Toruses

A torus, plural toruses or tori, is a doughnut-shaped surface generated by revolving a circle of radius r about an axis in the same plane that does not intersect the circle. In everyday language, it is the familiar donut shape.

In three-dimensional space, the standard embedded torus has major radius R and minor radius r, with R > r. It can be parameterized by x = (R + r cos v) cos u, y = (R + r cos v) sin u, z = r sin v, where u and v range from 0 to 2π. The surface is the Cartesian product of two circles S^1 × S^1.

Topologically, the torus is a compact, orientable surface of genus 1; its Euler characteristic is 0. Its fundamental group is isomorphic to Z × Z, reflecting two independent loops around the central hole and around the tube. Its homology groups are H0 = Z, H1 = Z^2, H2 = Z.

A solid torus is a three-dimensional region obtained by revolving a disk around an axis, equivalent to S^1 × D^2. It has a boundary that is a surface homeomorphic to a torus.

The Gaussian curvature of an embedded torus varies over the surface, with positive curvature on the outer regions and negative curvature on the inner regions. A flat torus, obtained as a quotient of the plane by a lattice, carries a flat metric and has zero curvature everywhere.

Tori appear in geometry, topology, physics, and computer graphics, and are used as examples of manifolds, in torus knot theory, and in the design of objects and architectural details.