Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral. Nautilus (from Greek ναυτίλος, 'sailor' is the common name of any marine creatures of the Cephalopod family Nautilidae, the sole Definition In Polar coordinates ( r, θ the curve can be written as r = ae^{b\theta}\ or \theta

In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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. A helix (pl helixes or helices) from the Greek word έλιξ, is a special kind of Space curve, i

A spiral is typically a planar curve (that is, flat), like the groove on a record or the arms of a spiral galaxy. A gramophone A spiral galaxy is a Galaxy belonging to one of the three main classes of galaxy originally described by Edwin Hubble in his 1936 work “The Realm of the A helix, on the other hand, is a three-dimensional coil that runs along the surface of a cylinder, like a screw. A screw is a shaft with a helical groove or thread formed on its surface and provision at one end to turn the 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. A colloquialism is an expression not used in formal speech, writing or Paralinguistics. Bookbinding is the process of physically assembling a Book from a number of folded or unfolded sheets of Paper or other material Mathematically this is incorrect but the terms are increasing in common usage.

In the side picture, the black curve at the bottom is an Archimedean spiral, while the green curve is a helix. The Archimedean spiral (also known as the arithmetic spiral) is a Spiral named after the 3rd century BC Greek Mathematician 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. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant). for the degeneracy of a Graph, see Arboricity#Related_concepts.

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 = 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 Archimedean spiral (also known as the arithmetic spiral) is a Spiral named after the 3rd century BC Greek Mathematician Fresnel integrals, S ( x) and C ( x) are two Transcendental functions named after Augustin-Jean Fresnel that are used Fermat's spiral (also known as a parabolic Spiral) follows the equation r = \pm\theta^{1/2}\ in Polar coordinates A hyperbolic spiral is a transcendental Plane curve also known as a reciprocal spiral. The word lituus originally meant a curved staff (" Crozier " or trumpet in the ancient Latin language and is used with several meanings in English Definition In Polar coordinates ( r, θ the curve can be written as r = ae^{b\theta}\ or \theta In Mathematics, the Fibonacci numbers are a Sequence of numbers named after Leonardo of Pisa, known as Fibonacci In Geometry, a golden spiral is a Logarithmic spiral whose growth factor b is related to &phi the Golden ratio.

Archimedean spiral

logarithmic spiral

Fermat's spiral

hyperbolic spiral

Three-dimensional spirals

For simple 3-d spirals, a third variable, h (height), is also a continuous, monotonic function of θ. The Archimedean spiral (also known as the arithmetic spiral) is a Spiral named after the 3rd century BC Greek Mathematician Definition In Polar coordinates ( r, θ the curve can be written as r = ae^{b\theta}\ or \theta Fermat's spiral (also known as a parabolic Spiral) follows the equation r = \pm\theta^{1/2}\ in Polar coordinates A hyperbolic spiral is a transcendental Plane curve also known as a reciprocal spiral. 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. A helix (pl helixes or helices) from the Greek word έλιξ, is a special kind of Space curve, i

The helix and vortex can be viewed as a kind of three-dimensional spiral. A helix (pl helixes or helices) from the Greek word έλιξ, is a special kind of Space curve, i V erification of the O rigins of R otation in T ornadoes Ex periment or VORTEX, is a field project that seeks to understand how a In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

For a helix with thickness, see spring (math). In Geometry, a spring is a Surface of revolution in the shape of a Helix with thickness generated by revolving a Circle about the path of a

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

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. A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex The Ouroboros (Greek grc Ουροβόρος from grc ουροβόρος όφις "tail-devouring snake" also spelled Ourorboros, Oroborus, Uroboros See also Great circle Small circle See also Great circle Small circle In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement e. keeping the same bearing. In Navigation, a bearing is the direction one object is from another object The curve has an infinite number of revolutions, with the distance between them decreasing as the curve approaches either of the poles. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star

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. See also Great circle Small circle

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 or in many Galician petroglyphs such as the one in Mogor. "Symbolic" redirects here For other uses see Symbolism (disambiguation and Symbolic (disambiguation. Newgrange (Dún Fhearghusa is one of the Passage tombs of the Brú na Bóinne complex in County Meath, one of the most famous See also triple spiral. The triple spiral or triskele is a Celtic and pre-Celtic symbol found on a number of Irish Megalithic 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). The Tang Dynasty ( Middle Chinese: dhɑng (June 18 618&ndashJune 4 907 was an imperial dynasty of China preceded by the Sui Dynasty and followed by Chang'an ( is an ancient Capital of more than ten dynasties in Chinese history.

