 Helix

This article is about the shape. For other uses, see Helix (disambiguation).
A helix (pl: helixes or helices) is a type of smooth space curve, i.e. a curve in threedimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filledin" helix – for example, a spiral ramp – is called a helicoid.^{[1]} 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".^{[2]}
Contents
Types
Helices can be either righthanded or lefthanded. 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 righthanded helix; if towards the observer then it is a lefthanded helix. Handedness (or chirality) is a property of the helix, not of the perspective: a righthanded helix cannot be turned or flipped to look like a lefthanded one unless it is viewed in a mirror, and vice versa.
Most hardware screw threads are righthanded helices. The alpha helix in biology as well as the A and B forms of DNA are also righthanded helices. The Z form of DNA is lefthanded.
The pitch of a helix is the width of one complete helix 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.^{[3]}
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 is the Corkscrew roller coaster at Cedar Point amusement park.
A circular helix, (i.e. one with constant radius) has constant band curvature and constant torsion.
A curve is called a general helix or cylindrical helix^{[4]} if its tangent makes a constant angle with a fixed line in space. A curve is a general helix if and only if the ratio of curvature to torsion is constant.^{[5]}
Mathematical description
In mathematics, a helix is a curve in 3dimensional space. The following parametrisation in Cartesian coordinates defines a helix:^{[6]}
As the parameter t increases, the point (x(t),y(t),z(t)) traces a righthanded helix of pitch 2π and radius 1 about the zaxis, in a righthanded coordinate system.
In cylindrical coordinates (r, θ, h), the same helix is parametrised by:
A circular helix of radius a and pitch 2πb is described by the following parametrisation:
Another way of mathematically constructing a helix is to plot a complex valued exponential function (e^{xi}) taking imaginary arguments (see Euler's formula).^{[vague]}
Except for rotations, translations, and changes of scale, all righthanded helices are equivalent to the helix defined above. The equivalent lefthanded 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 length of a circular helix of radius a and pitch 2πb expressed in rectangular coordinates as
equals , its curvature is and its torsion is
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
 Alpha helix
 Boerdijk–Coxeter helix
 Collagen
 Double helix
 Helical symmetry
 Helicoid
 Helix angle
 Seashell surface
 Solenoid
 Spiral
 Superhelix
 Triple helix
References
 ^ Weisstein, Eric W., "Helicoid" from MathWorld.
 ^ ἕλιξ, Henry George Liddell, Robert Scott, A GreekEnglish Lexicon, on Perseus
 ^ "Double Helix" by Sándor Kabai, Wolfram Demonstrations Project.
 ^ O'Neill, B. Elementary Differential Geometry, 1961 pg 72
 ^ O'Neill, B. Elementary Differential Geometry, 1961 pg 74
 ^ Weisstein, Eric W., "Helix" from MathWorld.
Categories: Helices
 Geometric shapes
 Curves
 Greek loanwords
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hélix — hélix … Dictionnaire des rimes
Helix — ? Helix … Википедия
hélix — [ eliks ] n. m. • 1690; gr. helix « spirale » 1 ♦ Anat. Ourlet du pavillon de l oreille, décrivant un demi cercle en partant de la conque jusqu à la partie supérieure du lobule. 2 ♦ Zool. Escargot. ● hélix nom masculin (latin scientifique helix,… … Encyclopédie Universelle
Helix SF — is a quarterly American speculative fiction online magazine edited by William Sanders and Lawrence Watt Evans. The poetry editor is Bud Webster.Sanders began the magazine as a place where writers could publish things that none of the regular… … Wikipedia
helix — HÉLIX s.n. (anat.) Margine a pavilionului urechii. [< fr. hélix, cf. gr. helix]. Trimis de LauraGellner, 20.04.2005. Sursa: DN HÉLIX s. n. margine a pavilionului urechii. (< fr. hélix) … Dicționar Român
hélix — m. anat. Repliegue semicircular en el borde del pabellón de la oreja. Medical Dictionary. 2011. hélix repliegue semicircular que forma el … Diccionario médico
helix — helix; helix·in; helix·om·e·ter; su·per·helix; … English syllables
Helix — He lix, n.; pl. L. {Helices}, E. {Helixes}. [L. helix, Gr. ?, ?, fr. ? to turn round; cf. L. volvere, and E. volute, voluble.] 1. (Geom.) A nonplane curve whose tangents are all equally inclined to a given plane. The common helix is the curve… … The Collaborative International Dictionary of English
Helix — Helix, OR U.S. city in Oregon Population (2000): 183 Housing Units (2000): 68 Land area (2000): 0.114338 sq. miles (0.296135 sq. km) Water area (2000): 0.000000 sq. miles (0.000000 sq. km) Total area (2000): 0.114338 sq. miles (0.296135 sq. km)… … StarDict's U.S. Gazetteer Places
Helix, OR — U.S. city in Oregon Population (2000): 183 Housing Units (2000): 68 Land area (2000): 0.114338 sq. miles (0.296135 sq. km) Water area (2000): 0.000000 sq. miles (0.000000 sq. km) Total area (2000): 0.114338 sq. miles (0.296135 sq. km) FIPS code:… … StarDict's U.S. Gazetteer Places
helix — a spiral thing, 1560s, from L. helix spiral, from Gk. helix (gen. helikos), related to eilein to turn, twist, roll, from PIE *wel ik , from root *wel to turn, revolve (see VULVA (Cf. vulva)) … Etymology dictionary