Turn (geometry)

Counterclockwise rotations about the center point where a complete rotation is equal to 1 turn

A turn is a unit of plane angle measurement equal to 2π radians, 360 degrees or 400 gradians. A turn is also referred to as a cycle (abbreviated cyc), revolution (abbreviated rev), complete rotation (abbreviated rot) or full circle.

Subdivisions of a turn include half turns, quarter turns, centiturns, milliturns, points, etc.

Subdivision of turns

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″. 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 ​¹⁄₂₅₆ turn.^[1] 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 2ⁿ equal parts for other values of n.^[2]

The notion of turn is commonly used for planar rotations.

History

The word turn originates via Latin and French from the Greek word τόρνος (tórnos – a lathe).

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.^[3][4] 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.^[5] Euler adopted the symbol with that meaning in 1737, leading to its widespread use.

Percentage protractors have existed since 1922,^[6] but the terms centiturns and milliturns were introduced much later by Fred Hoyle.^[7]

The German standard DIN 1315 (1974-03) proposed the unit symbol pla (from Latin: plenus angulus "full angle") for turns.^[8][9] Since 2011, the HP 39gII and HP Prime support the unit symbol tr for turns. In 2016, support for turns was also added to newRPL for the HP 50g.^[10] In June 2017, for release 3.6, the Python programming language adopted the name tau to represent the number of radians in a turn.^[11]

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.

Unit conversion

The circumference of the unit circle (whose radius is one) is 2π.

One turn is equal to 2π (≈ 6.283185307179586)^[12] radians.

Tau proposals

An arc of a circle with the same length as the radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to a full turn, or approximately 6.28 radians, which are expressed here using the Greek letter tau (Template:Tau).

In 2001, Robert Palais proposed 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 "pi with three legs" symbol to denote the constant ([math]\displaystyle{ \pi\!\;\!\!\!\pi }[/math] = 2π).^[13]

In 2010, Michael Hartl proposed to use tau to represent Palais' circle constant: Template:Tau = 2π. He offered two reasons. First, Template:Tau 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 3/4Template:Tau rad instead of 3/2π rad. Second, Template:Tau visually resembles π, whose association with the circle constant is unavoidable.^[14] Hartl's Tau Manifesto^[15] gives many examples of formulas that are asserted to be clearer where tau is used instead of pi.^[16][17][18]

The proposal is implemented in the Google calculator and in several computer programs like Python^[19], Perl^[20], Processing^[21], and Nim^[22]. It has also been used in at least one mathematical research article,^[23] authored by the Template:Tau-promoter P. Harremoës.^[24]

However, none of these proposals have received widespread acceptance by the mathematical and scientific communities.^[25]

Examples of use

• As an angular unit, the turn or revolution is particularly useful for large angles, such as in connection with electromagnetic coils and rotating objects. See also winding number.
• The angular speed of rotating machinery, such as automobile engines, is commonly measured in revolutions per minute or RPM.
• Turn is used in complex dynamics for measure of external and internal angles. The sum of external angles of a polygon equals one turn. Angle doubling map is used.
• Pie charts illustrate proportions of a whole as fractions of a turn. Each one percent is shown as an angle of one centiturn.

Kinematics of turns

In kinematics, a turn is a rotation less than a full revolution. A turn may be represented in a mathematical model that uses expressions of complex numbers or quaternions. In the complex plane every non-zero number has a polar coordinate expression z = r cis(a) = r cos(a) + ri sin(a) where r > 0 and a is in [0, 2π). A turn of the complex plane arises from multiplying z = x + iy by an element u = e^bi that lies on the unit circle:

z ↦ uz.

Frank Morley consistently referred to elements of the unit circle as turns in the book Inversive Geometry, (1933) which he coauthored with his son Frank Vigor Morley.^[26]

The Latin term for turn is versor, which is a quaternion that can be visualized as an arc of a great circle. The product of two versors can be compared to a spherical triangle where two sides add to the third. For the kinematics of rotation in three dimensions, see quaternions and spatial rotation.

See also

• Angle of rotation
• Revolutions per minute
• Repeating circle
• Spat (unit) — the 3D counterpart of the turn, equivalent to 4π steradians.
• Unit interval
• Turn (rational trigonometry)
• Spread (rational trigonometry)
• Modulo operation

Notes and references

External links

• "Pi is Wrong". The Mathematical Intelligencer (New York, USA: Springer-Verlag) 23 (3): 7–8. 2001. doi:10.1007/bf03026846. http://www.math.utah.edu/%7Epalais/pi.pdf. 

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