Helix

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File:Brick helix near Tsomo.jpg
A helix constructed in brick.
Crystal structure of a folded molecular helix reported by Lehn and coworkers in Helv. Chim. Acta., 2003, 86, 1598-1624.
File:DirkvdM natural spiral.jpg
A natural left-handed helix, used by a climber plant.

A helix (pl: helices), from the Greek word έλικας/έλιξ, is a three-dimensional, twisted shape. Common objects formed like a helix are a spring, a screw, and a spiral staircase (though the last would be more correctly called helical). 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.

Types

Helices can be either right-handed or left-handed. With the line of sight being the helical axis, if clockwise movement of the helix corresponds to axial movement away from the observer, then it is a right-handed helix. If counter-clockwise movement corresponds to axial movement away from the observer, 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 or flipped to look like a left-handed one unless it is viewed through a mirror, and vice versa.

Most hardware screws 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.

A double helix typically consists geometrically of two congruent helices with the same axis, differing by a translation along the axis, which may or may not be half-way.

A conic helix 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. An example of a helix would be the Corkscrew roller coaster at Cedar Point amusement park.

A circular helix has constant curvature and constant torsion. The pitch of a helix is the width of one complete helix turn, measured along the helix axis.

Mathematics

In mathematics, a helix is a curve in 3-dimensional space. The following three equations in rectangular coordinates define a helix:

$ x = \cos(t),\, $
$ y = \sin(t),\, $
$ z = t.\, $

As the parameter t increases, the point (x,y,z) traces a right-handed helix of pitch 2π about the z-axis, in a right-handed coordinate system.

In cylindrical coordinates (r, θ, h), the same helix is described by:

$ r = 1,\, $
$ \theta = t,\, $
$ h = t.\, $

Another way of mathematically constructing a helix is to plot a complex valued exponential function (e^xi) taking imaginary arguments (see Euler's formula).

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 either the x, y or z component.

The length of a general helix expressed in rectangular coordinates as

$ t\mapsto (a\cos t, a\sin t, bt), t\in [0,T] $

equals $ T\cdot \sqrt{a^2+b^2} $, its curvature is $ \frac{a}{a^2+b^2} $.

Examples

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.

See also

cs:Šroubovice da:Skruelinje de:Helixgl:Hélice (xeometría) id:Heliks it:Elica (geometria) hu:Csavarvonal nl:Helix (wiskunde) no:Helix nn:Helikssr:Хеликс sh:Heliks sv:Helix


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