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.

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 Euler spiral, 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 approximation of the Archimedean spiral composed of contiguous right triangles

Image:Archimedean spiral.svg|Archimedean spiral Image:Cornu Spiral.svg|Cornu spiral Image:Fermat's spiral.svg|Fermat's spiral Image:Hyperspiral.svg|hyperbolic spiral Image:Lituus.svg|lituus Image:Logarithmic spiral.png|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.

For a helix with thickness, see spring (math).

Another kind of spiral is a conic spiral along a circle. This spiral is formed along the surface of a cone whose axis is bent and restricted to a circle: