FermatSpirale

FermatSpirale, commonly referred to as Fermat's spiral, is a plane curve named after Pierre de Fermat. In polar coordinates, it is defined by r = a sqrt(theta) for theta ≥ 0, equivalently r^2 = a^2 theta, where a > 0 is a scaling constant. The curve begins at the origin and winds outward as theta increases.

The Fermat spiral is distinct from the Archimedean spiral r = a theta and from the logarithmic spiral r = b e^{k theta}. In a Fermat spiral, radial growth grows with the square root of the angle, so the curve expands gradually and the spacing between turns changes with theta. The derivative dr/dtheta = a/(2 sqrt(theta)) describes how the radius changes with angle.

Discrete constructions based on Fermat's spiral are used to model and analyze phyllotaxis in plants, where seed positions in sunflowers and pinecones follow the pattern r_n = c sqrt(n) with angular steps around the origin. The arrangement often uses nearly constant angular increments, such as the golden angle, to achieve near-uniform distribution of points along the spiral. Beyond biology, Fermat's spiral appears in computer graphics, optics, and design as a simple, smooth curve with aesthetically pleasing winding.

The term FermatSpirale is the stylized variant of the standard Fermat's spiral and is used in some texts to reflect its French lineage or to distinguish a specific presentation of the curve. It remains a standard example of a planar spiral in mathematics.