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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 hand ...
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 Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
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 Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
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 i ...
, as the DNA molecule is formed as two intertwined helices, and many
protein Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, respo ...
s 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 ear ...
. 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 similarit ...
''.


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 Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
) 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 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. For curves, the canonic ...
and constant
torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
. A ''
conic helix In mathematics, a conical spiral, also known as a conical helix, is a space curve on a right circular cone, whose floor plan is a plane spiral. If the floor plan is a logarithmic spiral, it is called '' conchospiral'' (from conch). Parametr ...
'', also known as a ''conic spiral'', may be defined as a
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:curvature 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. For curves, the canonic ...
to
torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a helix is a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
in 3-
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
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 in t ...
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 Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...
). 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 ''transla ...
, 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 ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
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 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. For curves, the canonic ...
is \frac and its
torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
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 hand ...
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 op ...
amusement park. Some curves found in nature consist of multiple helices of different handedness joined together by transitions known as
tendril perversion Tendril perversion is a geometric phenomenon sometimes observed in helical structures in which the direction of the helix transitions between left-handed and right-handed. Such a reversal of chirality is commonly seen in helical plant tendri ...
s. 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 a ...
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 ...
, so as to represent
octave equivalency In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
. 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) An aircraft in flight is free to rotate in three dimensions: ''Yaw (rotation), yaw'', nose left or right about an axis running up and down; ''pitch'', nose up or down about an axis running from wing to wing; and ''roll'', rotation about an axi ...
. 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 Fib ...


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 e ...
*
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 The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices. There are t ...
*
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 t ...
*
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 triple ...
*
Helical symmetry In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). Thus, a symme ...
* 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 cylinder, ...
*
Helical axis A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation (geometry), translation of a body occurs. Chasles' theorem (kinematics), Chasles' theorem shows that each Euclidea ...
*
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. Althou ...
*
Seashell surface In mathematics, a seashell surface is a surface made by a circle which spirals up the ''z''-axis while decreasing its own radius and distance from the ''z''-axis. Not all seashell surfaces describe actual seashells found in nature. Parametriza ...
*
Solenoid upright=1.20, An illustration of a solenoid upright=1.20, Magnetic field created by a seven-loop solenoid (cross-sectional view) described using field lines A solenoid () is a type of electromagnet formed by a helix, helical coil of wire whose ...
*
Superhelix A superhelix is a molecular structure in which a helix is itself coiled into a helix. This is significant to both proteins and genetic material, such as overwound circular DNA. The earliest significant reference in molecular biology is from 1971 ...
*
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 d ...


References

{{Spirals Geometric shapes Curves