View source for Helix

{{short description|Space curve that winds around a line}}
{{other uses}}
[[File:Springs 016.JPG|thumb|upright=1.35|(l-r) Tension, compression and torsion coil springs]]
[[File:Senkschraube.jpg|thumb|upright|A machine screw]]
[[File:Helix.svg|thumb|The right-handed helix {{math|(cos ''t'', sin ''t'', ''t'')}} for {{math|0 ≤ ''t'' ≤ 4''π''}} with arrowheads showing direction of increasing {{mvar|t}}]]

A '''helix''' ({{IPAc-en|'|h|iː|l|ɪ|k|s}}; {{plural form|'''helices'''}}) is a shape like a cylindrical [[coil spring]] or the thread of a [[machine screw]]. It is a type of [[smoothness (mathematics)|smooth]] [[skew curve]] with [[tangent line]]s at a constant [[angle]] to a fixed axis. 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 {{lang|grc|ἕλιξ}}, "twisted, curved".<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3De%28%2Flic1 {{lang|grc|ἕλιξ}}] {{webarchive|url=https://web.archive.org/web/20121016155956/http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=e(%2Flic1 |date=2012-10-16 }}, Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref> 
A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called a ''[[helicoid]]''.<ref>{{MathWorld | urlname=Helicoid | title=Helicoid}}</ref>

==Properties and types==

The '''pitch''' of a helix is the height of one complete helix [[turn (angle)|turn]], measured parallel to the axis of the helix.

A '''double helix''' consists of two (typically [[congruence (geometry)|congruent]]) helices with the same axis, differing by a translation along the axis.<ref>"[http://demonstrations.wolfram.com/DoubleHelix/ Double Helix] {{webarchive|url=https://web.archive.org/web/20080430111608/http://demonstrations.wolfram.com/DoubleHelix/ |date=2008-04-30 }}" by Sándor Kabai, [[Wolfram Demonstrations Project]].</ref>

A '''circular helix''' (i.e. one with constant radius) has constant band [[curvature]] and constant [[Torsion of curves|torsion]]. The slope of a circular helix is commonly defined as the ratio of the circumference of the circular cylinder that it spirals around, and its pitch (the height of one complete helix turn).

A ''[[conic helix]]'', also known as a ''conic spiral'', 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.

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>

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] {{webarchive|url=https://web.archive.org/web/20160304004003/http://journals.tubitak.gov.tr/math/issues/mat-04-28-2/mat-28-2-6-0301-4.pdf |date=2016-03-04 }}, 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''. [https://arxiv.org/abs/1302.3175 arXiv:1302.3175] {{webarchive|url=https://web.archive.org/web/20180205164124/https://arxiv.org/abs/1302.3175 |date=2018-02-05 }}.</ref>

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 (mathematics)|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.

[[File:Two Types of Helix.svg|thumb|'''Two types of helix shown in comparison'''. This shows the two [[chirality (mathematics)|chiralities]] of helices. One is left-handed and the other is right-handed. Each row compares the two helices from a different perspective. The chirality is a property of the object, not of the [[Perspectivity|perspective]] (view-angle)]]

==Mathematical description==
[[File:Rising circular.gif|thumb|250px|A helix composed of sinusoidal {{mvar|x}} and {{mvar|y}} components]]
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 particular helix;<ref>{{MathWorld | urlname=Helix | title=Helix}}</ref> perhaps the simplest equations for one is

: <math>\begin{align}
x(t) &= \cos(t),\\
y(t) &= \sin(t),\\
z(t) &= t.
\end{align}</math>

As the [[parameter]] {{mvar|t}} increases, the point <math>(x(t), y(t), z(t))</math> traces a right-handed helix of pitch {{math|2''π''}} (or slope 1) and radius 1 about the {{mvar|z}}-axis, in a right-handed coordinate system.

