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| {{Hatnote|For other uses of this word, see [[spiral (disambiguation)]].}}
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| {{Refimprove|date=July 2007}}
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| [[Image:NautilusCutawayLogarithmicSpiral.jpg|right|thumb|Cutaway of a [[nautilus]] shell showing the chambers arranged in an approximately [[logarithmic spiral]]]]
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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.
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| == Spirals and helices {{anchor|Spiral or helix}} ==
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| [[Image:Schraube und archimedische Spirale.png|right|thumb|An Archimedean spiral, a helix, and a conic spiral]]
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| Two major definitions of "spiral" in a respected American dictionary are: <ref name=free>"[http://www.thefreedictionary.com/spiral Spiral]", ''The [[American Heritage Dictionary]] of the English Language'', Houghton Mifflin Company, Fourth Edition, 2009.</ref>
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| : '''a.''' A curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.<br />'''b.''' A three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a [[helix]].
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| Definition '''a''' describes a [[Plane (mathematics)|planar]] curve, that extends in both of the perpendicular directions within its plane; the groove on one side of a [[Gramophone record|record]] closely approximates a plane spiral (and it is by the finite width and depth of the groove, but ''not'' by the wider spacing between than within tracks, that it falls short of being a perfect example); note that successive loops ''differ'' in diameter. In another example, the "center lines" of the arms of a [[spiral galaxy]] trace [[logarithmic spiral]]s.
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| Definition '''b''' includes two kinds of 3-dimensional relatives of spirals:
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| * A conical or [[volute spring]] (including 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 is often described as a spiral, or as a conic helix.
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| * Quite explicitly, '''b '''also includes a cylindrical coil spring and a strand of [[DNA]], both of which are quite helical, so that "helix" is a more ''useful'' description than "spiral" for each of them; in general, "spiral" is seldom applied if successive "loops" of a curve have the same diameter.<ref name=free/>
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| In the side picture, the black curve at the bottom is an [[Archimedean spiral]], while the green curve is a helix. The curve shown in red is a conic helix.
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| ==Two-dimensional spirals==
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| A [[two-dimensional]] spiral may be described most easily using [[polar coordinates]], where the [[radius]] ''r'' is a [[monotonic]] [[continuous function]] of angle ''θ''. The circle would be regarded as a [[degenerate (mathematics)|degenerate]] case (the [[function(mathematics)|function]] not being strictly monotonic, but rather [[Constant (mathematics)|constant]]).
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| Some of the more important sorts of two-dimensional spirals include:
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| * The [[Archimedean spiral]]: <math>r=a+b \cdot \theta </math> (see also:[[Involute]])
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| * The [[Euler spiral]], [[Cornu spiral]] or ''clothoid''
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| * [[Fermat's spiral]]: <math>r= \theta^{1/2}</math>
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| * The [[hyperbolic spiral]]: <math>r = a/ \theta</math>
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| * The [[Lituus (mathematics)|lituus]]: <math>r = \theta^{-1/2}</math>
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| * The [[logarithmic spiral]]: <math>r=a\cdot e^{b\theta}</math>; approximations of this are found in nature
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| * The [[Fibonacci spiral]] and [[golden spiral]]: special cases of the logarithmic spiral
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| * The [[Spiral of Theodorus]]: an approximation of the Archimedean spiral composed of contiguous right triangles
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| <gallery>
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| Image:Archimedean spiral.svg|Archimedean spiral
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| Image:Cornu Spiral.svg|Cornu spiral
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| Image:Fermat's spiral.svg|Fermat's spiral
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| Image:Hyperspiral.svg|hyperbolic spiral
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| Image:Lituus.svg|lituus
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| Image:Logarithmic Spiral Pylab.svg|logarithmic spiral
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| Image:Spiral of Theodorus.svg|spiral of Theodorus
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| </gallery> | |
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| ==Three-dimensional spirals==
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| 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''.
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| The helix and [[vortex]] can be viewed as a kind of [[dimension|three-dimensional]] spiral.
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| For a helix with thickness, see [[spring (math)]].
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| ===Spherical spiral===
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| [[Image:KUGSPI-9 Loxodrome.gif|thumb|left|[[Rhumb line]]]]
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| [[Image:KUGSPI-5 Archimedische Kugelspirale.gif|thumb|right|Archimedean Spherical Spiral]]
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| 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 [[Infinity|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 radius changes and hence is not a rhumb line.
