Spirals

Spirals are curves that wind around a fixed point, called the origin, with the distance to the origin changing as the polar angle increases. In mathematics, spirals are studied as families of curves defined by a relationship between the radius r and the angle theta. Different spirals differ in how r grows (or shrinks) with theta, producing distinctive shapes and spacing between successive turns.

The Archimedean spiral is defined by the equation r = a + b theta. It has evenly spaced turns: the radial distance between consecutive arms is constant, equal to 2πb. This simple form is often used in modeling and engineering due to its uniform spacing.

The logarithmic spiral, given by r = a e^{b theta}, has the property that the angle between the radius vector and the tangent is constant. This self-similarity means the shape looks the same at different scales, and it occurs frequently in nature, notably in nautilus shells, some galaxies, and various biological patterns.

Fermat’s spiral is described by r^2 = c theta (or r = sqrt(c theta)). The spacing between turns increases with radius, giving a different curvature profile than the Archimedean form. The hyperbolic spiral, defined by r theta = a (equivalently r = a / theta), winds toward the origin as theta grows, with decreasing spacing between turns.

The Lituus spiral, defined by r^2 theta = c, presents another winding pattern where radius decreases rapidly with angle. These families illustrate the diversity of spiral shapes encountered in mathematics.

Spirals also appear in nature and science, including shell shapes, phyllotaxis in plants, and spiral galaxies. They are studied in geometry, calculus, and dynamical systems, emphasizing their equations, curvature, and symmetries.