Turn (geometry)

Encyclopedia

A

equal to a 360°

or 2

radian

s or

(tau) radians. A turn is also referred to as a

A turn can be subdivided in many different ways, into half turns, quarter turns, centiturns, milliturns, binary angles, points etc.

of 0.36°, which can also be written as 21'36".

Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 points

. The

(albeit to limited precision). Other measures of angle used in computing may be based on dividing one whole turn into 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 ( 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 geometric notion of a turn has its origin in the sailors terminology of knots where a turn

means one round of rope on a pin or cleat

, or one round of a coil

. For knots the English terms of single turn, round turn and double round turn do not translate directly into the geometric notion of turn, but in German the correspondence is exact.

In 1697 David Gregory used (pi/rho) to denote the

in 1647 for the ratio of

The idea of using centiturns and milliturns as units was introduced by Sir Fred Hoyle.

Robert Palais proposed in 2001 to use the number of radians in a turn as the fundamental circle constant instead of , in order to make mathematics simpler and more intuitive, using a "pi with three legs" symbol to denote 1 turn (). In 2010, Michael Hartl proposed to use the Greek letter

(tau) to represent the number

instead.

Similarly, half a turn is often identified with the mathematical constant because half a turn is (≈3.14) radians.

a

A turn may be represented in a mathematical model

that uses expressions of complex number

s or quaternion

s. In the complex plane

every non-zero number has a polar coordinate expression

A turn of the complex plane arises from multiplying

:

Frank Morley

consistently referred to elements of the unit circle as

The Latin term for

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

.

**turn**is an angleAngle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

equal to a 360°

Degree (angle)

A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

or 2

Pi

' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

radian

Radian

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s or

Tau (2π)

Tau is a mathematical constant equal to the ratio of any circle's circumference to its radius, and has a value of approximately 6.28318531. This number also appears in many common formulas, often because it is the period of some very common functions — sine, cosine, , and others that involve...

(tau) radians. 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 angleAngle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

of 0.36°, which can also be written as 21'36".

Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 points

Compass Point

Compass Point may refer to:* Compass point, a direction on a traditional compass* Compass Point * Compass Point Shopping Centre, a shopping mall in Singapore* Compass Point Studios, a studio in Nassau, Bahamas...

. 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 byteByte

The byte is a unit of digital information in computing and telecommunications that most commonly consists of eight bits. Historically, a byte was the number of bits used to encode a single character of text in a computer and for this reason it is the basic addressable element in many computer...

(albeit to limited precision). Other measures of angle used in computing may be based on dividing one whole turn into 2

^{n}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.

## History

The word turn originates via Latin and French from the Greek word τόρνος (tornos – a latheLathe

A lathe is a machine tool which rotates the workpiece on its axis to perform various operations such as cutting, sanding, knurling, drilling, or deformation with tools that are applied to the workpiece to create an object which has symmetry about an axis of rotation.Lathes are used in woodturning,...

).

The geometric notion of a turn has its origin in the sailors terminology of knots where a turn

Turn (knot)

A turn is one round of rope on a pin or cleat, or one round of a coil. Turns can be made around various objects, through rings, or around the standing part of the rope itself or another rope. A turn also denotes a component of a knot....

means one round of rope on a pin or cleat

Cleat

Cleat may refer to:* Cleat , a type or part of a shoe* Cleat , a fitting on ships, boats, and docks to which ropes are tied* Cleats , a comic strip by Bill Hinds...

, or one round of a coil

Coil

A coil is a series of loops. A coiled coil is a structure in which the coil itself is in turn also looping.-Electromagnetic coils:An electromagnetic coil is formed when a conductor is wound around a core or form to create an inductor or electromagnet...

. For knots the English terms of single turn, round turn and double round turn do not translate directly into the geometric notion of turn, but in German the correspondence is exact.

