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A helix () is a shape like a
corkscrew A corkscrew is a tool for drawing corks from wine bottles and other household bottles that may be sealed with corks. In its traditional form, a corkscrew simply consists of a pointed metallic helix (often called the "worm") attached to a han ...
or
spiral staircase Stairs are a structure designed to bridge a large vertical distance between lower and higher levels by dividing it into smaller vertical distances. This is achieved as a diagonal series of horizontal platforms called steps which enable passage ...
. It is a type of smooth
space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
with
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
s at a constant
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
to a fixed axis. Helices are important in
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as
alpha helices The alpha helix (α-helix) is a common motif in the secondary structure of proteins and is a right hand-helix conformation in which every backbone N−H group hydrogen bonds to the backbone C=O group of the amino acid located four residues ea ...
. The word ''helix'' comes from the
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
word ''ἕλιξ'', "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called ''
helicoid The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similari ...
''.


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. 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 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. A curve is called a slant helix if its principal normal makes a constant angle with a fixed line in space. It can be constructed by applying a transformation to the moving frame of a general helix. For more general helix-like space curves can be found, see
space spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:spherical spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:chirality Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...
) 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.


Mathematical description

In mathematics, a helix is a curve in 3-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
al space. The following parametrisation in
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...
defines a particular helix; perhaps the simplest equations for one is : x(t) = \cos(t),\, : y(t) = \sin(t),\, : z(t) = t.\, As the
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
''t'' increases, the point (''x''(''t''),''y''(''t''),''z''(''t'')) traces a right-handed helix of pitch 2''π'' (or slope 1) and radius 1 about the ''z''-axis, in a right-handed coordinate system. In
cylindrical coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
(''r'', ''θ'', ''h''), the same helix is parametrised by: : r(t) = 1,\, : \theta(t) = t,\, : h(t) = t.\, A circular helix of radius ''a'' and slope ''a''/''b'' (or pitch 2''πb'') is described by the following parametrisation: : x(t) = a\cos(t),\, : y(t) = a\sin(t),\, : z(t) = bt.\, 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
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s,
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
, 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 = 2''πb'') expressed in rectangular coordinates as :t\mapsto (a\cos t, a\sin t, bt), t\in ,T/math> equals T\cdot \sqrt, its curvature is \frac and its torsion is \frac. A helix has constant non-zero curvature and torsion. A helix is the vector-valued function \mathbf=a\cos t \mathbf+a\sin t \mathbf+ b t\mathbf \mathbf=-a\sin t \mathbf+a\cos t \mathbf+ b \mathbf \mathbf=-a\cos t \mathbf-a\sin t \mathbf+ 0\mathbf , \mathbf, =\sqrt=\sqrt , \mathbf, = \sqrt = a s(t) = \int_^\sqrtd\tau = \sqrt t So a helix can be reparameterized as a function of s, which must be unit-speed: \mathbf(s) = a\cos \frac \mathbf+a\sin \frac \mathbf+ \frac \mathbf The unit tangent vector is \frac = \mathbf = \frac\sin \frac \mathbf+\frac\cos \frac\mathbf+ \frac \mathbf The normal vector is \frac = \kappa \mathbf = \frac\cos \frac \mathbf+\frac \sin \frac\mathbf+ 0 \mathbf Its curvature is \left, \frac\= \kappa = \frac. The unit normal vector is \mathbf=-\cos \frac \mathbf - \sin \frac \mathbf + 0 \mathbf The binormal vector is \mathbf=\mathbf\times\mathbf = \frac \left b\sin \frac\mathbf - b\cos \frac\mathbf+ a \mathbf\right/math> \frac = \frac \left b\cos \frac \mathbf + b\sin \frac\mathbf+ 0 \mathbf \right/math> Its torsion is \tau = \left, \frac \ = \frac.


Examples

An example of double helix in molecular biology is the
nucleic acid double helix Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: *Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucle ...
. An example of conic helix is the
Corkscrew A corkscrew is a tool for drawing corks from wine bottles and other household bottles that may be sealed with corks. In its traditional form, a corkscrew simply consists of a pointed metallic helix (often called the "worm") attached to a han ...
roller coaster at
Cedar Point Cedar Point is a amusement park located on a Lake Erie peninsula in Sandusky, Ohio, United States. Opened in 1870, it is considered the second-oldest operating amusement park in the U.S. behind Lake Compounce. Cedar Point is owned and oper ...
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 thread A screw thread, often shortened to thread, is a helical structure used to convert between rotational and linear movement or force. A screw thread is a ridge wrapped around a cylinder or cone in the form of a helix, with the former being called ...
s 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 Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...
,
pitch space In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apa ...
is often modeled with helices or double helices, most often extending out of a circle such as the
circle of fifths In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval of ...
, 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). Image:Lehn Beautiful Foldamer HelvChimActa 1598 2003.jpg, Crystal structure of a folded molecular helix reported by Lehn ''et al.'' in ''Helv. Chim. Acta.'', 2003, 86, 1598–1624. Image:DirkvdM natural spiral.jpg, A natural left-handed helix, made by a climber plant Image:Magnetic_deflection_helical_path.svg, A charged particle in a uniform
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
following a helical path Image:Ressort de traction a spires non jointives.jpg, A helical
coil spring A selection of conical coil springs The most common type of spring is the coil spring, which is made out of a long piece of metal that is wound around itself. Coil springs were in use in Roman times, evidence of this can be found in bronze Fi ...


See also

*
Alpha helix The alpha helix (α-helix) is a common motif in the secondary structure of proteins and is a right hand-helix conformation in which every backbone N−H group hydrogen bonds to the backbone C=O group of the amino acid located four residues ...
*
Arc spring The arc spring (also known as - bow spring, curved spring, circular spring or "banana" spring) is a special form of coil spring which was originally developed for use in the dual-mass flywheel of internal combustion engine drive trains. The term ...
* Boerdijk–Coxeter helix *
Circular polarization In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to ...
*
Collagen helix In molecular biology, the collagen triple helix or type-2 helix is the main secondary structure of various types of fibrous collagen, including type I collagen. In 1954, Ramachandran & Kartha (13, 14) advanced a structure for the collagen tr ...
* Helical symmetry * Helicity *
Helix angle In mechanical engineering, a helix angle is the angle between any helix and an axial line on its right, circular cylinder or cone. Common applications are screws, helical gears, and worm gears. The helix angle references the axis of the cylinde ...
* Helical axis *
Hemihelix A hemihelix is a curved geometric shape consisting of a series of helices with alternating chirality, connected by a perversion Perversion is a form of human behavior which deviates from what is considered to be orthodox or normal. Although ...
* Seashell surface * Solenoid * Superhelix *
Triple helix In the fields of geometry and biochemistry, a triple helix (plural triple helices) is a set of three congruent geometrical helices with the same axis, differing by a translation along the axis. This means that each of the helices keeps the same ...


References

{{Spirals Geometric shapes Curves