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shell showing the chambers arranged in an approximately logarithmic spiral.

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

A spiral is typically a Plane (mathematics) curve (that is, flat), like the ridges of a Gramophone 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 a few instances where in colloquial usage spiral is used as a synonym for helix, notably Stairway#Spiral and helical stairs and Bookbinding#Modern commercial binding of books. Mathematically this is incorrect but the terms are 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.

Two-dimensional spirals A two-dimensional spiral may be described easiest using polar coordinates, where the radius r is a continuous function monotonic function of angle θ. The circle would be regarded as a degenerate (mathematics) 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 = θ1/2
• The hyperbolic spiral: r = a/θ
• The lituus: r = 1/θ1/2
• The logarithmic spiral: r = abθ; approximations of this are found in nature
• The Fibonacci spiral and golden spiral: special cases of the logarithmic spiral.

Image:Archimedean spiral.svg|Archimedean spiralImage:Logarithmic spiral.svg]Image:Fermat's spiral.png|Fermat's spiralImage:Hyperspiral.png]

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 dimension 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:



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



Spherical spiral A spherical spiral (rhumb line or loxodrome, left 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 (navigation). The curve has an infinite number of Orbital revolutions, with the distance between them decreasing as the curve approaches either of the poles.

The gap between the curves of an Archimedean spiral (right 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.

As a symbol The spiral plays a certain role in symbolism, and appears in megalithic art, notably in the Newgrange tomb. See also triple spiral.

While scholars are still debating the subject, there is a growing acceptance that the simple spiral, when found in Chinese art, is an early symbol for the sun. Roof tiles dating back to the Tang Dynasty with this symbol have been found west of the ancient city of Chang'an (modern-day Xian).

The spiral is the most ancient symbol found on every civilized continent. Due to its appearance at burial sites across the globe, the spiral most likely represented the "life-death-rebirth" cycle. Similarly, the spiral symbolized the sun, as ancient people thought the sun was born each morning, died each night, and was reborn the next morning..

Spirals are also a symbol of hypnosis, stemming from the cliché of people and cartoon characters being hypnotized by staring into a spinning spiral (One example being Kaa in Disney's The Jungle Book (1967 film)). They are also used as a symbol of dizziness, where the eyes of a cartoon character, especially in anime and manga, will turn into spirals to show they are dizzy or dazed.

In Nature The study of spirals in nature have a long history, Christopher Wren observed that many Animal shell form a logarithmic spiral. Jan Swammerdam observed the common mathematical characteristics of a wide range of shells from Helix (genus) to Spirula and Henry Nottidge Moseley described the mathematics of univalve Animal shell. D'Arcy Wentworth Thompson's On Growth and Form gives extensive treatment to these spirals. He describes how shells are formed by rotating a closed curve around a fixed axis, the shape of the curve remains fixed but its size grows in a geometric progression. In some shell such as Nautilus and ammonites the generating curve revolves in a plane pirpendicular to the axis and the shell will form a planer discoid shape. In others it follows a skew path forming a helix-spiral pattern.

Thompson also studied spirals occurring in Horn (anatomy)s, teeth, claws and plants.

Spirals in plants and animals are frequently described as whorls.

A model for the pattern of florets in the head of a sunflower was proposed by H Vogel. This has the form \theta = n \times 137.5^{\circ}, r = c \sqrt{n} where n is the index number of the floret and c is a constant scaling factor, and is a form of Fermat's spiral. The angle 137.5° is related to the golden ratio and gives a close packing of florets.{{cite book | last =Prusinkiewicz | first =Przemyslaw | authorlink =Przemyslaw Prusinkiewicz | coauthors =[Aristid Lindenmayer | title =[The Algorithmic Beauty of Plants | publisher =Springer-Verlag | date =1990 | location = | pages =101-107 | url =http://algorithmicbotany.org/papers/#webdocs | doi = | id = ISBN 978-0387972978 -->

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Spiral. Welcome to Spiral, the Digital Repository for research output of Imperial College. Spiral contains full text peer-reviewed versions of journal articles and conference ...

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

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spiral model < programming > A software life-cycle model which supposes incremental development, using the waterfall model for each step, with the aim of managing risk.

Definition: spiral from Online Medical Dictionary
The Online Medical Dictionary is a searchable dictionary of definitions from medicine, science and technology.