimedia.org/wikipedia/commons/4/45/Angle-fractions.png)
A turn is a unit of
plane 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 are ...
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 c ...
s, 360
degrees or 400
gradian
In trigonometry, the gradian, also known as the gon (from grc, γωνία, gōnía, angle), grad, or grade, is a unit of measurement of an angle, defined as one hundredth of the right angle; in other words, there are 100 gradians in 90 degree ...
s.
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 Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
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 us ...
protractor".
Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32
compass points
The points of the compass are a set of horizontal, radially arrayed compass directions (or azimuths) used in navigation and cartography. A compass rose is primarily composed of four cardinal directions—north, east, south, and west—each sepa ...
, which implicitly have an angular separation of 1/32 turn. The ''binary degree'', also known as the ''
binary radian
The terms binary angular measurement (BAM) and binary angular measurement system (BAMS) refer to certain methodologies for representing and manipulating angles using binary ( base 2) fixed-point arithmetic. The unit of angular measure used in t ...
'' (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 unit ...
. 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
Planar is an adjective meaning "relating to a plane (geometry)".
Planar may also refer to:
Science and technology
* Planar (computer graphics), computer graphics pixel information from several bitplanes
* Planar (transmission line technologies), ...
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 to c ...
).
In 1697,
David Gregory used (pi over rho) to denote the
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pract ...
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 to ...
) 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 peop ...
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 ma ...
adopted the symbol with that meaning in 1737, leading to its widespread use.
The Latin word for ''turn'' is
versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
, 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 (geometry), position of an element (i.e., Point (m ...
. Versors form points in
elliptic space
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
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 quatern ...
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
Satellite watching or satellite spotting is a hobby which consists of the observation and tracking of artificial satellites that are orbiting Earth. People with this hobby are variously called satellite watchers, trackers, spotters, observers, ...
carry milliturn scales.
Unit symbols
The German standard
DIN 1315 (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. E ...
. 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 ...
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 symbolic ...
,
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 symbolic ...
, 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 symbolic ...
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 science ...
as well,
but the calculator instead implements "MUL" (''
multiples of '') as mode and unit since 2019.
Unit conversion
![Degree-Radian Conversion tau](https://upload.wikimedia.org/wikipedia/commons/c/c2/Degree-Radian_Conversion_tau.svg)
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 c ...
s, 360
degrees, or 400
gradian
In trigonometry, the gradian, also known as the gon (from grc, γωνία, gōnía, angle), grad, or grade, is a unit of measurement of an angle, defined as one hundredth of the right angle; in other words, there are 100 gradians in 90 degree ...
s.
Proposals for a single letter to represent 2
![Circle_radians_tau](https://upload.wikimedia.org/wikipedia/commons/2/28/Circle_radians_tau.gif)
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 (
).
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.
Gr ...
, 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
Tau (uppercase Τ, lowercase τ, or \boldsymbol\tau; el, ταυ ) is the 19th letter of the Greek alphabet, representing the voiceless dental or alveolar plosive . In the system of Greek numerals, it has a value of 300.
The name in English ...
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 in ...
began accepting answers expressed in terms of .
* The constant is made available in the Google calculator and in several programming languages such as
Python
Python may refer to:
Snakes
* Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia
** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia
* Python (mythology), a mythical serpent
Computing
* Python (pro ...
,
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(OH ...
,
Java,
.NET,
and
Haskell
Haskell () is a general-purpose, statically-typed, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research and industrial applications, Haskell has pioneered a number of programming lan ...
.
* 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 Electrical conductivity, conductor such as a wire in the shape of a wiktionary:coil, coil (spiral or helix). Electromagnetic coils are used in electrical engineering, in applications where electric curre ...
s and
rotating
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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 turn ...
''.
*
Pie chart
A pie chart (or a circle chart) is a circular Statistical graphics, 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 are ...
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 con ...
*
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 on ...
(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 instance ...
*
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 dimensionl ...
*
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 spat (symbol sp), from the Latin ''spatium'' ("space"), is a Unit (measurement), unit of solid angle. 1 spat is equal to 4 steradians or approximately square degrees of solid angle . Thus it is the solid angle subtended by a comple ...
– 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
steradian
The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radian ...
s.
*
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 th ...
*
Twist (mathematics)
In differential geometry, the twist of a ''ribbon'' is its rate of axial rotation. Let a ribbon (X,U) be composted of space curve X=X(s), where s is the arc length of X, and U=U(s) the a unit normal vector, perpendicular at each point to X. Sinc ...
References
External links
Tau manifesto
{{DEFAULTSORT:Turn (Geometry)
Units of plane angle
Mathematical concepts
1 (number)