# Helix

A helix (pl: helices), from the Greek word έλικας/έλιξ, is a twisted shape like a spring, screw or a spiral staircase. Helixes are important in biology, as DNA is helical and many proteins have helical substructures, known as alpha helices.

File:Helix.png
A left-handed and a right-handed helix.
File:Climber natural spiral.jpg
A natural helix, used by a climber plant

Photograph by Dirk van der Made

Right-handed and left-handed helices can be distinguished from each other. If you move along a helix in the direction of your right hand's thumb, and the helix turns in the direction of your right hand's fingers, then it's a right-handed helix, otherwise a left-handed one. Another way to visualize this distinction: picture the helix vertical; if the front strands move from the lower left to the upper right, then it is a right-handed helix. Note that handedness (or chirality) is a property of the helix, not of the perspective: you can turn a right-handed helix around and it's still right-handed.

Most 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.

The pitch of a helix is the length 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:

x = cosine t
y = sine t
z = t

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 helixes with the same axis, differing by a translation along the axis, which may or may not be half-way.

In music pitch space is often modeled with helixes or double helixes, 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.