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| {{About|the shape}}
| | Alysha Jenkin is my name and I'm comfortable when people make use of the complete name. New Jersey could be the only area I've been surviving in. My spouse doesn't like it the way in which I do but what I actually like doing is to collect coins and I'm going to be starting something different along side it. I'm currently an information specialist.<br><br> |
| [[Image:Helix.svg|thumb|The helix (cos ''t'', sin ''t'', ''t'') from ''t'' = 0 to 4π with arrowheads showing direction of increasing ''t'']]
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| A '''helix''' (pl: '''helixes''' or '''helices''') is a type of [[Differentiable manifold|smooth]] [[space curve]], i.e. a curve in [[three-dimensional 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 "filled-in" helix – for example, a spiral ramp – is called a [[helicoid]].<ref>{{MathWorld | urlname=Helicoid | title=Helicoid}}</ref> Helices are important in [[biology]], as the [[DNA]] molecule is formed as [[double helix|two intertwined helices]], and many [[protein]]s have helical substructures, known as [[alpha helix|alpha helices]]. The word ''helix'' comes from the [[Greek language|Greek]] word ''ἕλιξ'', "twisted, curved".<ref>[http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3De%28%2Flic1 ἕλιξ], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref>
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| ==Types==
| | Also visit my web blog ... [http://20thstreetblockparty.com/2014/jordan-kurland-intro/ Jordan Kurland] |
| Helixes 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. Thus, a helix cannot be described as 'spinning clockwise or counter-clockwise.' Handedness (or [[chirality (mathematics)|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 in a mirror, and vice versa.
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| Most hardware [[screw thread]]s are right-handed helices. The alpha helix in biology as well as the [[A-DNA|A]] and [[B-DNA|B]] forms of DNA are also right-handed helices. The [[Z-DNA|Z form]] of DNA is left-handed.
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| The '''pitch''' of a helix is the width of one complete helix turn, measured parallel to the axis of the helix.
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| A [[double helix]] consists of two (typically congruent) helices with the same axis, differing by a translation along the axis.<ref>"[http://demonstrations.wolfram.com/DoubleHelix/ Double Helix]" by Sándor Kabai, [[Wolfram Demonstrations Project]].</ref>
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| 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 (Cedar Point)|Corkscrew]] roller coaster at Cedar Point amusement park.
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| A '''circular helix,''' (i.e. one with constant radius) has constant band [[curvature]] and constant [[Torsion of curves|torsion]].
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| A curve is called a '''general helix''' or '''cylindrical helix'''<ref>O'Neill, B. ''Elementary Differential Geometry,'' 1961 pg 72</ref> 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 of curves|torsion]] is constant.<ref>O'Neill, B. ''Elementary Differential Geometry,'' 1961 pg 74</ref>
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| A curve is called a '''slant helix''' if its principal normal makes a constant angle with a fixed line in space.<ref>Izumiya, S. and Takeuchi, N. (2004) ''New special curves and developable surfaces.'' [http://journals.tubitak.gov.tr/math/issues/mat-04-28-2/mat-28-2-6-0301-4.pdf Turk J Math], 28:153–163.</ref> It can be constructed by applying a transformation to the moving frame of a general helix.<ref>Menninger, T. (2013), ''An Explicit Parametrization of the Frenet Apparatus of the Slant Helix''. [http://arxiv.org/abs/1302.3175 arXiv:1302.3175].</ref>
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| ==Mathematical description==
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| [[File:Rising circular.gif|thumb|250px|A helix composed of sinusoidal ''x'' and ''y'' components]]
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| In [[mathematics]], a helix is a [[Differential geometry of curves|curve]] in 3-[[dimension]]al space. The following [[Parametric equation|parametrisation]] in [[Cartesian coordinate system|Cartesian coordinates]] defines a helix:<ref>{{MathWorld | urlname=Helix | title=Helix}}</ref>
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| : <math>x(t) = \cos(t),\,</math>
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| : <math>y(t) = \sin(t),\,</math>
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| : <math>z(t) = t.\,</math>
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| As the [[parameter]] ''t'' increases, the point (''x''(''t''),''y''(''t''),''z''(''t'')) traces a right-handed helix of pitch 2''[[pi|π]]'' and radius 1 about the ''z''-axis, in a right-handed coordinate system.
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| In [[cylindrical coordinates]] (''r'', ''θ'', ''h''), the same helix is parametrised by:
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| : <math>r(t) = 1,\,</math>
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| : <math>\theta(t) = t,\,</math>
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| : <math>h(t) = t.\,</math>
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| A circular helix of radius ''a'' and pitch 2''πb'' is described by the following parametrisation:
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| : <math>x(t) = a\cos(t),\,</math>
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| : <math>y(t) = a\sin(t),\,</math>
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| : <math>z(t) = bt.\,</math>
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| Another way of mathematically constructing a helix is to plot a complex valued exponential function (''e<sup>xi</sup>'') taking imaginary arguments (see [[Euler's formula]]).{{Vague|date=November 2009}}
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| Except for [[rotation]]s, [[translation (geometry)|translation]]s, 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.
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| ===Arc length, curvature and torsion===
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| The length of a circular helix of radius ''a'' and pitch 2''πb'' expressed in rectangular coordinates as
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| :<math>t\mapsto (a\cos t, a\sin t, bt), t\in [0,T]</math>
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| equals <math>T\cdot \sqrt{a^2+b^2}</math>, its [[Curvature#One dimension in three dimensions: Curvature of space curves|curvature]] is <math>\frac{|a|}{a^2+b^2}</math>
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| and its [[Torsion of a curve|torsion]] is <math>\frac{b}{a^2+b^2}.</math>
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| ==Examples==
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| 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]].
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| <gallery heights="225px" widths="225px">
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| Image:Lehn Beautiful Foldamer HelvChimActa 1598 2003.jpg|Crystal structure of a [[foldamer|folded molecular helix]] reported by [[Jean-Marie Lehn|Lehn]] ''et al.'' in ''Helv. Chim. Acta.'', 2003, 86, 1598–1624.
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| Image:DirkvdM natural spiral.jpg|A natural left-handed helix, made by a [[vine|climber]] plant
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| Image:Magnetic_deflection_helical_path.svg|A charged particle in a uniform [[magnetic field]] following a helical path
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| Image:Ressort de traction a spires non jointives.jpg|A helical coil spring
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| </gallery>
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| == See also ==
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| *[[Alpha helix]]
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| *[[Boerdijk–Coxeter helix]]
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| *[[Collagen]]
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| *[[Double helix]]
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| *[[Symmetry#Helical_symmetry|Helical symmetry]]
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| *[[Helicoid]]
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| *[[Helix angle]]
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| *[[Seashell surface]]
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| *[[Solenoid]]
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| *[[Spiral]]
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| *[[Superhelix]]
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| *[[Triple helix]]
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| == References ==
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| {{reflist|2}}
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| [[Category:Helices| ]]
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| [[Category:Geometric shapes]]
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| [[Category:Curves]]
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| [[ar:لولب (رياضيات)]]
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Alysha Jenkin is my name and I'm comfortable when people make use of the complete name. New Jersey could be the only area I've been surviving in. My spouse doesn't like it the way in which I do but what I actually like doing is to collect coins and I'm going to be starting something different along side it. I'm currently an information specialist.
Also visit my web blog ... Jordan Kurland