A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines 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 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 and constant torsion. A '' conic helix'', 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 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.

Handedness

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

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 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 (''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 ''e^xi'' 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 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 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. 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 Music is generally defined as the The arts, art of arranging sound to create some combination of Musical form, form, harmony, melody, rhythm or otherwise Musical expression, expressive content. Exact definition of music, definitions of mu ...

, pitch space 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 t ...

following a helical path Image:Ressort de traction a spires non jointives.jpg, A helical coil spring

References