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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
An Archimedean spiral, a helix, and a conic spiral.

While a "spiral" and a "helix" are distinct as technical terms, a helix is sometimes described as a spiral in non-technical usage. The two primary definitions of a spiral are provided by the American Heritage Dictionary:[1]

a. A curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
b. A three-dimensional curve that turns around an axis at a varying distance while moving parallel to the axis.

The first definition is for a planar curve that extends primarily in length and width, but not in height. A groove on a record[2] or the arms of a spiral galaxy (a Logarithmic spiral) are examples of a spiral.

The second definition is for the 3-Dimensional variant of a spiral, for example a conical spring (device) can be described as a spiral whereas a cylindrical spring or strand of a DNA are examples of a helix.[1]

The length and width of a helix typically remain static and do not grow like on a planar spiral. If they do, then the helix becomes a conic helix. You can make a conic helix with an Archimedean or equiangular spiral by giving height to the center point, thereby creating a cone-shape from the spiral.[3]

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. The spring used to hold and make contact with the negative terminals of AA or AAA batteries in remote controls and the vortex that is created when water is draining in a sink are examples of conic helices.
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: \( r=a+b \cdot \theta \) (see also:Involute)
The Euler spiral, Cornu spiral or clothoid
Fermat's spiral: \( r= \theta^{1/2} \)
The hyperbolic spiral: r = a/ \theta
The lituus: \( r = \theta^{-1/2} \)
The logarithmic spiral: \( r=a\cdot e^{b\theta} \); 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

Archimedean spiral

Cornu spiral

Fermat's spiral

hyperbolic spiral


logarithmic spiral

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:

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:

TORUSA-1 Torus mit variablem Ringdurchmesser.PNG
Spherical spiral
Rhumb line

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. The curve has an infinite number of 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 radius changes and hence is not a rhumb line.
As a symbol
The Newgrange entrance slab

The spiral plays a specific role in symbolism, and appears in megalithic art, notably in the Newgrange tomb or in many Galician petroglyphs such as the one in Mogor. See, for example, the 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).

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). 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. The spiral is also found in structures as small as the double helix structure of DNA and as large as the spiral structure of a galaxy.

The spiral is also a symbol of the process of dialectic.
In nature
The 53rd plate from Ernst Haeckel's Kunstformen der Natur (1904), depicting organisms classified as Prosobranchia (now known to be polyphyletic).

The study of spirals in nature have a long history, Christopher Wren observed that many shells form a logarithmic spiral. Jan Swammerdam observed the common mathematical characteristics of a wide range of shells from Helix to Spirula and Henry Nottidge Moseley described the mathematics of univalve shells. 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 perpendicular to the axis and the shell will form a planar discoid shape. In others it follows a skew path forming a helico-spiral pattern.

Thompson also studied spirals occurring in horns, teeth, claws and plants.[4]

Spirals in plants and animals are frequently described as whorls. This is also the name given to spiral shaped fingerprints.

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.[5]
In art

The spiral has inspired artists throughout the ages. Among the most famous of spiral-inspired art is Robert Smithson's earthwork, "Spiral Jetty", at the Great Salt Lake in Utah. The spiral theme is also present in David Wood's Spiral Resonance Field at the Balloon Museum in Albuquerque, as well as in the critically acclaimed Nine Inch Nails 1994 concept album The Downward Spiral. The Spiral is also a prominent theme in the anime Gurren Lagann, where it represents a philosophy and way of life.

Spirals was also the source of material for Japanese horror manga artist Junji Ito for his manga Uzumaki about a town obsessed with spirals, which was adapted to a feature film in 2000.
See also

Seashell surface
Celtic maze (straight-line spiral)


^ a b Spiral
^ Spirals by Jürgen Köller
^ Draw Helixes
^ Thompson, D'Arcy (1917,1942). On Growth and Form
^ Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0-387-97297-8.

External links, an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination.
Spirals by Jürgen Köller

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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