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). Hypnosis is often thought to be a wakeful state of focused attention and heightened suggestibility with diminished peripheral awareness A cliché (from French, klɪ'ʃe or cliche is a phrase expression or idea that has been overused to the point of losing its intended force Kaa was not a poison snake--in fact he rather despised the poison snakes as cowards--but his strength lay in his hug and when he had once lapped his huge coils round anybody there was no more to The Jungle Book is a 1967 animated Feature Film, released on October 18 1967. 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. Many different terms are often used to describe what is collectively known as dizziness. (anime in Japanese, ˈmɑŋgə is the Japanese word for Comics (sometimes called komikku コミック and print Cartoons In their modern form manga date from shortly

In nature

The 53rd plate from Ernst Haeckel's Kunstformen der Natur (1904), depicting organisms classified as Prosobranchia (now known to be polyphyletic). Ernst Heinrich Philipp August Haeckel ( February 16, 1834 — August 9, 1919)also written von Haeckel, was an eminent German Prosobranchiagif|thumb|The shell of a Harpa species a prosobranch gastropod]] Prosobranchia used to be a large Taxonomic Subclass of Snails

The study of spirals in nature have a long history, Christopher Wren observed that many shells form a logarithmic spiral. Nature, in the broadest sense is equivalent to the natural world, physical universe, material world or material universe. Sir Christopher Wren ( 20 October 1632 &ndash 25 February 1723) was a 17th century English Designer, Astronomer Definition In Polar coordinates ( r, θ the curve can be written as r = ae^{b\theta}\ or \theta 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. Jan Swammerdam ( February 12, 1637, Amsterdam - February 17, 1680) was a Dutch biologist and microscopist Helix is a Genus of terrestrial pulmonate Snails native to Europe and the regions around the Mediterranean Sea. Spirula spirula is a Species of deepwater Squid -like Cephalopod. Henry Nottidge Moseley ( 14 november 1844 - 10 november 1891) was a British naturalist. The class Gastropoda or the gastropods, also previously known as gasteropods, or univalves, and more commonly known as Snails D’Arcy Wentworth Thompson's On Growth and Form gives extensive treatment to these spirals. Sir D'Arcy Wentworth Thompson ( May 2, 1860, Edinburgh &ndash June 21, 1948 St 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. The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined In Mathematics, a geometric progression, also known as a geometric sequence, is a Sequence of Numbers where each term after the first is found 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. Nautilus (from Greek ναυτίλος, 'sailor' is the common name of any marine creatures of the Cephalopod family Nautilidae, the sole Ammonites are an extinct group of marine animals of the subclass Ammonoidea in the class Cephalopoda phylum In others it follows a skew path forming a helico-spiral pattern. A helix (pl helixes or helices) from the Greek word έλιξ, is a special kind of Space curve, i

Thompson also studied spirals occurring in horns, teeth, claws and plants. A horn is a pointed projection of the Skin on the head of various Mammals consisting of a covering of horn ( Keratin and other Proteins A claw is a curved pointed appendage found at the end of a toe or finger in most Mammals, Birds, and some Reptiles. Plants are living Organisms belonging to the kingdom Plantae. ^[1]

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. A flower, also known as a bloom or Blossom, is the reproductive structure found in Flowering plants (plants of the division Magnoliophyta, also The sunflower ( Helianthus annuus) is an Annual plant in the family Asteraceae and native to the Americas, with a large flowering This has the form

,

where n is the index number of the floret and c is a constant scaling factor, and is a form of Fermat's spiral. Fermat's spiral (also known as a parabolic Spiral) follows the equation r = \pm\theta^{1/2}\ in Polar coordinates The angle 137. 5° is related to the golden ratio and gives a close packing of florets. In Mathematics and the Arts two quantities are in the Golden ratio if the Ratio between the sum of those quantities and the larger one is the ^[2]

References

1. ^ Thompson, D'Arcy (1917,1942), On Growth and Form 
2. ^ Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). Przemyslaw Prusinkiewicz is a Polish Computer scientist who advanced the idea that Fibonacci numbers in nature can be in part understood as the expression Aristid Lindenmeyer ( November 17, 1925 – October 30, 1989) was an Hungarian Biologist. The Algorithmic Beauty of Plants. The Algorithmic Beauty of Plants is a book by Przemyslaw Prusinkiewicz and Aristid Lindenmayer. Springer-Verlag, 101-107. ISBN 978-0387972978.  

See also

• Seashell surface
• Celtic maze (straight-line spiral)

External links

• A Glimpse of the Spiral as a Symbol for the Transcendental Mystery of God by Paula Vaughan

• SpiralZoom.com, an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination. In mathematics a seashell surface is a surface made by a circle which spirals up the z -axis while decreasing its own radius and distance from the z -axis A Celtic maze is a straight-line Spiral Pattern drawn all over the world beginning in Prehistory and associated with double axes ( labrys