The term winding number may also refer to the rotation number of an iterated map.

This curve has winding number two around the point p.

In 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. The winding number depends on the orientation of the curve, and is negative if the curve travels around the point clockwise.

Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics, including string theory.

Intuitive description[link]

An object traveling along the red curve makes two counterclockwise turns around the person at the origin.

Suppose we are given a closed, oriented curve in the xy plane. We can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the winding number of the curve is equal to the total number of counterclockwise turns that the object makes around the origin.

When counting the total number of turns, counterclockwise motion counts as positive, while clockwise motion counts as negative. For example, if the object first circles the origin four times counterclockwise, and then circles the origin once clockwise, then the total winding number of the curve is three.

Using this scheme, a curve that does not travel around the origin at all has winding number zero, while a curve that travels clockwise around the origin has negative winding number. Therefore, the winding number of a curve may be any integer. The following pictures show curves with winding numbers between −2 and 3:

Failed to parse (Missing texvc executable; please see math/README to configure.): \cdots
	−2	−1	0
				Failed to parse (Missing texvc executable; please see math/README to configure.): \cdots
	1	2	3

Formal definition[link]

A curve in the xy plane can be defined by parametric equations:

Failed to parse (Missing texvc executable; please see math/README to configure.): x = x(t)\quad\text{and}\quad y=y(t)\qquad\text{for }0 \leq t \leq 1.

If we think of the parameter t as time, then these equations specify the motion of an object in the plane between t = 0 and t = 1. The path of this motion is a curve as long as the functions x(t) and y(t) are continuous. This curve is closed as long as the position of the object is the same at t = 0 and t = 1.

We can define the winding number of such a curve using the polar coordinate system. Assuming the curve does not pass through the origin, we can rewrite the parametric equations in polar form:

Failed to parse (Missing texvc executable; please see math/README to configure.): r = r(t)\quad\text{and}\quad \theta = \theta(t)\qquad\text{for }0 \leq t \leq 1.

The functions r(t) and θ(t) are required to be continuous, with r > 0. Because the initial and final positions are the same, θ(0) and θ(1) must differ by an integer multiple of 2π. This integer is the winding number:

Failed to parse (Missing texvc executable; please see math/README to configure.): \text{winding number} = \frac{\theta(1) - \theta(0)}{2\pi}.

This defines the winding number of a curve around the origin in the xy plane. By translating the coordinate system, we can extend this definition to include winding numbers around any point p.

Alternative definitions[link]

Winding number is often defined in different ways in various parts of mathematics. All of the definitions below are equivalent to the one given above:

Differential geometry[link]

In differential geometry, parametric equations are usually assumed to be differentiable (or at least piecewise differentiable). In this case, the polar coordinate θ is related to the rectangular coordinates x and y by the equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): d\theta = \frac{1}{r^2} \left( x\,dy - y\,dx \right)\quad\text{where }r^2 = x^2 + y^2.

By the fundamental theorem of calculus, the total change in θ is equal to the integral of dθ. We can therefore express the winding number of a differentiable curve as a line integral:

Failed to parse (Missing texvc executable; please see math/README to configure.): \text{winding number} = \frac{1}{2\pi} \oint_C \,\frac{x}{r^2}\,dy - \frac{y}{r^2} \,dx.

The one-form dθ (defined on the complement of the origin) is closed but not exact, and it generates the first De Rham cohomology group of the punctured plane. In particular, if ω is any closed differentiable one-form defined on the complement of the origin, then the integral of ω along closed loops gives a multiple of the winding number.

Complex analysis[link]

In complex analysis, the winding number of a closed curve C in the complex plane can be expressed in terms of the complex coordinate z = x + iy. Specifically, if we write z = re^iθ, then

Failed to parse (Missing texvc executable; please see math/README to configure.): dz = e^{i\theta} dr + ire^{i\theta} d\theta\!\,

and therefore

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{dz}{z} \;=\; \frac{dr}{r} + i\,d\theta \;=\; d[ \ln r ] + i\,d\theta.

The total change in ln(r) is zero, and thus the integral of dz ⁄ z is equal to i multiplied by the total change in θ. Therefore:

Failed to parse (Missing texvc executable; please see math/README to configure.): \text{winding number} = \frac{1}{2\pi i} \oint_C \frac{dz}{z}.

