Torus

See also torus (nuclear physics).

In geometry, a torus (pl. tori) is a solid of revolution generated by revolving a circle about an axis coplanar with the circle. The sphere is a special case of the torus obtained when the axis of rotation is a diameter of the circle. If the axis of rotation does not intersect the circle, the torus has a hole in the middle and resembles a ring doughnut, a hula hoop or an inflated tire (U.K. tyre). The other case, when the axis of rotation is a chord of the circle, produces a sort of squashed sphere resembling a round cushion. Torus was the Latin word for a cushion of this shape.

A torus can be defined parameterically by

x(u, v) = (a + b cos v) cos u

y(u, v) = (a + b cos v) sin u

z(u, v) = b sin v.

where u, v ∈ [0, 2π].

According to a broader definition, the generator of a torus need not be a circle but could also be an ellipse or any other conic section.

Generalization in topology

In topology, a torus or n-torus refers to a product of n circles. The torus discussed above is the 2-torus -- the product of just two circles. The 1-torus is just the circle, and the surface of a doughnut shape is a 2-torus. (In proper mathematical usage, a solid as described above would be spoken of as generated from a disk, i.e., a filled-in circle.) An n-torus is an example of an n-dimensional compact manifold.

The fundamental group and first homology group of an n-torus is a free abelian group of rank n.

When the unit circle is identified with the unit complex numbers with multiplication, the n-torus becomes a compact abelian Lie group. Toroidal groups play an important part in the theory of compact Lie groups. This is due in part to the fact that in any compact Lie group one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension.

See also

algebraic torus, n-sphere