A helix (pl: helixes or helices), from the Greek word έλιξ, is a special kind of space curve, i.e. a smooth curve in three-space. As a mental image of a helix one may take the spring (although the spring is not a curve, and so is technically not a helix, it does give a convenient mental picture). A helix is characterised by the fact that the tangent line at any point makes a constant angle with a fixed line. A filled in helix, for example a spiral staircase, is called a helicoid[1]. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices.
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Helices can be either right-handed or left-handed. With the line of sight being the helical axis, if clockwise movement of the helix corresponds to axial movement away from the observer, then it is called a right-handed helix. If anti-clockwise movement corresponds to axial movement away from the observer, it is a left-handed helix. Handedness (or chirality) is a property of the helix, not of the perspective: a right-handed helix cannot be turned or flipped to look like a left-handed one unless it is viewed through a mirror, and vice versa.
Most hardware screws are right-handed helices. The alpha helix in biology as well as the A and B forms of DNA are also right-handed helices. The Z form of DNA is left-handed.
A double helix typically consists geometrically of two congruent helices with the same axis, differing by a translation along the axis, which may or may not be half-way.[2]
A conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of the angle indicating direction from the axis. An example of a helix would be the Corkscrew roller coaster at Cedar Point amusement park.
A circular helix has constant band curvature and constant torsion. The pitch of a helix is the width of one complete helix turn, measured along the helix axis.
A curve is called a general helix if its tangent makes a constant angle with a fixed line in space.
In mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a helix[3]:
As the parameter t increases, the point (x(t),y(t),z(t)) traces a right-handed helix of pitch 2π about the z-axis, in a right-handed coordinate system.
In cylindrical coordinates (r, θ, h), the same helix is parametrised by:
The above example is an example of circular helix of radius 1 and pitch 2π.
Circular helix of radius a and pitch 2πb is described by the following parametrisation:
Another way of mathematically constructing a helix is to plot a complex valued exponential function (e^xi) taking imaginary arguments (see Euler's formula).
Except for rotations, translations, and changes of scale, all right-handed helices are equivalent to the helix defined above. The equivalent left-handed helix can be constructed in a number of ways, the simplest being to negate either the x, y or z component.
The length of a circular helix of radius a and pitch 2πb expressed in rectangular coordinates as
![t\mapsto (a\cos t, a\sin t, bt), t\in [0,T]](http://upload.wikimedia.org/math/b/3/3/b330929a1f1e9a4e8a1c6b79f346fc8b.png)
equals 
, its curvature is 
and its torsion is 
In music, pitch space is often modeled with helices or double helices, most often extending out of a circle such as the circle of fifths, so as to represent octave equivalency.