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 tha ...
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 hereditar ...
, 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, respon ...
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 earli ...
. 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 similar ...
''.
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
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 can ...
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).
Parametric ...
'', 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 can ...]
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 i ...]
) 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 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 ...
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 coor ...
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
:
:
:
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 d ...
(''r'', ''θ'', ''h''), the same helix is parametrised by:
:
:
:
A circular helix of radius ''a'' and slope ''a''/''b'' (or pitch 2''πb'') is described by the following parametrisation:
:
:
:
Another way of mathematically constructing a helix is to plot the complex-valued function ''e
xi'' 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 ...
).
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 (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
, 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
: