annulus

An annulus is a ring-shaped region or object. The term is used in geometry, complex analysis, and applied sciences to describe the space between two concentric circles with the same center. In its simplest planar form, it is specified by inner radius r and outer radius R with 0 < r < R.

Its area equals π(R^2 − r^2); the boundary consists of two circles with circumferences 2πR and 2πr. As a topological space, an annulus is homeomorphic to S^1 × [0,1] and has fundamental group isomorphic to the integers, reflecting a single hole. In complex analysis, annuli are domains {z : r < |z| < R}, important for Laurent series, which converge in annuli but not in disks.

In engineering and manufacturing, an annulus describes the hollow region between two coaxial cylinders, such as a pipe, sleeve, or shaft, and is a standard consideration in tolerances, seals, and heat exchangers.

In biology and anatomy, annulus denotes ring-like structures. Examples include the annulus fibrosus of intervertebral discs, a tough ring around the disc, and the annulus on certain mushrooms—the ring on the stalk formed by the partial veil.

Etymology and usage: the word derives from Latin annulus, meaning a small ring, and is used across disciplines to denote any ring-shaped structure or region.