Helix

Overview

**helix**is a type of smooth

Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

space curve, i.e. a curve in three-dimensional space

Three-dimensional space

Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

. It has the property that the tangent line at any point makes a constant angle

Angle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

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

Helicoid

The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through...

. Helices are important in biology

Biology

Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

, as the DNA

DNA

Deoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms . The DNA segments that carry this genetic information are called genes, but other DNA sequences have structural purposes, or are involved in...

molecule is formed as two intertwined helices, and many protein

Protein

Proteins are biochemical compounds consisting of one or more polypeptides typically folded into a globular or fibrous form, facilitating a biological function. A polypeptide is a single linear polymer chain of amino acids bonded together by peptide bonds between the carboxyl and amino groups of...

s have helical substructures, known as alpha helices

Alpha helix

A common motif in the secondary structure of proteins, the alpha helix is a right-handed coiled or spiral conformation, in which every backbone N-H group donates a hydrogen bond to the backbone C=O group of the amino acid four residues earlier...

.

Unanswered Questions

Encyclopedia

A

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

. Helices are important in biology

, as the DNA

molecule is formed as two intertwined helices, and many protein

s have helical substructures, known as alpha helices

. The word

word

) 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.

Most hardware screw thread

s 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.

The

A double helix consists of two (typically congruent) helices with the same axis, differing by a translation along the axis.

A

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

and constant torsion

.

A curve is called a

to torsion

is constant.

, a helix is a curve

in 3-dimension

al space. The following parametrisation

in Cartesian coordinates

defines a helix:

As the parameter

In cylindrical coordinates (

A circular helix of radius

Another way of mathematically constructing a helix is to plot a complex valued exponential function (

).

Except for rotation

s, 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

equals , its curvature is

and its torsion is

, 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.

**helix**is a type of smoothDifferentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

space curve, i.e. a curve in three-dimensional space

Three-dimensional space

Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

. It has the property that the tangent line at any point makes a constant angle

Angle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

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 helicoidHelicoid

The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through...

. Helices are important in biology

Biology

Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

, as the DNA

DNA

Deoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms . The DNA segments that carry this genetic information are called genes, but other DNA sequences have structural purposes, or are involved in...

molecule is formed as two intertwined helices, and many protein

Protein

Proteins are biochemical compounds consisting of one or more polypeptides typically folded into a globular or fibrous form, facilitating a biological function. A polypeptide is a single linear polymer chain of amino acids bonded together by peptide bonds between the carboxyl and amino groups of...

s have helical substructures, known as alpha helices

Alpha helix

A common motif in the secondary structure of proteins, the alpha helix is a right-handed coiled or spiral conformation, in which every backbone N-H group donates a hydrogen bond to the backbone C=O group of the amino acid four residues earlier...

. The word

*helix*comes from the GreekGreek language

Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

word

*ἕλιξ*, "twisted, curved".## Types

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 chiralityChirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...

) 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.

Most hardware screw thread

Screw thread

A screw thread, often shortened to thread, is a helical structure used to convert between rotational and linear movement or force. A screw thread is a ridge wrapped around a cylinder or cone in the form of a helix, with the former being called a straight thread and the latter called a tapered thread...

s are right-handed helices. The alpha helix in biology as well as the A

A-DNA

A-DNA is one of the many possible double helical structures of DNA. A-DNA is thought to be one of three biologically active double helical structures along with B- and Z-DNA. It is a right-handed double helix fairly similar to the more common and well-known B-DNA form, but with a shorter more...

and B forms of DNA are also right-handed helices. The Z form

Z-DNA

Z-DNA is one of the many possible double helical structures of DNA. It is a left-handed double helical structure in which the double helix winds to the left in a zig-zag pattern...

of DNA is left-handed.

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.

A

**conic helix**may be defined as a spiralSpiral

In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.-Spiral or helix:...

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

Corkscrew (Cedar Point)

Corkscrew is a roller coaster at the Cedar Point amusement park in Sandusky, Ohio. When built in 1976, it was the first roller coaster in the world with 3 inversions....

roller coaster at Cedar Point amusement park.

A

**circular helix,**(i.e. one with constant radius) has constant band curvatureCurvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

and constant torsion

Torsion of curves

In the elementary differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting. Taken together,the curvature and the torsion of a space curve are analogous to the curvature of a plane curve...

.

A curve is called a

**general helix**or**cylindrical helix**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 curvatureCurvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

to torsion

Torsion of curves

In the elementary differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting. Taken together,the curvature and the torsion of a space curve are analogous to the curvature of a plane curve...

is constant.

## Mathematical description

In mathematicsMathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a helix is a curve

Differential geometry of curves

Differential geometry of curves is the branch of geometry that dealswith smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus....

in 3-dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

al space. The following parametrisation

Parametric equation

In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....

in Cartesian coordinates

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

defines a helix:

As the parameter

Parameter

Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

*t*increases, the point (*x*(*t*),*y*(*t*),*z*(*t*)) traces a right-handed helix of pitch 2*π*

and radius 1 about thePi

' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

*z*-axis, in a right-handed coordinate system.In cylindrical coordinates (

*r*,*θ*,*h*), the same helix is parametrised by:A circular helix of radius

*a*and pitch 2*πb*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 formulaEuler's formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...

).

Except for rotation

Rotation

A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

s, translation

Translation (geometry)

In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

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.### Arc length, curvature and torsion

The length of a circular helix of radius*a*and pitch 2*πb*expressed in rectangular coordinates asequals , its curvature is

and its torsion is

## Examples

In musicMusic

Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, pitch space

Pitch space

In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apart. Depending on the complexity of the relationships...

is often modeled with helices or double helices, most often extending out of a circle such as the circle of fifths

Circle of fifths

In music theory, the circle of fifths shows the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys...

, so as to represent octave equivalency.

## See also

- Alpha helixAlpha helixA common motif in the secondary structure of proteins, the alpha helix is a right-handed coiled or spiral conformation, in which every backbone N-H group donates a hydrogen bond to the backbone C=O group of the amino acid four residues earlier...
- Boerdijk–Coxeter helixBoerdijk–Coxeter helixThe Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra. There are two chiral forms, with either clockwise or counterclockwise windings. Contrary to any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally...
- CollagenCollagenCollagen is a group of naturally occurring proteins found in animals, especially in the flesh and connective tissues of mammals. It is the main component of connective tissue, and is the most abundant protein in mammals, making up about 25% to 35% of the whole-body protein content...
- Double helix
- Helical symmetry
- HelicoidHelicoidThe helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through...
- Helix angleHelix angleIn mechanical engineering, a helix angle is the angle between any helix and an axial line on its right, circular cylinder or cone. Common applications are screws, helical gears, and worm gears....
- Seashell surfaceSeashell surfaceIn mathematics, a seashell surface is a surface made by a circle which spirals up the z-axis while decreasing its own radius and distance from the z-axis...
- SolenoidSolenoidA solenoid is a coil wound into a tightly packed helix. In physics, the term solenoid refers to a long, thin loop of wire, often wrapped around a metallic core, which produces a magnetic field when an electric current is passed through it. Solenoids are important because they can create...
- SpiralSpiralIn mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.-Spiral or helix:...
- SuperhelixSuperhelixA superhelix is a molecular structure in which a helix is itself coiled into a helix. This is significant to both proteins and genetic material, such as overwound circular DNA....
- Triple helixTriple helixIn geometry, a triple helix is a set of three congruent geometrical helices with the same axis, differing by a translation along the axis. Structures in the form of a triple helix include:* collagen helix...