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| ==In nature==
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| [[File:Helianthus whorl.jpg|thumb|right|Sunflower head displaying florets in spirals of 34 and 55 around the outside]]
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| The study of spirals in [[nature]] has 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]] 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 perpendicular to the axis and the shell will form a planar 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>{{Cite journal|first=D'Arcy|last=Thompson|title=On Growth and Form|date=1917,1942|postscript=<!--None-->}}</ref>
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| A model for the pattern of [[floret]]s in the head of a [[sunflower]] was proposed by H Vogel. This has the form
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| :<math>\theta = n \times 137.5^{\circ},\ r = c \sqrt{n}</math>
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| 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 | year =1990 | location = | pages =101–107 | url =http://algorithmicbotany.org/papers/#webdocs | doi = | isbn = 978-0-387-97297-8 }}</ref>
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| Spirals in plants and animals are frequently described as [[whorl (botany)|whorls]]. This is also the name given to spiral shaped [[fingerprint]]s.
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| ==As a symbol==
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| [[Image:Newgrange Entrance Stone.jpg|thumb|The [[Newgrange]] entrance slab]][[File:Phoenix-Deer Valley Rock Art Center- Petroglyph - 1.JPG|thumb|250px|This [[Petroglyph]] with a spiral figure carved into it was made by the [[Hohokam]]s, a [[Native Americans in the United States|Native American]] tribe in the United States, over a 1000 years ago.]]
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| The [[triple spiral]] motif is a [[Neolithic]] symbol in Europe. While it is considered a [[Celts|Celtic]] symbol the triple spiral is in fact a pre-Celtic symbol.<ref>Anthony Murphy and Richard Moore, ''Island of the Setting Sun: In Search of Ireland's Ancient Astronomers,'' 2nd ed., Dublin: The Liffey Press, 2008, pp. 168-169</ref> It is carved into the rock of a stone lozenge near the main entrance of the prehistoric [[Newgrange]] monument in [[County Meath]], [[Republic of Ireland|Ireland]]. Newgrange was built around 3200 BCE predating the Celts and the triple spirals were carved at least 2,500 years before the Celts reached Ireland but has long since been incorporated into Celtic culture.<ref name="knowth.com"/><ref name="knowth.com">{{cite web|url=http://knowth.com/newgrange.htm |title=Newgrange Ireland - Megalithic Passage Tomb - World Heritage Site |publisher=Knowth.com |date=2007-12-21 |accessdate=2013-08-16}}</ref> The [[triskelion]] symbol, consisting of three interlocked spirals or three bent human legs, appears in many early cultures, including [[Mycenaean Greece|Mycenaean]] vessels, on coinage in [[Lycia]], on [[stater]]s of [[Pamphylia]] (at [[Aspendos]], 370–333 BC) and [[Pisidia]], as well as on the [[heraldic]] emblem on warriors' shields depicted on Greek pottery.<ref>For example, the trislele on [[Achilles]]' round shield on an Attic late sixth-century ''[[hydria]]'' at the [[Boston Museum of Fine Arts]], illustrated in John Boardman, Jasper Griffin and Oswyn Murray, ''Greece and the Hellenistic World'' (Oxford History of the Classical World) vol. I (1988), p. 50.</ref>
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| Spirals can be found throughout pre-Columbian art in Latin and Central America. The more than 1,400 [[petroglyphs]] (rock engravings) in [[Las Plazuelas]], [[Guanajuato]] [[Mexico]], dating 750-1200 AD, predominantly depict spirals, dot figures and scale models.<ref>{{cite web| last = | first = | title = Rock Art Of Latin America & The Caribbean | publisher = International Council on Monuments & Sites| date = June 2006| url = http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf| page = 5 | accessdate = 4 January 2014}}</ref> In Colombia monkeys, frog and lizard like figures depicted in petroglyphs or as gold offering figures frequently includes spirals, for example on the palms of hands.<ref>{{cite web| last = | first = | title = Rock Art Of Latin America & The Caribbean | publisher = International Council on Monuments & Sites| date = June 2006| url = http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf| page = 99 | accessdate = 4 January 2014}}</ref> In Lower Central America spirals along with circles, wavy lines, crosses and points are universal petroglyphs characters.<ref>{{cite web| last = | first = | title = Rock Art Of Latin America & The Caribbean | publisher = International Council on Monuments & Sites| date = June 2006| url = http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf| page = 17 | accessdate = 4 January 2014}}</ref> Spirals can also be found among the [[Nazca Lines]] in the coastal desert of Peru, dating from 200 BC to 500 AD. The [[geoglyphs]] number in the thousands and depict animals, plants and geometric motifs, including spirals.<ref>{{cite web| last = Jarus| first = Owen| title = Nazca Lines: Mysterious Geoglyphs in Peru| publisher = LiveScience | date = 14 August 2012| url = http://www.livescience.com/22370-nazca-lines.html| accessdate = 4 January 2014}}</ref>
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| 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).
