Cutaway of a nautilus 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 can refer to Spiral: a curve in mathematics an extremely common but technically incorrect term for another curve, the helix; e. ... Download high resolution version (1095x862, 107 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1095x862, 107 KB) Wikipedia does not have an article with this exact name. ... Genera Allonautilus Nautilus Nautilus (from Greek ναυτίλος, sailor) is the common name of any marine creatures of the cephalopod family Nautilidae, the sole family of the suborder Nautilina. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

Spiral or helix?

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

A "spiral" and a "helix" are two terms that are easily confused, but represent different objects. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... A helix (pl: helices), from the Greek word έλικας/έλιξ, is a twisted shape like a spring, screw or a spiral (correctly termed helical) staircase. ...

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. There are a few 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 in common usage. Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ... It has been suggested that Childrens gramophone records be merged into this article or section. ... It has been suggested that spiral nebula be merged into this article or section. ... Screws come in a variety of shapes and sizes for different purposes. ... A colloquialism is an informal expression, that is, an expression not used in formal speech or writing. ... Spiral (double helix) stairway in the Vatican Museum Stairs, staircase, stairway, stairwell, and flight of stairs are all names for a construction designed to bridge a large vertical distance by dividing it into smaller vertical distances, called steps. ... Old book binding and cover Bookbinding is the process of physically assembling a book from a number of folded or unfolded sheets of paper or other material. ...

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 Archimedean spiral is a curve which in polar coordinates (r, θ) can be described by the equation with real numbers a and b. ...

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). Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... Circle illustration In classical geometry, a radius (plural: radii) of a circle or sphere is any line segment from its center to its boundary. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. ...

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.

Archimedean spiral An Archimedean spiral is a curve which in polar coordinates (r, θ) can be described by the equation with real numbers a and b. ... In mathematics and optics, the two Fresnel integrals, S(x) and C(x) arise in the description of near field Fresnel diffraction phenomena, and are the integrals defined as follows: . Some may use π t2/2 instead of t2, in which case the S(x) and C(x) above should... Fermats spiral (also known as a parabolic spiral) follows the equation in polar coordinates. ... A hyperbolic spiral is a transcendental plane curve also known as a reciprocal spiral. ... A lituus (Latin for crozier) is a spiral in which (in polar coordinates) the angle is inversely proportional to the square of the radius. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral. ... A golden spiral is a logarithmic spiral whose growth factor b is related to phi, the golden ratio. ... Image File history File links This is a lossless scalable vector image. ... An Archimedean spiral is a curve which in polar coordinates (r, θ) can be described by the equation with real numbers a and b. ...

logarithmic spiral Image File history File links This is a lossless scalable vector image. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ...

Fermat's spiral Image File history File links Fermat's_spiral. ... Fermats spiral (also known as a parabolic spiral) follows the equation in polar coordinates. ...

hyperbolic spiral Image File history File links Size of this preview: 201 × 201 pixelsFull resolution (201 × 201 pixel, file size: 2 KB, MIME type: image/png)Hyperbolic spiral, rθ=2, made by me. ... A hyperbolic spiral is a transcendental plane curve also known as a reciprocal spiral. ...

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. In mathematics, functions between ordered sets are monotonic (or monotone, or even isotone) if they preserve the given order. ... A helix (pl: helices), from the Greek word έλικας/έλιξ, is a twisted shape like a spring, screw or a spiral (correctly termed helical) staircase. ...

The helix and vortex can be viewed as a kind of three-dimensional spiral. A helix (pl: helices), from the Greek word έλικας/έλιξ, is a twisted shape like a spring, screw or a spiral (correctly termed helical) staircase. ... Vortex created by the passage of an aircraft wing, revealed by coloured smoke A vortex (pl. ... 2-dimensional renderings (ie. ...

For a helix with thickness, see spring (math). A Spring A left-handed and a right-handed spring. ...

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: Look up cone in Wiktionary, the free dictionary. ...

Image File history File links Size of this preview: 309 × 149 pixelsFull resolution (309 × 149 pixel, file size: 8 KB, MIME type: image/png) Source: de:Bild:TORUSA-4_Konische_Spirale_entlang_eines_Kreises. ...

