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The right-handed helix (cos t, sin t, t) from t = 0 to 4p with arrowheads showing direction of increasing t Helix.svg
The right-handed helix (cos t, sin t, t) from t = 0 to 4π with arrowheads showing direction of increasing t

A helix ( /ˈhlɪks/ ) is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. 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". [1] A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called helicoid . [2]


Properties and types

The pitch of a helix is the height 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 circular helix (i.e. one with constant radius) has constant band curvature and constant torsion.

A conic helix , also known as a conic spiral, 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.

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]

For more general helix-like space curves can be found, see space spiral; e.g., spherical spiral.


Helices can be either right-handed or left-handed. 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 right-handed helix; if towards the observer, then 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 to look like a left-handed one unless it is viewed in a mirror, and vice versa.

Two types of helix shown in comparison. This shows the two chiralities of helices. One is left-handed and the other is right-handed. Each row compares the two helices from a different perspective. The chirality is a property of the object, not of the perspective (view-angle) Two Types of Helix.svg
Two types of helix shown in comparison. This shows the two chiralities of helices. One is left-handed and the other is right-handed. Each row compares the two helices from a different perspective. The chirality is a property of the object, not of the perspective (view-angle)

Mathematical description

A helix composed of sinusoidal x and y components Rising circular.gif
A helix composed of sinusoidal x and y components

In mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a particular helix; [8] perhaps the simplest equations for one is

As the parameter t increases, the point (x(t),y(t),z(t)) traces a right-handed helix of pitch 2π (or slope 1) and radius 1 about the z-axis, in a right-handed coordinate system.

In cylindrical coordinates (r, θ, h), the same helix is parametrised by:

A circular helix of radius a and slope a/b (or pitch 2πb) is described by the following parametrisation:

Another way of mathematically constructing a helix is to plot the complex-valued function exi 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 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 any one of the x, y or z components.

Arc length, curvature and torsion

The arc length of a circular helix of radius a and slope a/b (or pitch 2πb) expressed in rectangular coordinates as

equals , its curvature is and its torsion is A helix has constant non-zero curvature and torsion.

A helix is the vector-valued function

So a helix can be reparameterized as a function of , which must be unit-speed:

The unit tangent vector is

The normal vector is

Its curvature is .

The unit normal vector is

The binormal vector is

Its torsion is .


An example of double helix in molecular biology is the nucleic acid double helix.

An example of conic helix is the Corkscrew roller coaster at Cedar Point amusement park.

Some curves found in nature consist of multiple helices of different handedness joined together by transitions known as tendril perversions.

Most hardware screw threads 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.

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.

In aviation, geometric pitch is the distance an element of an airplane propeller would advance in one revolution if it were moving along a helix having an angle equal to that between the chord of the element and a plane perpendicular to the propeller axis; see also: pitch angle (aviation).

See also

Related Research Articles

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<span class="mw-page-title-main">Polar coordinate system</span> Coordinates determined by distance and angle

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.

<span class="mw-page-title-main">Spherical coordinate system</span> 3-dimensional coordinate system

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<span class="mw-page-title-main">Curvature</span> Measure of the property of a curve or a surface to be "bended"

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

<span class="mw-page-title-main">Archimedean spiral</span> Spiral with constant distance from itself

The Archimedean spiral is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation

<span class="mw-page-title-main">Spiral</span> Curve that winds around a central point

In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.

<span class="mw-page-title-main">Law of sines</span> Property of all triangles on a Euclidean plane

In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,

<span class="mw-page-title-main">Hyperbolic spiral</span> Spiral asymptotic to a line

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<span class="mw-page-title-main">Fermat's spiral</span> Spiral that surrounds equal area per turn

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  1. ἕλιξ Archived 2012-10-16 at the Wayback Machine , Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  2. Weisstein, Eric W. "Helicoid". MathWorld .
  3. "Double Helix Archived 2008-04-30 at the Wayback Machine " by Sándor Kabai, Wolfram Demonstrations Project.
  4. O'Neill, B. Elementary Differential Geometry, 1961 pg 72
  5. O'Neill, B. Elementary Differential Geometry, 1961 pg 74
  6. Izumiya, S. and Takeuchi, N. (2004) New special curves and developable surfaces. Turk J Math Archived 2016-03-04 at the Wayback Machine , 28:153–163.
  7. Menninger, T. (2013), An Explicit Parametrization of the Frenet Apparatus of the Slant Helix. arXiv:1302.3175 Archived 2018-02-05 at the Wayback Machine .
  8. Weisstein, Eric W. "Helix". MathWorld .