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| 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. The spiral is also found in structures as small as the [[double helix]] of [[DNA]] and as large as a [[spiral galaxy|galaxy]]. Because of this frequent natural occurrence, the spiral is the official symbol of the [[World Pantheist Movement]].<ref name=WPM />
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| [[File:Panthéisme.JPG|thumb|The [[World Pantheist Movement]]'s spiral-based logo<ref name=WPM>{{cite web|last=Harrison|first=Paul|title=Pantheist Art|url=http://www.pantheism.net/pan/free/pan9.pdf|publisher=World Pantheist Movement|accessdate=7 June 2012}}</ref>]] The spiral is also a symbol of the process of [[dialectic]].
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| ==In art==
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| The spiral has inspired artists throughout the ages. Among the most famous of spiral-inspired art is [[Robert Smithson]]'s [[earthworks (art)|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 [[Anderson-Abruzzo Albuquerque International Balloon Museum|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. It also central in Mario Merz and Andy Goldsworthy's work.
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| ==See also==
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| *[[Celtic maze]] (straight-line spiral)
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| *[[Fibonacci number]]
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| *[[Patterns in nature]]
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| *[[Seashell surface]]
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| *[[Spirangle]]
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| ==References==
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| {{reflist}}
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| == Related publications ==
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| * Cook, T., 1903. ''Spirals in nature and art''. Nature 68 (1761), 296.
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| * Cook, T., 1979. ''The curves of life''. Dover, New York.
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| * Habib, Z., Sakai, M., 2005. ''Spiral transition curves and their applications''. Scientiae Mathematicae Japonicae 61 (2), 195 – 206.
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| * Dimulyo, S., Habib, Z., Sakai, M., 2009. ''Fair cubic transition between two circles with one circle inside or tangent to the other''. Numerical Algorithms 51, 461–476 [http://www.springerlink.com/content/113644325114041q/].
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| * Harary, G., Tal, A., 2011. ''The natural 3D spiral''. Computer Graphics Forum 30 (2), 237 – 246 [http://webee.technion.ac.il/~ayellet/Ps/11-HararyTal.pdf].
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| * Xu, L., Mould, D., 2009. ''Magnetic curves: curvature-controlled aesthetic curves using magnetic fields''. In: Deussen, O., Hall, P. (Eds.), Computational Aesthetics in Graphics, Visualization, and Imaging. The Eurographics Association [http://gigl.scs.carleton.ca/sites/default/files/ling_xu/artn-cae.pdf].
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| * Wang, Y., Zhao, B., Zhang, L., Xu, J., Wang, K., Wang, S., 2004. ''Designing fair curves using monotone curvature pieces''. Computer Aided Geometric Design 21 (5), 515–527 [http://www.sciencedirect.com/science/article/pii/S0167839604000470].
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| * A. Kurnosenko. ''Applying inversion to construct planar, rational spirals that satisfy two-point G2 Hermite data''. Computer Aided Geometric Design, 27(3), 262–280, 2010 [http://www.sciencedirect.com/science/article/pii/S0167839609001423].
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| * A. Kurnosenko. ''Two-point G2 Hermite interpolation with spirals by inversion of hyperbola''. Computer Aided Geometric Design, 27(6), 474–481, 2010.
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| * Miura, K.T., 2006. ''A general equation of aesthetic curves and its self-affinity''. Computer-Aided Design and Applications 3 (1–4), 457–464 [http://ktm11.eng.shizuoka.ac.jp/profile/ktmiura/pdf/KTMiura-CAD06Final.pdf].
