## ::Torus

### ::concepts

**Torus**::''n'' Group::space Surface::circle ''R''::which Product::**torus** Topology::mathbf

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In geometry, a **torus** (plural **tori**) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a *torus of revolution".*

Real-world examples of (approximately) toroidal objects include inner tubes, swim rings, and the surface of a doughnut or bagel.

A torus should not be confused with a solid torus, which is formed by rotating a disk, rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. Real-world approximations include doughnuts, vadai or vada, many lifebuoys, and O-rings.

In topology, a ring torus is homeomorphic to the Cartesian product of two circles: *S*^{1} × *S*^{1}, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into three-dimensional Euclidean space, but another way to do this is the Cartesian product of the embedding of *S*^{1} in the plane. This produces a geometric object called the Clifford torus, a surface in 4-space.

In the field of topology, a torus is any topological space that is topologically equivalent to a torus.

**Torus sections**

Intro Geometry Topology Two-sheeted cover

*n*-dimensional torus Flat torus

*n*-fold torus Toroidal polyhedra Automorphisms Cutting a torus See also Notes References External links

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