Fermat's spiral

Fermat's spiral

Fermat's spiral (also known as a parabolic spiral) follows the equation

:$r = pm heta^\left\{1/2\right\},$

in polar coordinates (the more general Fermat's spiral follows "r" 2 = "a" 2"&theta;".) It is a type of Archimedean spiral. [mathworld|urlname=FermatsSpiral |title=Fermat Spiral]

In disc phyllotaxis (sunflower, daisy), the mesh of spirals occurs in Fibonacci numbers because divergence (angle of succession in a single spiral arrangement) approaches the golden ratio. The shape of the spirals depends on the growth of the elements generated sequentially. In mature-disc phyllotaxis, when all the elements are the same size, the shape of the spirals is that of Fermat spirals&mdash;ideally. That is because Fermat's spiral traverses equal annuli in equal turns. The full model proposed by H Vogel in 1979 [Citation
last =Vogel
first =H
title =A better way to construct the sunflower head
journal =Mathematical Biosciences
issue =44
pages =179–189
year =1979
] is

:$r = c sqrt\left\{n\right\},$

:$heta = n imes 137.508^circ,$

where "θ" is the angle, "r" is the radius or distance from the center, and "n" is the index number of the floret and "c" is a constant scaling factor. The angle 137.5° is the golden angle which is approximated by ratios of Fibonacci numbers. [cite book
last =Prusinkiewicz
first =Przemyslaw
coauthors =Lindenmayer, Aristid
title =The Algorithmic Beauty of Plants
publisher =Springer-Verlag
date =1990
location =
pages =101&ndash;107
url =http://algorithmicbotany.org/papers/#webdocs
doi =
id = ISBN 978-0387972978
]

References

jordan ratliff

References

*

Wikimedia Foundation. 2010.

Look at other dictionaries:

• Spiral — In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.piral or helixA spiral and a helix are two terms that are easily confused, but represent different objects …   Wikipedia

• Fermat, Pierre de — born Aug. 17, 1601, Beaumont de Lomagne, France died Jan. 12, 1665, Castres French mathematician. Of Basque origin, Fermat studied law at Toulouse and developed interests in foreign languages, Classical literature, ancient science, and… …   Universalium

• Archimedean spiral — The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point… …   Wikipedia

• Pierre de Fermat — Born August 17, 1601( …   Wikipedia

• List of curves — This is a list of curves, by Wikipedia page. See also list of curve topics, list of surfaces, Riemann surface. Algebraic curves*Cubic plane curve *Quartic plane curve *Quintic plane curve *Sextic plane curveRational curves*Ampersand curve… …   Wikipedia

• List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… …   Wikipedia

• Phyllotaxis — Crisscrossing spirals found in nature In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem (from Ancient Greek phýllon leaf and táxis arrangement ).[1] …   Wikipedia

• Fibonacci number — A tiling with squares whose sides are successive Fibonacci numbers in length …   Wikipedia

• Sunflower — For other uses, see Sunflower (disambiguation). Helianthus annuus Sunflower Scientific classification Kingdom: Plantae …   Wikipedia

• Golden angle — In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden section; that is, into two arcs such that the ratio of the length of the larger arc to the smaller is the… …   Wikipedia