More generally, the winding number of C around any complex number a is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2\pi i} \oint_C \frac{dz}{z - a}.

This is a special case of the famous Cauchy integral formula. Winding numbers play a very important role throughout complex analysis (c.f. the statement of the residue theorem).

Topology[link]

In topology, the winding number is an alternate term for the degree of a continuous mapping. In physics, winding numbers are frequently called topological quantum numbers. In both cases, the same concept applies.

The above example of a curve winding around a point has a simple topological interpretation. The complement of a point in the plane is homotopy equivalent to the circle, such that maps from the circle to itself are really all that need to be considered. It can be shown that each such map can be continuously deformed to (is homotopic to) one of the standard maps Failed to parse (Missing texvc executable; please see math/README to configure.): S^1 \to S^1 : s \mapsto s^n , where multiplication in the circle is defined by identifying it with the complex unit circle. The set of homotopy classes of maps from a circle to a topological space is called the first homotopy group or fundamental group of that space. The fundamental group of the circle is the integers Z and the winding number of a complex curve is just its homotopy class.

Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes Pontryagin index.

Polygons[link]

The boundary of the regular Enneagram {9/4} winds around its centre 4 times, so it has a density of 4.

In polygons, the winding number is referred to as the polygon density. For convex polygons, and more generally simple polygons (not self-intersecting), the density is 1, by the Jordan curve theorem. By contrast, for a regular star polygon {p/q}, the density is q.

Turning number[link]

One can also consider the winding number of the path with respect to the tangent of the path itself. As a path followed through time, this would be the winding number with respect to the origin of the velocity vector. In this case the example illustrated on the right has a winding number of 4 (or −4), because the small loop is counted.

This is only defined for immersed paths (i.e., for differentiable paths with nowhere vanishing derivatives), and is the degree of the tangential Gauss map.

This is called the turning number, and can be computed as the total curvature divided by 2π.

Winding number and Heisenberg ferromagnet equations[link]

Finally, note that the winding number is closely related with the (2 + 1)-dimensional continuous Heisenberg ferromagnet equations and its integrable extensions: the Ishimori equation etc. Solutions of the last equations are classified by the winding number or topological charge (topological invariant and/or topological quantum number).

See also[link]

• Argument principle
• Linking coefficient
• Polygon density
• Residue theorem
• Topological degree theory
• Topological quantum number
• Wilson loop
• Nonzero-rule

External links[link]

• Winding number, PlanetMath.org.

Detail of William and Mary from the ceiling of the Painted Hall

The phrase William and Mary usually refers to the coregency over the Kingdoms of England, Scotland and Ireland, of first-cousins King William III & II and Queen Mary II. Their joint reign began in February 1689, when they were offered the throne by the Parliament of England, replacing James II & VII, Mary's father and William's uncle/father-in-law, who was "deemed to have fled" the country in the Glorious Revolution of 1688. After Mary died in 1694, William ruled alone until his death in 1702. William and Mary were childless and were ultimately succeeded by Mary's younger sister, Anne.

Historic influence[link]

To end the Glorious Revolution, William and Mary signed the English Bill of Rights and began a new co-operation between the Parliament and the monarchs, leading to a greater measure of personal liberty and democracy in Britain. This action both signaled the end of several centuries of tension and conflict between the crown and parliament, and the end of the idea that England would be restored to Roman Catholicism, King William being a Protestant leader. The English Bill of Rights also inspired the English colonists in North America to revolt against the rule of James II and his proposed changes in colonial governance. These colonial revolts occurred in Massachusetts, New York, and Maryland.^[1]

See also[link]

• Philip II of Spain and Mary I of England
• Francis II of France and Mary, Queen of Scots

References[link]

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Ed Turner was a CNN executive vice president and one of the first people Ted Turner hired in 1980 to help make his dream of a 24/7 news network a reality.

Ed—no relation to Ted Turner—was a well-known figure throughout the industry for his often novel, always daring approach to news. A veteran journalist, Turner served as Metromedia's corporate news director before joining United Press International Television News. After UPITN, Turner moved to CBS where he produced the "CBS Morning News" show. Turner served at CNN until 1998,when he left his position as vice president in charge of newsgathering. He died of cancer in 2002 at the age of 66.

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