Template:Short description Template:About
A helix (Template:IPAc-en; Template:Plural form) is a shape like a cylindrical coil spring or the thread of a machine screw. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. 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 {{#invoke:Lang|lang}}, "twisted, curved".<ref>{{#invoke:Lang|lang}} Template:Webarchive, 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>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Helicoid%7CHelicoid.html}} |title = Helicoid |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref>
Properties and typesEdit
The pitch of a helix is the height 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.<ref>"Double Helix Template:Webarchive" by Sándor Kabai, Wolfram Demonstrations Project.</ref>
A circular helix (i.e. one with constant radius) has constant band curvature and constant 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 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. Turk J Math Template:Webarchive, 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. arXiv:1302.3175 Template:Webarchive.</ref>
For more general helix-like space curves can be found, see space spiral; e.g., spherical spiral.
HandednessEdit
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.
Mathematical descriptionEdit
In mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a particular helix;<ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Helix%7CHelix.html}} |title = Helix |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</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 Template:Mvar increases, the point <math>(x(t), y(t), z(t))</math> traces a right-handed helix of pitch Template:Math (or slope 1) and radius 1 about the Template:Mvar-axis, in a right-handed coordinate system.
In cylindrical coordinates Template:Math, 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 Template:Mvar and slope Template:Math (or pitch Template:Math) 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 Template:Math as a function of the real number Template:Mvar (see Euler's formula). The value of Template:Mvar 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 Template:Mvar, Template:Mvar or Template:Mvar components.
Arc length, curvature and torsionTemplate:AnchorEdit
A circular helix of radius <math>a>0</math> and slope Template:Math (or pitch Template:Math) 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 of
- <math>\frac{a}{a^2+b^2},</math>
and a torsion of
- <math>\frac{b}{a^2+b^2}.</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 Template:Mvar, 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>
ExamplesEdit
An example of a double helix in molecular biology is the nucleic acid double helix.
An example of a 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, 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).
- Lehn Beautiful Foldamer HelvChimActa 1598 2003.jpg
Crystal structure of a folded molecular helix reported by Lehn et al.<ref>Template:Cite journal</ref>
- DirkvdM natural spiral.jpg
A natural left-handed helix, made by a climber plant
- Magnetic deflection helical path.svg
A charged particle in a uniform magnetic field following a helical path
- Ressort de traction a spires non jointives.jpg
A helical coil spring
See alsoEdit
Template:Sister project Template:Div col
- Alpha helix
- Arc spring
- Boerdijk–Coxeter helix
- Circular polarization
- Collagen helix
- Helical symmetry
- Helicity
- Helix angle
- Helical axis
- Hemihelix
- Seashell surface
- Solenoid
- Superhelix
- Triple helix