Fermat's spiral

Fermat's spiral: a>0, one branch $$r=+a\sqrt{\phi }$$

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, modeling plant growth and the shapes of certain spiral galaxies, and the design of variable capacitors, solar power reflector arrays, and cyclotrons.

Coordinate representation

Polar

The representation of the Fermat spiral in polar coordinates Template:Math is given by the equation $$r=\pm a\sqrt{\phi }$$ for Template:Math.

The parameter $a$ is a scaling factor affecting the size of the spiral but not its shape.

The two choices of sign give the two branches of the spiral, which meet smoothly at the origin. If the same variables were reinterpreted as Cartesian coordinates, this would be the equation of a parabola with horizontal axis, which again has two branches above and below the axis, meeting at the origin.

Cartesian

The Fermat spiral with polar equation $$r=\pm a\sqrt{\phi }$$ can be converted to the Cartesian coordinates Template:Math by using the standard conversion formulas Template:Math and Template:Math. Using the polar equation for the spiral to eliminate Template:Mvar from these conversions produces parametric equations for one branch of the curve:

$\{\begin{array}{c}x(\phi )=+a\sqrt{\phi }\cos (\phi )\\ y(\phi )=+a\sqrt{\phi }\sin (\phi )\end{array}$

and the second one

$\{\begin{array}{c}x(\phi )=-a\sqrt{\phi }\cos (\phi )\\ y(\phi )=-a\sqrt{\phi }\sin (\phi )\end{array}$

They generate the points of branches of the curve as the parameter Template:Mvar ranges over the positive real numbers.

For any Template:Math generated in this way, dividing Template:Mvar by Template:Mvar cancels the Template:Math parts of the parametric equations, leaving the simpler equation Template:Math. From this equation, substituting Template:Mvar by Template:Math (a rearranged form of the polar equation for the spiral) and then substituting Template:Mvar by Template:Math (the conversion from Cartesian to polar) leaves an equation for the Fermat spiral in terms of only Template:Mvar and Template:Mvar: $$\frac{x}{y}=\cot \left(\frac{x^2+y^2}{a^2}\right).$$ Because the sign of Template:Mvar is lost when it is squared, this equation covers both branches of the curve.

Geometric properties

A Fermat's spiral divides the plane into two connected and congruent regions (diagram: black and white)

Division of the plane

A complete Fermat's spiral (both branches) is a smooth double point free curve, in contrast with the Archimedean and hyperbolic spiral. Like a line or circle or parabola, it divides the plane into two connected regions.

Definition of sector (light blue) and polar slope angle Template:Mvar

Polar slope

From vector calculus in polar coordinates one gets the formula

$\tan \alpha =\frac{r^{\prime }}{r}$

for the polar slope and its angle Template:Mvar between the tangent of a curve and the corresponding polar circle (see diagram).

For Fermat's spiral Template:Math one gets

$\tan \alpha =\frac{1}{2\phi }.$

Hence the slope angle is monotonely decreasing.

Curvature

From the formula

$\kappa =\frac{r^2+2(r^{\prime }{)}^2-r{r}^{″}}{{(r^2+(r^{\prime }{)}^2)}^{\frac{3}{2}}}$

for the curvature of a curve with polar equation Template:Math and its derivatives

$\begin{array}{cc}{r}^{\prime }& =\frac{a}{2\sqrt{\phi }}=\frac{a^2}{2r}\\ r^{″}& =-\frac{a}{4{\sqrt{\phi }}^3}=-\frac{a^4}{4r^3}\end{array}$

one gets the curvature of a Fermat's spiral: $$\kappa (r)=\frac{2r(4r^4+3a^4)}{{(4r^4+a^4)}^{\frac{3}{2}}}.$$

At the origin the curvature is 0. Hence the complete curve has at the origin an inflection point and the Template:Mvar-axis is its tangent there.

Area between arcs

The area of a sector of Fermat's spiral between two points Template:Math and Template:Math is

$\begin{array}{cc}\underset{\_}{A}& =\frac{1}{2}\underset{{\phi }_1}{\overset{{\phi }_2}{\int }}r(\phi {)}^{2}d\phi \\ & =\frac{1}{2}\underset{{\phi }_1}{\overset{{\phi }_2}{\int }}{a}^2\phi d\phi \\ & =\frac{a^2}{4}({\phi }_2^2-{\phi }_1^2)\\ & =\frac{a^2}{4}({\phi }_2+{\phi }_1)({\phi }_2-{\phi }_1).\end{array}$

Fermat's spiral: area between neighbored arcs

After raising both angles by Template:Math one gets

$\bar{A}=\frac{a^2}{4}({\phi }_2+{\phi }_1+4\pi )({\phi }_2-{\phi }_1)=\underset{\_}{A}+a^2\pi ({\phi }_2-{\phi }_1).$

Hence the area Template:Mvar of the region between two neighboring arcs is $$A=a^2\pi ({\phi }_2-{\phi }_1).$$ Template:Mvar only depends on the difference of the two angles, not on the angles themselves.

For the example shown in the diagram, all neighboring stripes have the same area: Template:Math.

This property is used in electrical engineering for the construction of variable capacitors.

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

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

Arc length

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

$$\begin{array}{cc}L& =\underset{{\phi }_1}{\overset{{\phi }_2}{\int }}\sqrt{{(r^{\prime }(\phi ))}^2+r^2(\phi )}d\phi \\ & =\frac{a}{2}\underset{{\phi }_1}{\overset{{\phi }_2}{\int }}\sqrt{\frac{1}{\phi }+4\phi }d\phi .\end{array}$$

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 Template:Math and the incomplete beta function Template:Math:

$$\begin{array}{cc}L& =a\cdot \sqrt{\phi }\cdot {}_{\mathrm{2}}{\mathrm{F}}_{\mathrm{1}}(-\frac{1}{2},\frac{1}{4};\frac{5}{4};-4\cdot {\phi }^2)\\ & =a\cdot \frac{1-i}{8}\cdot \mathrm{B}(-4\cdot {\phi }^2;\frac{1}{4},\frac{3}{2})\\ \end{array}$$

The inversion of Fermat's spiral (green) is a lituus (blue)

Circle inversion

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

• The image of Fermat's spiral Template:Math under the inversion at the unit circle is a lituus spiral with polar equation $$r=\frac{1}{a\sqrt{\phi }}.$$ When Template:Math, both curves intersect at a fixed point on the unit circle.
• The tangent (Template:Mvar-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

$$\begin{array}{cc}r& =c\sqrt{n},\\ \theta & =n\times 137.508^{\circ },\end{array}$$

where Template:Mvar is the angle, Template:Mvar is the radius or distance from the center, and Template:Mvar is the index number of the floret and Template:Mvar is a constant scaling factor. The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers. Template:Wide image

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. Template:Clear right

See also

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

Further reading

• Template:Cite book

External links

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