Turn (angle)
A turn is a unit of plane angle measurement equal to radians, 360 degrees or 400 gradians. A turn is also referred to as a cycle (abbreviated cyc. or cyl.), revolution (abbreviated rev.), complete rotation (abbreviated rot.) or full circle. Subdivisions of a turn include half-turns, quarter-turns, centiturns, milliturns, points, etc. Subdivisions. 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". 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 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 notion of turn is commonly used for planar rotations. History. The word turn originates via Latin and French from the Greek word ( – a lathe). In 1697, David Gregory  used (pi over rho) to denote the perimeter of a circle (i.e., the circumference) divided by its radius. However, earlier in 1647, William Oughtred 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 mathematician William Jones. Euler 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. Versors form points  in elliptic space and motivate the study of quaternions, 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 in 1962. Some measurement devices for artillery and satellite watching 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. However, it is a legal unit of measurement in the EU  and Switzerland. 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". 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. Unit conversion. One turn is equal to (≈ ) radians, 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 (i.e. Pi is approx. 6.28...) of a circle. 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 (formula_1). In 2008, Thomas Colignatus proposed the uppercase Greek letter theta, Θ, to represent 2. 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. 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: The following table shows how various identities appear if was used instead of . For a more complete list, see List of formulae involving π. ! Formula !! Using !! Using !! Notes