## ::Torus

### ::concepts

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

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A torus
As the distance to the axis of revolution decreases, the ring torus becomes a horn torus, then a spindle torus, and finally degenerates into a sphere.

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: S1 × S1, 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 S1 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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