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The torus is a doughnut-shaped surface that appears frequently in geometry and topology. In its standard geometric construction, it is formed by revolving a circle of radius r about an axis in the same plane at a distance R from the circle’s center, with R > r. The result is a compact, orientable surface of genus 1, which is topologically equivalent to the cartesian product S^1 × S^1.

A common coordinate description uses two angles u and v:

x = (R + r cos v) cos u

y = (R + r cos v) sin u

z = r sin v

with 0 ≤ u < 2π and 0 ≤ v < 2π. Equivalently, it satisfies the implicit equation (√(x^2 + y^2) − R)^2 + z^2 = r^2. The torus has a smooth embedding in three-dimensional Euclidean space, and its surface is orientable with varying Gaussian curvature across its surface.

Topologically, the torus is the product S^1 × S^1, has Euler characteristic 0, and has a fundamental group isomorphic to Z × Z. A related object is the solid torus, the three-dimensional region bounded by the torus surface; its boundary is the torus itself.

Variants and notable examples include the flat torus, obtained as the quotient of the Euclidean plane by a lattice (R^2 / Λ), which provides a canonical flat metric and plays a central role in complex analysis as a basic Riemann surface. In the context of the three-sphere S^3, the Clifford torus is a standard flat torus embedded in S^3 and is a minimal surface.

Applications span mathematics, physics, computer graphics, and architecture, where toroidal shapes model periodic structures, magnetic confinement devices, and aesthetic or functional designs.