This image is reminiscent of a Ouroboros symbol and could be mistaken for a torus with a continuously-increasing diameter: 1478 drawing by Theodoros Pelecanos, in alchemical tract titled Synosius. ...

Image File history File links Size of this preview: 310 × 185 pixelsFull resolution (310 × 185 pixel, file size: 16 KB, MIME type: image/png) Source: de:Bild:TORUSA-1_Torus_mit_variablem_Ringdurchmesser. ...

Spherical spiral

Rhumb line

Archimedean 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. The curve has an infinite number of revolutions, with the distance between them decreasing as the curve approaches either of the poles. Image File history File links Loxodrome Description: The Loxodrome hat im Gegensatz zur archimedischen Kugelspirale einen konstanten Winkel zwischen den Längengraden und den Breitengraden, der hier das Verhältnis von 10 Länge zu 1 Breite hat. ... Image File history File links Loxodrome Description: The Loxodrome hat im Gegensatz zur archimedischen Kugelspirale einen konstanten Winkel zwischen den Längengraden und den Breitengraden, der hier das Verhältnis von 10 Länge zu 1 Breite hat. ... Example of pole-to-pole loxodrome In navigation, a rhumb line (or loxodrome) is a line crossing all meridians at the same angle, i. ... Image File history File links Size of this preview: 200 × 200 pixelsFull resolution (200 × 200 pixel, file size: 62 KB, MIME type: image/gif) Source: de:Bild:KUGSPI-5_Archimedische_Kugelspirale. ... Image File history File links Size of this preview: 200 × 200 pixelsFull resolution (200 × 200 pixel, file size: 62 KB, MIME type: image/gif) Source: de:Bild:KUGSPI-5_Archimedische_Kugelspirale. ... Example of pole-to-pole loxodrome In navigation, a rhumb line (or loxodrome) is a line crossing all meridians at the same angle, i. ... An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ... Longitude, sometimes denoted by the Greek letter λ (lambda),[1][2] describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ... In navigation, a bearing is the clockwise angle between a reference direction (or a datum line) and the direction to an object. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Orbital revolution is the cyclical path taken by one object around another object (or point, line, etc. ...

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. Example of pole-to-pole loxodrome In navigation, a rhumb line (or loxodrome) is a line crossing all meridians at the same angle, i. ...

As a symbol

The Newgrange entrance slab

The spiral plays a certain role in symbolism, and appears in megalithic art, notably in the Newgrange tomb. See also triple spiral. Image File history File links The entrance slab to Newgrange tomb, Ireland File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links The entrance slab to Newgrange tomb, Ireland File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Megalithic tomb, Mane Braz, Brittany A megalith is a large stone which has been used to construct a structure or monument either alone or with other stones. ... Newgrange, which is located at , is one of the passage tombs of the Brú na Bóinne complex in County Meath, and the most famous of all Irish prehistoric sites. ... A modern form of the triple spiral symbol Triple spiral visible on entrance stone at Newgrange The triple spiral or triskele is a Celtic and pre-Celtic symbol found on a number of Irish Megalithic and Neolithic sites, most notably inside the Newgrange passage tomb, on the entrance stone, and...

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). China under the Tang Dynasty (yellow) and its sphere of influence Capital Changan (618–904) Luoyang (904-907) Language(s) Middle Chinese Religion Buddhism, Taoism, Confucianism Government Monarchy Emperor  - 618-626 Emperor Gaozu  - 684, 705-710 Emperor Zhongzong  - 684, 710-712 Emperor Ruizong  - 904-907 Emperor Ai History  - Li... Changan â–¶(?) (Simplified Chinese: 长安; Traditional Chinese: 長安; pinyin: ; Wade-Giles: Chang-an) is the ancient capital of more than ten dynasties in China. ...

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. Professor Charcot was well-known for showing, during his lessons at the Salpêtrière hospital, hysterical woman patients – here, his favorite patient, Blanche (Marie) Wittman, supported by Joseph BabiÅ„ski. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Mowgli with Kaa in a Russian cartoon (1969). ... // Pre-syncope is a sensation of feeling faint. ... The main cast of the anime Cowboy Bebop (1998) (L to R: Spike Spiegel, Jet Black, Ed Tivrusky, Faye Valentine, and Ein the dog) For the oleo-resin, see Animé (oleo-resin). ... Manga )   (pl. ...

