# Helix

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A helix () is a shape like a corkscrew or spiral staircase. 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 ἕλιξ, "twisted, curved". [1] A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called helicoid . [2]

## Properties and types

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. [3]

A circular helix (i.e. one with constant radius) has constant band curvature and constant torsion.

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 [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]

A curve is called a slant helix if its principal normal makes a constant angle with a fixed line in space. [6] It can be constructed by applying a transformation to the moving frame of a general helix. [7]

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) 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 description

In mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a particular helix; [8] perhaps the simplest equations for one is

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

As the parameter t increases, the point (x(t),y(t),z(t)) traces a right-handed helix of pitch 2π (or slope 1) and radius 1 about the z-axis, in a right-handed coordinate system.

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

${\displaystyle r(t)=1,\,}$
${\displaystyle \theta (t)=t,\,}$
${\displaystyle h(t)=t.\,}$

A circular helix of radius a and slope a/b (or pitch 2πb) is described by the following parametrisation:

${\displaystyle x(t)=a\cos(t),\,}$
${\displaystyle y(t)=a\sin(t),\,}$
${\displaystyle z(t)=bt.\,}$

Another way of mathematically constructing a helix is to plot the complex-valued function exi as a function of the real number x (see Euler's formula). The value of x 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 x, y or z components.

### Arc length, curvature and torsion

The arc length of a circular helix of radius a and slope a/b (or pitch 2πb) expressed in rectangular coordinates as

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

equals ${\displaystyle T\cdot {\sqrt {a^{2}+b^{2}}}}$, its curvature is ${\displaystyle {\frac {|a|}{a^{2}+b^{2}}}}$ and its torsion is ${\displaystyle {\frac {b}{a^{2}+b^{2}}}.}$ A helix has constant non-zero curvature and torsion.

A helix is the vector-valued function

${\displaystyle \mathbf {r} =a\cos t\mathbf {i} +a\sin t\mathbf {j} +bt\mathbf {k} }$
${\displaystyle \mathbf {v} =-a\sin t\mathbf {i} +a\cos t\mathbf {j} +b\mathbf {k} }$
${\displaystyle \mathbf {a} =-a\cos t\mathbf {i} -a\sin t\mathbf {j} +0\mathbf {k} }$
${\displaystyle |\mathbf {v} |={\sqrt {(-a\sin t)^{2}+(a\cos t)^{2}+b^{2}}}={\sqrt {a^{2}+b^{2}}}}$
${\displaystyle |\mathbf {a} |={\sqrt {(-a\sin t)^{2}+(a\cos t)^{2}}}=a}$
${\displaystyle s(t)=\int _{0}^{t}{\sqrt {a^{2}+b^{2}}}d\tau ={\sqrt {a^{2}+b^{2}}}t}$

So a helix can be reparameterized as a function of ${\displaystyle s}$, which must be unit-speed:

${\displaystyle \mathbf {r} (s)=a\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +a\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +{\frac {bs}{\sqrt {a^{2}+b^{2}}}}\mathbf {k} }$

The unit tangent vector is

${\displaystyle {\frac {d\mathbf {r} }{ds}}=\mathbf {T} ={\frac {-a}{\sqrt {a^{2}+b^{2}}}}\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +{\frac {a}{\sqrt {a^{2}+b^{2}}}}\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +{\frac {b}{\sqrt {a^{2}+b^{2}}}}\mathbf {k} }$

The normal vector is

${\displaystyle {\frac {d\mathbf {T} }{ds}}=\kappa \mathbf {N} ={\frac {-a}{a^{2}+b^{2}}}\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +{\frac {-a}{a^{2}+b^{2}}}\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +0\mathbf {k} }$

Its curvature is ${\displaystyle \left|{\frac {d\mathbf {T} }{ds}}\right|=\kappa ={\frac {|a|}{a^{2}+b^{2}}}}$.

The unit normal vector is

${\displaystyle \mathbf {N} =-\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} -\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +0\mathbf {k} }$

The binormal vector is

${\displaystyle \mathbf {B} =\mathbf {T} \times \mathbf {N} ={\frac {1}{\sqrt {a^{2}+b^{2}}}}\left[b\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} -b\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +a\mathbf {k} \right]}$

${\displaystyle {\frac {d\mathbf {B} }{ds}}={\frac {1}{a^{2}+b^{2}}}\left[b\cos {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {i} +b\sin {\frac {s}{\sqrt {a^{2}+b^{2}}}}\mathbf {j} +0\mathbf {k} \right]}$

Its torsion is ${\displaystyle \tau =\left|{\frac {d\mathbf {B} }{ds}}\right|={\frac {b}{a^{2}+b^{2}}}}$.

## Examples

An example of double helix in molecular biology is the nucleic acid double helix.

An example of 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).

## Related Research Articles

A centripetal force is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

The Archimedean spiral is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation

In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.

In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,

A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation

A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance from the spiral center, contrasting with the Archimedean spiral and the logarithmic spiral. Fermat spirals are named after Pierre de Fermat.

The Fresnel integralsS(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function. They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization of the object.

In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space R3, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: Jean Frédéric Frenet, in his thesis of 1847, and Joseph Alfred Serret, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet available at the time of their discovery.

The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known.

In the differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the osculating plane. Taken together, the curvature and the torsion of a space curve are analogous to the curvature of a plane curve. For example, they are coefficients in the system of differential equations for the Frenet frame given by the Frenet–Serret formulas.

A parametric surface is a surface in the Euclidean space which is defined by a parametric equation with two parameters . Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof.

The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices. There are two chiral forms, with either clockwise or counterclockwise windings. Unlike any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally repetitive in 3-dimensional space. Even in an infinite string of stacked tetrahedra, no two tetrahedra will have the same orientation, because the helical pitch per cell is not a rational fraction of the circle. However, modified forms of this helix have been found which are rotationally repetitive, and in 4-dimensional space this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the 3-sphere surface of the 600-cell, one of the six regular convex polychora.

In geometry, a generalized helicoid is a surface in Euclidean space generated by rotating and simultaneously displacing a curve, the profile curve, along a line, its axis. Any point of the given curve is the starting point of a circular helix. If the profile curve is contained in a plane through the axis, it is called the meridian of the generalized helicoid. Simple examples of generalized helicoids are the helicoids. The meridian of a helicoid is a line which intersects the axis orthogonally.

A proper reference frame in the theory of relativity is a particular form of accelerated reference frame, that is, a reference frame in which an accelerated observer can be considered as being at rest. It can describe phenomena in curved spacetime, as well as in "flat" Minkowski spacetime in which the spacetime curvature caused by the energy–momentum tensor can be disregarded. Since this article considers only flat spacetime—and uses the definition that special relativity is the theory of flat spacetime while general relativity is a theory of gravitation in terms of curved spacetime—it is consequently concerned with accelerated frames in special relativity.

## References

1. ἕλιξ Archived 2012-10-16 at the Wayback Machine , Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
2. Weisstein, Eric W. "Helicoid". MathWorld .
3. "Double Helix Archived 2008-04-30 at the Wayback Machine " by Sándor Kabai, Wolfram Demonstrations Project.
4. O'Neill, B. Elementary Differential Geometry, 1961 pg 72
5. O'Neill, B. Elementary Differential Geometry, 1961 pg 74
6. Izumiya, S. and Takeuchi, N. (2004) New special curves and developable surfaces. Turk J Math Archived 2016-03-04 at the Wayback Machine , 28:153–163.
7. Menninger, T. (2013), An Explicit Parametrization of the Frenet Apparatus of the Slant Helix. arXiv:1302.3175 Archived 2018-02-05 at the Wayback Machine .