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A helix () is a shape like a corkscrew or spiral staircase. 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 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". A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called '' helicoid''.


Properties and types

The ''pitch'' of a helix is the height of one complete helix
turn Turn may refer to: Arts and entertainment Dance and sports * Turn (dance and gymnastics), rotation of the body * Turn (swimming), reversing direction at the end of a pool * Turn (professional wrestling), a transition between face and heel * 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 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. 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 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; e.g.,
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- dimensional 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 ''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 (''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, 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 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 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 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 tendril ...
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, 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 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


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 * Circular polarization * Collagen helix *
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 * Triple helix


References

{{Spirals Geometric shapes Curves