Spiral 61563 225818650 2008-07-15T15:28:44Z MattieTK 434929 Reverted edits by [[Special:Contributions/Time for action|Time for action]] to last version by Pbroks13 (using [[WP:HG|Huggle]]) {{Refimprove|date=July 2007}} {{dablink|For other senses of this word, see [[spiral (disambiguation)]].}} [[Image:NautilusCutawayLogarithmicSpiral.jpg|right|thumb|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 or helix== [[Image:TORUSJ Schraube und archimedische Spirale.PNG|right|thumb|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 spiral is typically a [[Plane (mathematics)|planar]] curve (that is, flat), like the groove on a [[Gramophone record|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 many instances where in [[colloquial]] usage spiral is used as a synonym for helix, notably [[Stairway#Spiral and helical stairs|spiral staircase]] and [[Bookbinding#Modern commercial binding|spiral binding]] of books. 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. 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 function|continuous]] [[monotonic]] function of angle θ. The circle would be regarded as a [[degenerate (mathematics)|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''θ * The [[Cornu spiral]] or ''clothoid'' * [[Fermat's spiral]]: ''r'' = θ<sup>1/2</sup> * The [[hyperbolic spiral]]: ''r'' = ''a''/θ * The [[lituus]]: ''r'' = θ<sup>-1/2</sup> * The [[logarithmic spiral]]: ''r'' = ''ab''<sup>θ</sup>; approximations of this are found in nature * The [[Fibonacci spiral]] and [[golden spiral]]: special cases of the logarithmic spiral. <gallery> Image:Archimedean spiral.svg|Archimedean spiral Image:Cornu Spiral.svg|Cornu spira Image:Fermat's spiral.svg|Fermat's spiral Image:Hyperspiral.svg|hyperbolic spiral Image:Lituus.svg|lituus Image:Logarithmic spiral.svg|logarithmic spiral </gallery> ==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|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 (geometry)|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]] ===Spherical spiral=== [[Image:KUGSPI-9_Loxodrome.gif|thumb|left|[[Rhumb line]]]] [[Image:KUGSPI-5 Archimedische Kugelspirale.gif|thumb|right|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 (navigation)|bearing]]. The curve has an [[infinite]] number of [[Orbital revolution|revolution]]s, 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== [[Image:Newgrange Entrance Stone.jpg|thumb|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. 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)|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. ==In nature== [[Image:Haeckel Prosobranchia.jpg|thumb|300px|right|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 [[Animal shell|shells]] form a [[logarithmic spiral]]. [[Jan Swammerdam]] observed the common mathematical characteristics of a wide range of shells from ''[[Helix (genus)|Helix]]'' to ''[[Spirula]]'' and [[Henry Nottidge Moseley]] described the mathematics of [[univalve]] [[Animal shell|shells]]. [[D'Arcy Wentworth Thompson|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 [[ammonite]]s 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|helico]]-spiral pattern. Thompson also studied spirals occurring in [[Horn (anatomy)|horn]]s, [[teeth]], [[claw]]s and [[plant]]s. <ref>{{Citation|first=D'Arcy|last=Thompson|title=On Growth and Form|date=1917,1942}}</ref> Spirals in plants and animals are frequently described as [[whorl]]s. A model for the pattern of [[floret]]s in the head of a [[sunflower]] was proposed by H Vogel. This has the form :<math>\theta = n \times 137.5^{\circ}</math>, <math>r = c \sqrt{n}</math> 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.<ref>{{cite book | last =Prusinkiewicz | first =Przemyslaw | authorlink =Przemyslaw Prusinkiewicz | coauthors =[[Aristid Lindenmayer|Lindenmayer, Aristid]] | 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 }}</ref> ==References== <references/> ==See also== *[[Seashell surface]] *[[Celtic maze]] (straight-line spiral) ==External links== *[http://onewomansmind.net/center/spirals.html ''A Glimpse of the Spiral as a Symbol for the Transcendental Mystery of God'' by Paula Vaughan] * [http://SpiralZoom.com ''SpiralZoom.com''], an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination. {{commons|Category:Spirals|{{PAGENAME}}}} [[Category:Spirals| ]] <!-- interwiki --> [[bs:Spirala]] [[bg:Спирала]] [[cs:Spirála]] [[da:Spiral]] [[de:Spirale]] [[es:Espiral]] [[eo:Spiralo]] [[fr:Spirale]] [[hr:Spirala]] [[it:Spirale]] [[he:ספירלה]] [[hu:Spirál]] [[nl:Spiraal]] [[ja:螺旋]] [[no:Spiral]] [[pt:Espiral]] [[ru:Спираль]] [[sk:Špirála (matematika)]] [[sl:Spirala]] [[fi:Spiraali]] [[sv:Spiral]] [[zh:螺线]]