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 revolution or complete rotation or full circle or cycle or rev or rot.

A turn can be subdivided in many different ways: into half turns, quarter turns, centiturns, milliturns, binary angles, 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. Two special rotations have acquired appellations of their own: a rotation through 180° is commonly referred to as a half-turn (π radians),^[3] a rotation through 90° is referred to as a quarter-turn.

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.^[4][5] 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.^[6] Euler adopted the symbol with that meaning in 1737, leading to its widespread use.

Percentage protractors have existed since 1922,^[7] but the terms centiturns and milliturns were introduced much later by Sir Fred Hoyle.^[8]

The German standard DIN 1315 (1974-03) proposed the unit symbol pla (from Latin: plenus angulus "full angle") for turns.^[9][10] 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.^[11] In June 2017, for release 3.6, the Python programming language adopted the name tau to represent the number of radians in a turn.^[12]

Unit conversion

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

One turn is equal to 2π (≈ 6.283185307179586)^[13] radians.

Tau proposal

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 (τ).

In 1958, Albert Eagle proposed the Greek letter tau τ as a symbol for 1/2π, selecting the new symbol because π resembles two τ symbols conjoined (ττ).^[14]

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 ( = 2π).^[15]

In 2010, Michael Hartl proposed to use tau to represent Palais' circle constant (not Eagle's): τ = 2π. He offered two reasons. First, τ 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 3/4τ rad instead of 3/2π rad. Second, τ visually resembles π, whose association with the circle constant is unavoidable.^[16] Hartl's Tau Manifesto gives many examples of formulas that are intuitively clearer where tau is used instead of pi, particularly in cases where a factor of 2 is thus not canceled with another, unrelated factor.^[17][18][19]

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.^[20]

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)
• Unit interval
• Turn (rational trigonometry)
• Spread

Notes and references

External links

• Palais, Robert (2001). "Pi is Wrong" (PDF). The Mathematical Intelligencer. New York, USA: Springer-Verlag. 23 (3): 7–8. doi:10.1007/bf03026846.