Fermat's spiral

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

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

in polar coordinates (the more general Fermat's spiral follows "r" ² = "a" ²"θ".) 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—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

:$r\; =\; c\; sqrt\{n\},$

:$heta\; =\; n\; imes\; 137.508^circ,$

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
authorlink =Przemyslaw Prusinkiewicz
coauthors =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 ]

References

jordan ratliff

References

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