Helix

Helix
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Helix".

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, used by a climber plant.

A helix (pl: helices), from the Greek word έλικας/έλιξ, is a three-dimensional, twisted shape. Common objects formed like a helix are a spring, a screw, and a spiral staircase (though the last would be more correctly called helical).^[1]^[2] 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.

content

1 Types
2 Mathematics
3 Examples
4 References
5 See also

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 a right-handed helix. If counter-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.

Here is another test for handedness: first grip the helix with your right hand and direct your thumb parallel to the axis of the helix. Then curl your fingers toward your palm, following the path of the spiral as if the helix were a set of rails that your fingers must slide along. If this causes your entire hand to move in the same direction as your thumb is pointing, then the helix is right-handed. If not, it is left-handed. Try this test on the left-handed helix in the picture below; in this case, your hand should move in the direction opposite to the way your thumb points.

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.^[3]

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 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 if its tangent makes a constant angle with a fixed line in space.

Mathematics

In mathematics, a helix is a curve in 3-dimensional space. The following three equations in rectangular coordinates define a helix^[4]:

$x = \cos(t),\,$

$y = \sin(t),\,$

$z = t.\,$

As the parameter t increases, the point (x,y,z) 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 described by:

$r = 1,\,$

$\theta = t,\,$

$h = t.\,$

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 equations:

$x = a\cos(t),\,$

$y = a\sin(t),\,$

$z = bt.\,$

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 either the x, y or z component.

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 $\frac{b}{a^2+b^2}.$

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.

References

^ "Helices" by Sándor Kabai, The Wolfram Demonstrations Project.
^ "Helical Staircase" by Sándor Kabai, The Wolfram Demonstrations Project.
^ "Double Helix" by Sándor Kabai, The Wolfram Demonstrations Project.
^ Eric W. Weisstein, Helix at MathWorld.

Contents

Types

Mathematics

Examples

References

See also