In 1697 David Gregory used (pi/rho) to denote the

**p**erimeter of a circle (i.e. the circumference) divided by its**r**adius, though (delta/pi) had been used by William OughtredWilliam Oughtred

William Oughtred was an English mathematician.After John Napier invented logarithms, and Edmund Gunter created the logarithmic scales upon which slide rules are based, it was Oughtred who first used two such scales sliding by one another to perform direct multiplication and division; and he is...

in 1647 for the ratio of

**d**iameter to**p**erimeter. The first use of on its own with its present meaning of perimeter/diameter was by William Jones in 1706. Euler adopted the symbol with that meaning in 1737, leading to its widespread use.The idea of using centiturns and milliturns as units was introduced by Sir Fred Hoyle.

Robert Palais proposed in 2001 to use the number of radians in a turn as the fundamental circle constant instead of , in order to make mathematics simpler and more intuitive, using a "pi with three legs" symbol to denote 1 turn (). In 2010, Michael Hartl proposed to use the Greek letter

Tau (2π)

Tau is a mathematical constant equal to the ratio of any circle's circumference to its radius, and has a value of approximately 6.28318531. This number also appears in many common formulas, often because it is the period of some very common functions — sine, cosine, , and others that involve...

(tau) to represent the number

Tau (2π)

Tau is a mathematical constant equal to the ratio of any circle's circumference to its radius, and has a value of approximately 6.28318531. This number also appears in many common formulas, often because it is the period of some very common functions — sine, cosine, , and others that involve...

instead.

## Mathematical constants

One turn can be identified with (≈6.28) radians.Similarly, half a turn is often identified with the mathematical constant because half a turn is (≈3.14) radians.

## Conversion of some common angles

Units | | Values | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Turns |
0 | 1/12 | 1/10 | 1/8 | 1/6 | 1/5 | 1/4 | 1/2 | 3/4 | 1 |

RadianRadian Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit... s |
0 | |||||||||

Degrees Degree (angle) A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians... |
0° | 30° | 36° | 45° | 60° | 72° | 90° | 180° | 270° | 360° |

GradsGrad (angle) The gradian is a unit of plane angle, equivalent to of a turn. It is also known as gon, grad, or grade . One grad equals of a degree or of a radian... |
0^{g} |
33⅓^{g} |
40^{g} |
50^{g} |
66⅔^{g} |
80^{g} |
100^{g} |
200^{g} |
300^{g} |
400^{g} |

## Examples of use

- As an angular unit it is particularly useful for large angles, such as in connection with coilCoilA coil is a series of loops. A coiled coil is a structure in which the coil itself is in turn also looping.-Electromagnetic coils:An electromagnetic coil is formed when a conductor is wound around a core or form to create an inductor or electromagnet...

s and rotatingRotationA rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

objects. See also winding numberWinding numberIn mathematics, the winding number 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...

. - Turn is used in complex dynamicsComplex dynamicsComplex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions.-Techniques:*General** Montel's theorem...

for measure of externalExternal rayAn external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.This curve is only sometimes a half-line but is called ray because it is image of ray....

and internal angles. The sum of external angles of a polygon equals one turn. - Pie chartPie chartA pie chart is a circular chart divided into sectors, illustrating proportion. In a pie chart, the arc length of each sector , is proportional to the quantity it represents. When angles are measured with 1 turn as unit then a number of percent is identified with the same number of centiturns...

s 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 kinematicsKinematics

Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....

a

**turn**is a rotation less than a full revolution.A turn may be represented in a mathematical model

Mathematical model

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

that uses expressions of complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s or quaternion

Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

s. In the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

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*= e^{b i}that lies on the unit circleUnit circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

:

Frank Morley

Frank Morley

Frank Morley was a leading mathematician, known mostly for his teaching and research in the fields of algebra and geometry...

consistently referred to elements of the unit circle as

*turns*in the book*Inversive Geomety*(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 arcArc (geometry)

In geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle...

of a great circle

Great circle

A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

. 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

Quaternions and spatial rotation

Unit quaternions provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock. Compared to rotation matrices they are more numerically stable and may...

.