In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.

Spiral or helix

A spiral is typically a planar curve (that is, flat), like the groove on a record or the arms of a spiral galaxy. A helix, on the other hand, is a three-dimensional coil that runs along the surface of a cylinder, like a screw. There are many instances where in colloquial usage spiral is used as a synonym for helix, notably spiral staircase and spiral binding of books. Mathematically this is incorrect but the terms are increasing in common usage.

In the side picture, the black curve at the bottom is an Archimedean spiral, while the green curve is a helix. A cross between a spiral and a helix, such as the curve shown in red, is known as a conic helix. An example of a conic helix is the spring used to hold and make contact with the negative terminals of AA or AAA batteries in remote controls.

Two-dimensional spirals

A two-dimensional spiral may be described most easily using polar coordinates, where the radius r is a continuous monotonic function of angle θ. The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant).

Some of the more important sorts of two-dimensional spirals include:

• The Archimedean spiral: r = a + bθ
• The Cornu spiral or clothoid
• Fermat's spiral: r = θ^1/2
• The hyperbolic spiral: r = a/θ
• The lituus: r = θ^-1/2
• The logarithmic spiral: r = ab^θ; approximations of this are found in nature
• The Fibonacci spiral and golden spiral: special cases of the logarithmic spiral
• The Spiral of Theodorus: an aproximation of the Archimedean spiral composed of contiguous right triangles

Image:Archimedean spiral.svg|Archimedean spiral Image:Cornu Spiral.svg|Cornu spira Image:Fermat's spiral.svg|Fermat's spiral Image:Hyperspiral.svg|hyperbolic spiral Image:Lituus.svg|lituus Image:Logarithmic spiral.svg|logarithmic spiral Image:Spiral of Theodorus.svg|spiral of Theodorus

Three-dimensional spirals

For simple 3-d spirals, a third variable, h (height), is also a continuous, monotonic function of θ. For example, a conic helix may be defined as a spiral on a conic surface, with the distance to the apex an exponential function of θ.

The helix and vortex can be viewed as a kind of three-dimensional spiral.