In [[cylindrical coordinates]] {{math|(''r'', ''θ'', ''h'')}}, the same helix is parametrised by:
: <math>\begin{align}
r(t) &= 1,\\
\theta(t) &= t,\\
h(t) &= t.
\end{align}</math>

A circular helix of radius {{mvar|a}} and slope {{math|{{sfrac|''a''|''b''}}}} (or pitch {{math|2''πb''}}) is described by the following parametrisation:

: <math>\begin{align}
x(t) &= a\cos(t),\\
y(t) &= a\sin(t),\\
z(t) &= bt.
\end{align}</math>

Another way of mathematically constructing a helix is to plot the complex-valued function {{math|''e<sup>xi</sup>''}} as a function of the real number {{mvar|x}} (see [[Euler's formula]]).
The value of {{mvar|x}} and the real and imaginary parts of the function value give this plot three real dimensions.

Except for [[rotation]]s, [[translation (geometry)|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 {{mvar|x}}, {{mvar|y}} or {{mvar|z}} components.

===Arc length, curvature and torsion{{anchor|Arc length|Curvature|Torsion}}===

A circular helix of radius <math>a>0</math> and slope {{math|{{sfrac|''a''|''b''}}}} (or pitch {{math|2''πb''}}) expressed in Cartesian coordinates as the [[parametric equation]]
:<math>t\mapsto (a\cos t, a\sin t, bt), t\in [0,T]</math>
has an [[arc length]] of
:<math>A = T\cdot \sqrt{a^2+b^2},</math>
a [[Curvature#One dimension in three dimensions: Curvature of space curves|curvature]] of
:<math>\frac{a}{a^2+b^2},</math>
and a [[Torsion of a curve|torsion]] of
:<math>\frac{b}{a^2+b^2}.</math>


Arc length per revolution (<math> T=2 \pi </math>):
<math>A = \sqrt{(2\pi a)^2+(2\pi b)^2}</math>
or
<math>A = \sqrt{(2\pi a)^2+p^2}</math>
where <math>p = </math>pitch.

Twisted length per unit straight length (axial length):
<math>A = \frac{ \sqrt{(2\pi a)^2+p^2}}{p}</math>


A helix has constant non-zero curvature and torsion.

A helix is the vector-valued function

<math display="block">\begin{align}
\mathbf{r}&=a\cos t \mathbf{i}+a\sin t \mathbf{j}+ b t\mathbf{k}\\[6px]

\mathbf{v}&=-a\sin t \mathbf{i}+a\cos t \mathbf{j}+ b \mathbf{k}\\[6px]

\mathbf{a}&=-a\cos t \mathbf{i}-a\sin t \mathbf{j}+ 0\mathbf{k}\\[6px]

|\mathbf{v}|&=\sqrt{(-a\sin t )^2 +(a\cos t)^2 + b^2}=\sqrt{a^2 +b^2}\\[6px]

|\mathbf{a}| &= \sqrt{(-a\sin t )^2 +(a\cos t)^2 } = a\\[6px]

s(t) &= \int_{0}^{t}\sqrt{a^2 +b^2}d\tau = \sqrt{a^2 +b^2} t
\end{align}</math>

So a helix can be reparameterized as a function of {{mvar|s}}, which must be unit-speed:

<math display="block">\mathbf{r}(s) = a\cos \frac{s}{\sqrt{a^2 +b^2} } \mathbf{i}+a\sin \frac{s}{\sqrt{a^2 +b^2}} \mathbf{j}+ \frac{bs}{\sqrt{a^2 +b^2}} \mathbf{k}</math>

The unit tangent vector is

<math display="block">\frac{d \mathbf{r}}{d s} = \mathbf{T} = \frac{-a}{\sqrt{a^2 +b^2} }\sin \frac{s}{\sqrt{a^2 +b^2} } \mathbf{i}+\frac{a}{\sqrt{a^2 +b^2} }\cos \frac{s}{\sqrt{a^2 +b^2} }\mathbf{j}+ \frac{b}{\sqrt{a^2 +b^2}} \mathbf{k}</math>

