A turn can be subdivided in many different ways: into half turns, quarter turns, centiturns, milliturns, binary angles, points etc.
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. The binary degree is used in computing so that an angle can be efficiently represented in a single byte (albeit to limited precision). Other measures of angle used in computing may be based on dividing one whole turn into 2n equal parts for other values of n.
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 ( radians), a rotation through 90° is referred to as a quarter-turn. A half-turn is often referred to as a reflection in a point since these are identical for transformations in two-dimensions.
The word turn originates via Latin and French from the Greek word τόρνος (tornos – a lathe).
In 1697, David Gregory used (pi/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/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.
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, using a "pi with three legs" symbol to denote the constant (). In 2010, Michael Hartl proposed to use the Greek letter τ (tau) instead for two reasons. First, τ is the radian angle measure for one turn of a circle, which allows fractions of a turn to be expressed, such as for a turn or . Second, τ visually resembles π, whose association with the circle constant is unavoidable. Hartl's Tau Manifesto gives many examples of formulas that are simpler if tau is used instead of pi.
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 cos a + r i sin a where r > 0 and a is in [0, 2π). A turn of the complex plane arises from multiplying z = x + i y by an element u = eb i that lies on the unit circle:
Frank Morley consistently referred to elements of the unit circle as turns in the book Inversive Geometry (1933) that he coauthored with his son Frank Vigor Morley.
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.