# Torus

In geometry, a torus (pl. 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 ring torus or simply torus if the ring shape is implicit.

When the axis is tangent to the circle, the resulting surface is called a horn torus; when the axis is a chord of the circle, it is called a spindle torus. A degenerate case is when the axis is a diameter of the circle, which simply generates the surface of a sphere. The ring torus bounds a solid known as a toroid. The adjective toroidal can be applied to tori, toroids or, more generally, any ring shape as in toroidal inductors and transformers. Real-world examples of (approximately) toroidal objects include doughnuts, vadais, inner tubes, many lifebuoys, O-rings and vortex 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 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, surface in 4-space.

The word torus comes from the Latin word meaning cushion.

### Other articles related to "torus":

Systoles Of Surfaces - Torus
... In 1949 Loewner proved his inequality for metrics on the torus T2, namely that the systolic ratio SR(T2) is bounded above by, with equality in the flat (constant curvature) case ...
Proof of Loewner's Torus Inequality
... Loewner's torus inequality can be proved most easily by using the computational formula for the variance, Namely, the formula is applied to the probability measure defined by the measure of the unit ... This approach therefore produces the following version of Loewner's torus inequality with isosystolic defect where ƒ is the conformal factor of the metric with respect to a unit ...
Cutting A Torus
... A standard torus (specifically, a ring torus) can be cut with n planes into at most parts ...