Helix
From Wikipedia, the free encyclopedia
- This article is about the shape. See helix (disambiguation) for other meanings.
A helix (pl: helices), from the Greek word έλικας/έλιξ, is a twisted shape like a spring, screw or a spiral (correctly termed helical) staircase. Helices are important in biology, as DNA is helical and many proteins have helical substructures, known as alpha helices.
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.
Most hardware screws are right-handed helices. Points on the surface of an (American) football thrown by a right-handed quarterback (usually) travel in a right-handed helix. 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 pitch of a helix is the width of one complete helix turn, measured along the helix axis.
In mathematics, a helix is a curve in 3-dimensional space. The following three equations in rectangular coordinates define a helix:
Here t is a real parameter. As 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
- θ = h
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.
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.
For a helix with thickness, see Spring (math).
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.
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.
