Helix
A helix (pl: helixes or helices) is a type of smooth space curve, i.e. a curve in threedimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helices are coil springs and the handrails of spiral staircases. A "filledin" helix – for example, a spiral ramp – 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. The word helix comes from the Greek word ἕλιξ, "twisted, curved".^{[2]}
Contents
Types
Helices can be either righthanded or lefthanded. With the line of sight along the helix's axis, if a clockwise screwing motion moves the helix away from the observer, then it is called a righthanded helix; if towards the observer, then it is a lefthanded helix. Handedness (or chirality) is a property of the helix, not of the perspective: a righthanded helix cannot be turned to look like a lefthanded one unless it is viewed in a mirror, and vice versa.
Most hardware screw threads are righthanded helices. The alpha helix in biology as well as the A and B forms of DNA are also righthanded helices. The Z form of DNA is lefthanded.
The pitch of a helix is the width of one complete helix turn, measured parallel to the axis of the helix.
A double helix consists of two (typically congruent) helices with the same axis, differing by a translation along the axis.^{[3]}
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 is the Corkscrew roller coaster at Cedar Point amusement park.
A circular helix, (i.e. one with constant radius) has constant band curvature and constant torsion.
A curve is called a general helix or cylindrical helix^{[4]} if its tangent makes a constant angle with a fixed line in space. A curve is a general helix if and only if the ratio of curvature to torsion is constant.^{[5]}
A curve is called a slant helix if its principal normal makes a constant angle with a fixed line in space.^{[6]} It can be constructed by applying a transformation to the moving frame of a general helix.^{[7]}
Some curves found in nature consist of multiple helices of different handedness joined together by transitions known as tendril perversions.
Mathematical description
In mathematics, a helix is a curve in 3dimensional space. The following parametrisation in Cartesian coordinates defines a particular helix,^{[8]} probably the simplest equations for one:
As the parameter t increases, the point (x(t),y(t),z(t)) traces a righthanded helix of pitch (i.e., slope) 1 and radius 1 about the zaxis, in a righthanded coordinate system.
In cylindrical coordinates (r, θ, h), the same helix is parametrised by:
A circular helix of radius a and pitch (or slope) b/a is described by the following parametrisation:
Another way of mathematically constructing a helix is to plot the complexvalued function e^{xi} as a function of the real number x (see Euler's formula). The value of x and the real and imaginary parts of the function value give this plot three real dimensions.
Except for rotations, translations, and changes of scale, all righthanded helices are equivalent to the helix defined above. The equivalent lefthanded helix can be constructed in a number of ways, the simplest being to negate any one of the x, y or z components.
Arc length, curvature and torsion
The length of a circular helix of radius a and pitch (or slope) b/a expressed in rectangular coordinates as
equals , its curvature is and its torsion is
Examples
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.

Crystal structure of a folded molecular helix reported by Lehn et al. in Helv. Chim. Acta., 2003, 86, 1598–1624.

A natural lefthanded helix, made by a climber plant

A charged particle in a uniform magnetic field following a helical path
See also
References
 ↑ Weisstein, Eric W., "Helicoid", MathWorld.
 ↑ ἕλιξ, Henry George Liddell, Robert Scott, A GreekEnglish Lexicon, on Perseus
 ↑ "Double Helix" by Sándor Kabai, Wolfram Demonstrations Project.
 ↑ O'Neill, B. Elementary Differential Geometry, 1961 pg 72
 ↑ O'Neill, B. Elementary Differential Geometry, 1961 pg 74
 ↑ Izumiya, S. and Takeuchi, N. (2004) New special curves and developable surfaces. Turk J Math, 28:153–163.
 ↑ Menninger, T. (2013), An Explicit Parametrization of the Frenet Apparatus of the Slant Helix. arXiv:1302.3175.
 ↑ Weisstein, Eric W., "Helix", MathWorld.