page_title: Fermat's spiral
section_title: The golden ratio and the golden angle
hierarchical_section_title: Fermat's spiral / The golden ratio and the golden angle
caption_attribution_description: English: Illustration of pattern of florets in a sunflower following Vogles model. Made with SingSurf and JavaView, post processed with gimp/inkscape. Author: R Morris
context_page_description: A Fermat's spiral or parabolic spiral is a plane curve named after Pierre de Fermat. Its polar coordinate representation is given by
  which describes a parabola with horizontal axis.
  Fermat's spiral is similar to the Archimedean spiral. But an Archimedean spiral has always the same distance between neighboring arcs, which is not true for Fermat's spiral.
  Like other spirals Fermat's spiral is used for curvature continuous blending of curves.
context_section_description: In disc phyllotaxis, as in the sunflower and 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 is
  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 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers.
  The resulting spiral pattern of unit disks should be distinguished from the Doyle spirals, patterns formed by tangent disks of geometrically increasing radii placed on logarithmic spirals.
caption: The pattern of florets produced by Vogel's model (central image). The other two images show the patterns for slightly different values of the angle.
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