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.
In Cartesian coordinates
Fermat's spiral with polar equation
can be described in Cartesian coordinates (x = r cos φ, y = r sin φ) by the parametric representation
From the parametric representation and φ = r2/a2, r = √x2 + y2 one gets a representation by an equation:
A Fermat's spiral divides the plane into two connected 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. It divides the plane (like a line or circle or parabola) into two connected regions. But this division is less obvious than the division by a line or circle or parabola. It is not obvious to which side a chosen point belongs.
Definition of sector (light blue) and polar slope angle α
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 φ1 = 0, φ2 = 2π; then the area of the black region (see diagram) is A0 = a2π2, which is half of the area of the circle K0 with radius r(2π). The regions between neighboring curves (white, blue, yellow) have the same area A = 2a2π2. Hence:
The area between two arcs of the spiral after a full turn equals the area of the circle K0.
The length of the arc of Fermat's spiral between two points (r(φ1), φ1) can be calculated by the integral:
The image of Fermat's spiral r = a√φ under the inversion at the unit circle is a lituus spiral with polar equation
When φ = 1/a2, both curves intersect at a fixed point on the unit circle.
The tangent (x-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
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 pattern of florets produced by Vogel's model (central image). The other two images show the patterns for slightly different values of the angle.
^ abAnastasios M. Lekkas, Andreas R. Dahl, Morten Breivik, Thor I. Fossen: "Continuous-Curvature Path Generation Using Fermat's Spiral". In: Modeling, Identification and Control. Vol. 34, No. 4, 2013, pp. 183–198, ISSN 1890-1328.
^Fritz Wicke: Einführung in die höhere Mathematik. Springer-Verlag, 2013, ISBN 978-3-662-36804-6, p. 414.
^Lettre de Fermat à Mersenne du 3 juin 1636, dans Paul Tannery. In: Oeuvres de Fermat. T. III, S. 277, Lire en ligne.
^Vogel, H (1979). "A better way to construct the sunflower head". Mathematical Biosciences. 44 (44): 179–189. doi:10.1016/0025-5564(79)90080-4.
^Noone, Corey J.; Torrilhon, Manuel; Mitsos, Alexander (December 2011). "Heliostat Field Optimization: A New Computationally Efficient Model and Biomimetic Layout". Solar Energy. doi:10.1016/j.solener.2011.12.007.
J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 31, 186. ISBN 0-486-60288-5.