Helix

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This article is about the shape. For other uses, see Helix (disambiguation).
A helix constructed in brick.
Crystal structure of a folded molecular helix reported by Lehn and coworkers in Helv. Chim. Acta., 2003, 86, 1598-1624.
A natural left-handed helix, made by a climber plant.

A helix (pl: helixes or helices) is a special kind of space curve, i.e. a smooth curve in three-space. As a mental image of a helix one may take the spring (although the spring is not a curve, and so is technically not a helix, it does give a convenient mental picture). A helix is characterised by the fact that the tangent line at any point makes a constant angle with a fixed line. A filled in helix, for example a spiral staircase, is called a helicoid[1]. 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.

Contents

[edit] Types

Helices can be either right-handed or left-handed. With the line of sight being the helical axis, if clockwise movement of the helix corresponds to axial movement away from the observer, then it is called a right-handed helix. If anti-clockwise movement corresponds to axial movement away from the observer, 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 or flipped to look like a left-handed one unless it is viewed through a mirror, and vice versa.

Most hardware screws 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.

A double helix typically consists geometrically of two congruent helices with the same axis, differing by a translation along the axis, which may or may not be half-way.[2]

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 of a helix would be the Corkscrew roller coaster at Cedar Point amusement park.

A circular helix has constant band curvature and constant torsion. The pitch of a helix is the width of one complete helix turn, measured along the helix axis.

A curve is called a general helix or cylindrical helix[3] 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[4].

[edit] Mathematics

In mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a helix[5]:

x(t) = \cos(t),\,
y(t) = \sin(t),\,
z(t) = t.\,
The helix (cos t, sin t, t) from t = 0 to 4π with arrowheads showing direction of increasing t.

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

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

r(t) = 1,\,
\theta(t) = t,\,
h(t) = t.\,

The above example is an example of circular helix of radius 1 and pitch 2π.

Circular helix of radius a and pitch 2πb is described by the following parametrisation:

x(t) = a\cos(t),\,
y(t) = a\sin(t),\,
z(t) = bt.\,

Another way of mathematically constructing a helix is to plot a complex valued exponential function (e^xi) taking imaginary arguments (see Euler's formula).

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 of the x, y or z components.

The length of a circular helix of radius a and pitch 2πb expressed in rectangular coordinates as

t\mapsto (a\cos t, a\sin t, bt), t\in [0,T]

equals T\cdot \sqrt{a^2+b^2}, its curvature is \frac{|a|}{a^2+b^2} and its torsion is \frac{b}{a^2+b^2}.

[edit] Examples

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.

[edit] See also

[edit] References

  1. ^ Weistein, Eric W., "Helicoid" from MathWorld.
  2. ^ "Double Helix" by Sándor Kabai, Wolfram Demonstrations Project.
  3. ^ O'Neill, B. Elementary Differential Geometry, 1961 pg 72
  4. ^ O'Neill, B. Elementary Differential Geometry, 1961 pg 74
  5. ^ Weistein, Eric W., "Helix" from MathWorld.
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