A helix (; pl. helices) is a shape like a cylindrical coil spring or the thread of a machine screw. It is a type of smooth skew curve with tangent lines at a constant angle to a fixed axis. 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, "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is a surface called a helicoid.
Properties and types The pitch of a helix is the height 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 circular helix (i.e. one with constant radius) has constant band curvature and constant torsion. The slope of a circular helix is commonly defined as the ratio of the circumference of the circular cylinder that it spirals around, and its pitch (the height of one complete helix turn). A conic helix, also known as a conic spiral, 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. 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 curvature to torsion is constant. A curve is called a slant helix if its principal normal makes a constant angle with a fixed line in space. It can be constructed by applying a transformation to the moving frame of a general helix. For more general helix-like space curves can be found, see space spiral; e.g., spherical spiral.
Handedness 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 chirality) is a property of the helix, not of the perspective: a right-handed helix cannot be turned to look like a left-handed one unless it is viewed in a mirror, and vice versa.
Mathematical description
In mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a particular helix; perhaps the simplest equations for one is
x ( t ) = cos ( t ) , y ( t ) = sin ( t ) , z ( t ) = t . {\displaystyle {\begin{aligned}x(t)&=\cos(t),\\y(t)&=\sin(t),\\z(t)&=t.\end{aligned}}}
As the parameter t increases, the point ( x ( t ) , y ( t ) , z ( t ) ) {\displaystyle (x(t),y(t),z(t))} traces a right-handed helix of pitch 2π (or slope 1) and radius 1 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 , θ ( t ) = t , h ( t ) = t . {\displaystyle {\begin{aligned}r(t)&=1,\\\theta (t)&=t,\\h(t)&=t.\end{aligned}}}
A circular helix of radius a and slope a/b (or pitch 2πb) is described by the following parametrisation:
x ( t ) = a cos ( t ) , y ( t ) = a sin ( t ) , z ( t ) = b t . {\displaystyle {\begin{aligned}x(t)&=a\cos(t),\\y(t)&=a\sin(t),\\z(t)&=bt.\end{aligned}}}
Another way of mathematically constructing a helix is to plot the complex-valued function exi as a function of the real number x (see Euler's formula). The value of x and the real and imaginary parts of the function value give this plot three real dimensions. 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 one of the x, y or z components.
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