The normal vector is

<math display="block">\frac{d \mathbf{T}}{d s} = \kappa \mathbf{N} = \frac{-a}{a^2 +b^2 }\cos \frac{s}{\sqrt{a^2 +b^2} } \mathbf{i}+\frac{-a}{a^2 +b^2} \sin \frac{s}{\sqrt{a^2 +b^2} }\mathbf{j}+ 0 \mathbf{k}</math>

Its curvature is

<math display="block">\kappa = \left|\frac{d\mathbf{T}}{ds}\right|= \frac{a}{a^2 +b^2 }</math>.

The unit normal vector is

<math display="block">\mathbf{N}=-\cos \frac{s}{\sqrt{a^2 +b^2} } \mathbf{i} - \sin \frac{s}{\sqrt{a^2 +b^2} } \mathbf{j} + 0 \mathbf{k}</math>

The binormal vector is

<math display="block"> \begin{align}
\mathbf{B}=\mathbf{T}\times\mathbf{N} &= \frac{1}{\sqrt{a^2 +b^2 }} \left( b\sin \frac{s}{\sqrt{a^2 +b^2}}\mathbf{i} - b\cos \frac{s}{\sqrt{a^2 +b^2}}\mathbf{j}+ a \mathbf{k}\right)\\[12px]
\frac{d\mathbf{B}}{ds} &= \frac{1}{a^2 +b^2} \left( b\cos \frac{s}{\sqrt{a^2 +b^2}} \mathbf{i} + b\sin \frac{s}{\sqrt{a^2 +b^2}}\mathbf{j}+ 0 \mathbf{k} \right)
\end{align}</math>

Its torsion is
<math display="block">\tau = \left| \frac{d\mathbf{B}}{ds} \right| = \frac{b}{a^2 +b^2}.</math>

==Examples==
An example of a double helix in molecular biology is the [[nucleic acid double helix]].

An example of a conic helix is the [[Corkscrew (Cedar Point)|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 perversion]]s.

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.

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]].

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)]].

<gallery heights="225px" widths="225px">
Image:Lehn Beautiful Foldamer HelvChimActa 1598 2003.jpg|Crystal structure of a [[foldamer|folded molecular helix]] reported by [[Jean-Marie Lehn|Lehn]] ''et al.''<ref>{{cite journal|last1=Schmitt |first1=J.-L. |last2=Stadler |first2=A.-M. |last3=Kyritsakas |first3=N. |last4=Lehn |first4=J.-M. |date=2003 |title=Helicity-Encoded Molecular Strands: Efficient Access by the Hydrazone Route and Structural Features |journal=Helvetica Chimica Acta |volume=86 |issue=5 |pages=1598–1624 |doi=10.1002/hlca.200390137}}</ref>
Image:DirkvdM natural spiral.jpg|A natural left-handed helix, made by a [[vine|climber]] plant
Image:Magnetic_deflection_helical_path.svg|A charged particle in a uniform [[magnetic field]] following a helical path
Image:Ressort de traction a spires non jointives.jpg|A helical [[coil spring]]
Image:DNA orbit animated.gif|A [[DNA double helix]]
</gallery>

==See also==
{{wiktionary}}
{{div col}}
* [[Alpha helix]]
* [[Arc spring]]
* [[Boerdijk–Coxeter helix]]
* [[Circular polarization]]
* [[Collagen helix]]
* [[Helical symmetry]]
* [[Helicity (disambiguation)|Helicity]]
* [[Helix angle]]
* [[Helical axis]]
* [[Hemihelix]]
* [[Seashell surface]]
* [[Solenoid]]
* [[Superhelix]]
* [[Triple helix]]
{{div col end}}

== References ==
{{Reflist|2}}

{{Spirals}}
{{Authority control}}

[[Category:Helices| ]]
[[Category:Geometric shapes]]
[[Category:Curves]]