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| * Miura, K., Sone, J., Yamashita, A., Kaneko, T., 2005. ''Derivation of a general formula of aesthetic curves''. In: 8th International Conference on Humans and Computers (HC2005). Aizu-Wakamutsu, Japan, pp. 166 – 171 [http://ktm11.eng.shizuoka.ac.jp/profile/ktmiura/pdf/acurveHC0.pdf].
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| * Meek, D., Walton, D., 1989. ''The use of Cornu spirals in drawing planar curves of controlled curvature''. Journal of Computational and Applied Mathematics 25 (1), 69–78 [http://www.sciencedirect.com/science/article/pii/0377042789900769].
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| * Farin, G., 2006. ''Class A Bézier curves''. Computer Aided Geometric Design 23 (7), 573–581 [http://www.sciencedirect.com/science/article/pii/S016783960600032X].
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| * Farouki, R.T., 1997. ''Pythagorean-hodograph quintic transition curves of monotone curvature''. Computer-Aided Design 29 (9), 601–606.
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| * Yoshida, N., Saito, T., 2006. ''Interactive aesthetic curve segments''. The Visual Computer 22 (9), 896–905 [http://www.yoshida-lab.net/aesthetic/ias2006pg.pdf].
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| * Yoshida, N., Saito, T., 2007. ''Quasi-aesthetic curves in rational cubic Bézier forms''. Computer-Aided Design and Applications 4 (9–10), 477–486 [http://www.yoshida-lab.net/aesthetic/cad07yoshida.pdf].
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| * Ziatdinov, R., Yoshida, N., Kim, T., 2012. ''Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions''. Computer Aided Geometric Design 29 (2), 129 – 140 [http://www.sciencedirect.com/science/article/pii/S0167839611001452].
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| * Ziatdinov, R., Yoshida, N., Kim, T., 2012. ''Fitting G2 multispiral transition curve joining two straight lines'', Computer-Aided Design 44(6), 591–596 [http://www.sciencedirect.com/science/article/pii/S001044851200019X].
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| * Ziatdinov, R., 2012. ''Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function''. Computer Aided Geometric Design 29(7): 510–518 [http://www.sciencedirect.com/science/article/pii/S0167839612000325].
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| * Ziatdinov, R., Miura K.T., 2012. ''On the Variety of Planar Spirals and Their Applications in Computer Aided Design''. European Researcher 27(8–2), 1227-–1232 [http://www.erjournal.ru/pdf.html?n=1345307278.pdf].
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| ==External links==
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| {{Commons category|Spirals|Spiral}}
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| * [http://SpiralZoom.com ''SpiralZoom.com''], an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination.
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| *[http://www.mathematische-basteleien.de/spiral.htm Spirals by Jürgen Köller]
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| * [http://eol.org/collections/10796 Spirals] – an Encyclopedia of Life collection with examples of spirals in nature.
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| *[http://oeis.org/A202407 Archimedes' spiral transforms into Galileo's spiral. Mikhail Gaichenkov, OEIS]
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| [[Category:Spirals]]
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Like all the opposite world leaders in the tire trade, Goodyear has been reeling from the effects of overcapacity and low worldwide demand for tires. For the first six months of this yr, Goodyear reported a internet lack of $sixty seven.6 million, in contrast with internet earnings of $eleven.5 million within the comparable interval in 1990. Tires are the third largest expense for fleet managers, use Yokohama's Inflation Pressure Calculator and Gas Financial savings Calculator to see how you can optimize your fleet's efficiency. BEGIN NOW Photographs, specifications, features and benefits for our full line of sunshine truck business and medium truck tires. OBTAIN PDF Take away expensive hubcaps prior to taking the vehicle to restore shop to ensure they are not damaged. Free tire stress check
of only 2 tire recycling permits within the state of MS and Free delivery offer doesn't apply to tire and wheel purchases. Free Lifetime Rotation with four Tire Purchase Free Tire Air Strain and Put on Test Tire Gas Efficiency Client Information Program If you want to buy items on sale, at a store you should hold onto them for somewhat bit of time, generally even months. Seasonal items like winter coats can be bought at a 50% low cost in early spring, when winter rolls around you'll be able to promote them at close to full worth. It's a must to store them safely for a few months, however when you've got the time it is a good deal. Aug 14 Tonneau cowl (ARE) $350 (vallejo / benicia) pic auto elements - by proprietor Aug 14 CONSIDERED ONE OF A SORT HOT ROD CLOCK $200 (san jose downtown) pic map auto parts - by owner
One factor you must do for getting money off at Tire Discounters must be to hunt for coupons and other deals. There are many places the place you possibly can search out coupons. For probably the most half local papers and publications have a coupons segment, and which means you actually ought to look by means of these commonly to see what kind of coupons and particular deals that they and other businesses at the moment have in the mean time. A lot of cities and cities also have an area coupon publication that you could both buy or is given away for nothing. You might discover that Tire Discounters place coupons as well as other affords into these books.
