Spiral

From Wikipedia, the free encyclopedia

For other senses of this word, see spiral (disambiguation).

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

1 Spiral vs. helix
2 Two-dimensional spirals
3 Three-dimensional spirals
- 3.1 Spherical spiral
4 As a symbol
5 Popular culture
6 In Nature
7 External links

[edit] Spiral vs. helix

Image:TORUSJ Schraube und archimedische Spirale.PNG

A "spiral" and a "helix" are two terms that are easily confused, but represent different objects.

A spiral is typically a planar curve (that is, flat), like the ridges of 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.

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.

[edit] Two-dimensional spirals

A two-dimensional spiral may be described easiest 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 = 1/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.

[edit] 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:

Image:TORUSA-4 Konische Spirale entlang eines Kreises.PNG

This image is reminiscent of a Ouroboros symbol and could be mistaken for a torus with a continuously-increasing diameter:

Image:TORUSA-1 Torus mit variablem Ringdurchmesser.PNG

[edit] Spherical spiral

Image:KUGSPI-9 Loxodrome.gif

A spherical spiral (rhumb line or loxodrome, upper picture) is the curve on a sphere traced by a ship traveling from one pole to the other while keeping a fixed angle (unequal to 0° and to 90°) with respect to the meridians of longitude, i.e. keeping the same bearing. The curve has an infinite number of revolutions, with the distance between them decreasing as the curve approaches either of the poles.

Image:KUGSPI-5 Archimedische Kugelspirale.gif

The gap between the curves of an Archimedean spiral (lower picture) remains constant as the curve progresses across the surface of the sphere. Therefore, this line has finite length. Notice that this is not the same thing as the rhumb line described earlier.