In Nature

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 pirpendicular to the axis and the shell will form a planer discoid shape. In others it follows a skew path forming a helico-spiral pattern. Image File history File links Download high resolution version (2351x3298, 1521 KB) Wikipedia links on Talk: Because the description below is a Wikimedia Commons description page, the links are to Wikimedia Commons pages also. ... Sir Christopher Wren, (20 October 1632–25 February 1723) was a 17th century English designer, astronomer, geometrician, and the greatest English architect of his time. ... Various seashells Danielle A shell is the hard, rigid outer covering, or integument, allanimals. ... A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... Jan Swammerdam (February 12, 1637 - February 17, 1680) was a Dutch biologist and microscopist. ... Species See text. ... Binomial name Spirula spirula (Linnaeus, 1758) Rams Horn Squid (Spirula spirula) is a unique and peculiar species of mollusc which constitutes a family (Spirulidae) and order (Spirulida) by itself. ... Henry Nottidge Moseley (14 November 1844 - 10 November 1891) was a British naturist. ... Subclass Subclass Eogastropoda     Patellogastropoda Subclass Orthogastropoda   Superorder Cocculiniformia   Superorder Hot Vent Taxa     Neomphaolida   Superorder Vetigastropoda   Superorder Neritaemorphi     Neritopsina   Superorder Caenogastropoda     Architaenioglossa     Sorbeoconcha   Superorder Heterobranchia     Heterostropha     Opisthobranchia     Pulmonata The gastropods, gasteropods, or univalves, are the largest and most successful class of mollusks, with 60,000-75,000 extant species known, comprising... Various seashells Danielle A shell is the hard, rigid outer covering, or integument, allanimals. ... DArcy Wentworth Thompson (May 2, 1860- June 21, 1948) was a biologist and mathematician and the author of the 1917 book, On Growth and Form, an influential work of striking originality. ... Shape (OE. sceap Eng. ... Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ... which converges to 2. ... Genera Allonautilus Nautilus Nautilus (from Greek ναυτίλος, sailor) is the common name of any marine creatures of the cephalopod family Nautilidae, the sole family of the suborder Nautilina. ... For other uses, see Ammonite (disambiguation). ... A helix (pl: helices), from the Greek word έλικας/έλιξ, is a twisted shape like a spring, screw or a spiral (correctly termed helical) staircase. ...

Thompson also studied spirals occurring in horns, teeth, claws and plants. ^[1] Highland cow, a very old long-horned breed from Scotland. ... Types of teeth Molars are used for grinding up foods Carnassials are used for slicing food. ... A claw is a curved pointed appendage, found at the end of a toe or finger or, in arthropods, of the tarsus. ... Divisions Green algae Chlorophyta Charophyta Land plants (embryophytes) Non-vascular plants (bryophytes) Marchantiophyta—liverworts Anthocerotophyta—hornworts Bryophyta—mosses Vascular plants (tracheophytes) †Rhyniophyta—rhyniophytes †Zosterophyllophyta—zosterophylls Lycopodiophyta—clubmosses †Trimerophytophyta—trimerophytes Pteridophyta—ferns and horsetails Seed plants (spermatophytes) †Pteridospermatophyta—seed ferns Pinophyta—conifers Cycadophyta—cycads Ginkgophyta—ginkgo Gnetophyta—gnetae Magnoliophyta—flowering plants...

Spirals in plants and animals are frequently described as whorls. Look up whorl in Wiktionary, the free dictionary. ...

A model for the pattern of florets in the head of a sunflower was proposed by H Vogel. This has the form Wildflowers A flower is the reproductive organ of those plants classified as angiosperms (flowering plants; Division Magnoliophyta). ... Binomial name L. The sunflower (Helianthus annuus) is an annual plant native to the Americas in the family Asteraceae, with a large flowering head (inflorescence). ...

,

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.^[2] Fermats spiral (also known as a parabolic spiral) follows the equation in polar coordinates. ... // Articles with similar titles include Golden mean (philosophy), the felicitous middle between two extremes, and Golden numbers, an indicator of years in astronomy and calendar studies. ...

References

1. ^ Thompson, D'Arcy (1917,1942), On Growth and Form
2. ^ Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag, 101-107. ISBN 978-0387972978. 

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Spiral