Youngster needs to be carry out the Boat Trailer Tires search accordance with all of your desires make use of the web to perform the idea, this can drastically help you in comparing the auto tires with distinct manufacturers and select the the auto tires that will simplest suits to your desires. Now, quite a few the auto tires producers that have the endorsed internet page on the net, there gave the impression to be in addition quite a lot of them as well as have an internet shop to advertise their particular merchandise and solutions. That is certainly obviously very helpful due to the fact throughout these websites it is attainable to easily receive the data as well as whole descriptions to the Trailer Tires Concerning the Writer
Schwab was great in explaining and allowed me to observe the Highway Power balancing. I took notes and gave a replica to Costco Customer Service - each tire was mounted incorrectly, had flawed quantity of weights to compensate for improper mounting. What I mean is that the tire producer puts a yellow dot on the sidewall - that is advisable to line up with the valve stem. Costco had the yellow dots opposite the valve stem, then over- weighted to compensate.
The primary three digits (230) symbolize the width of the tire measured in millimeters (mm), the following two digits (forty) symbolize the peak to width ratio often called the facet ratio, the next alphabet denotes the type of tire (right here R stands for radial tire), the subsequent two digits (15) characterize the rim diameter measured in inches. The following two digits (95) stand for the load rating that signifies how much load the tire is designed to hold, and the alphabet (U) denotes the velocity ranking of the tire. Word that the velocity ranking is represented by the alphabets A to Z, with Z denoting the best and A denoting the lowest worth. The final alphabet(s) (M+S) indicates the type of tire, which in this case is mud (M) and snow (S).
Nevertheless, when you plan to promote your previous motorbike, there are issues you should do in an effort to guarantee a great sale. What are these things? The company stated the tires, which the corporate has examined for 4 years, are getting used in the area by about 20 small freight trucking companies. Goodyear said it expected the tires to be widely available on the market and distribution by the first quarter of subsequent year. If the 1963 Chevrolet Biscayne available as traditional vehicles for sale in South Africa was not modified, that is what you possibly can count on. Would you take into account a 1963 Chevrolet Biscayne as classic automobiles for sale in South Africa ? Industrial Kings gives a large variety of commercial cars Melbourne including all kinds of forklifts on the market Find out how to Compare Prices on Tires
If you're talking tires, shoppers usually stand to lose a lot of money. You want to drive safely, however do not need to break the bank just to place a new set of tires on your automotive. To keep you informed about how the tire enterprise works, we talked to a tire industry professional. This insider's account will help information you thru this essential automotive transaction. When the tire man has your automobile up on the rack, it's an ideal time for them to sell you a wheel alignment, brake job or shocks. Alignment is essential, but your tires will tell you when you need an alignment as a result of they may put on erratically (and the car may "pull" to at least one aspect or one other, too). The best way to Change a Tire Rim on a Automotive Methods to Seal a Tire on a Rim Information on Cooper Tire Plants Completely different Rear Tire Measurement
The customers, who want to improve their tires or simply give a brand new look to their vehicles or vans promote them. Therefore, you get prime-notch used tires at a much-lowered price Typically, you simply cannot differentiate between the new tires and the used tires sold by us. Most web tire shops shouldn't have a bodily location so you don't pay any gross sales taxes. Based on several on-line tire retailers, the typical worth for a set of new automotive tires is roughly $375. So not having to pay sales tax on $375 will save you a lot of money. Ask a truck owner and you'll come to know the way a lot thought and analysis goes into buying mild truck tires. These are those that make for a smooth trip. read extra Dick Cepek Tires Save 5% Immediately on All Tires DIY Cheap Basement