thumb|Fermat's spiral: a>0, one branch <math display="block">r=+a\sqrt{\varphi}</math>

thumb|Fermat's spiral, both branches

A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance from the spiral center, contrasting with the Archimedean spiral (for which this distance is invariant) and the logarithmic spiral (for which the distance between turns is proportional to the distance from the center). Fermat spirals are named after Pierre de Fermat.

Their applications include curvature continuous blending of curves,

thumb|the regions in between (white, blue, yellow) have all the same area, which is equal to the area of the drawn circle.

Special case due to Fermat

In 1636, Fermat wrote a letter to Marin Mersenne which contains the following special case:

Let ; then the area of the black region (see diagram) is , which is half of the area of the circle with radius . The regions between neighboring curves (white, blue, yellow) have the same area . Hence:

  • The area between two arcs of the spiral after a full turn equals the area of the circle .

Arc length

The length of the arc of Fermat's spiral between two points can be calculated by the integral:

<math display="block">\begin{align}

L&=\int_{\varphi_1}^{\varphi_2}\sqrt{\left(r^\prime(\varphi)\right)^2+r^2(\varphi)}\,d\varphi \\

&=\frac{a}{2}\int_{\varphi_1}^{\varphi_2}\sqrt{\frac{1}{\varphi}+4\varphi}\,d\varphi .

\end{align}</math>

This integral leads to an elliptical integral, which can be solved numerically.

The arc length of the positive branch of the Fermat's spiral from the origin can also be defined by hypergeometric functions and the incomplete beta function :

<math display="block">\begin{align}

L &= a \cdot \sqrt{\varphi} \cdot \operatorname{_{2}F_{1\left( -\tfrac12,\, \tfrac14;\, \tfrac54;\, -4 \cdot \varphi^{2} \right)\\

&= a \cdot\frac{1 - i}{8} \cdot \operatorname{B}\left( -4 \cdot \varphi^{2};\, \tfrac14,\, \tfrac32 \right)\\

\end{align}</math>

thumb|The inversion of Fermat's spiral (green) is a [[lituus (mathematics)|lituus (blue)]]

Circle inversion

The inversion at the unit circle has in polar coordinates the simple description .

  • The image of Fermat's spiral under the inversion at the unit circle is a lituus spiral with polar equation <math display="block">r=\frac{1}{a\sqrt{\varphi.</math> When , both curves intersect at a fixed point on the unit circle.
  • The tangent (-axis) at the inflection point (origin) of Fermat's spiral is mapped onto itself and is the asymptotic line of the lituus spiral.

The golden ratio and the golden angle

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

<math display="block">\begin{align}

r &= c \sqrt{n},\\

\theta &= n \times 137.508^\circ,

\end{align}</math>

where is the angle, is the radius or distance from the center, and is the index number of the floret and 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.

Solar plants

Fermat's spiral has also been found to be an efficient layout for the mirrors of concentrated solar power plants.

See also

  • List of spirals
  • Patterns in nature
  • Spiral of Theodorus

References

Further reading

  • Online exploration using JSXGraph (JavaScript)
  • Fermat's Natural Spirals, in sciencenews.org