Torus

See also torus (nuclear physics).

In geometry, a torus (pl. tori) is a doughnut shaped surface 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) = (R + r cos v) cos u

y(u, v) = (R + r cos v) sin u

z(u, v) = r sin v.

where u, v ∈ [0, 2π], R is the distance from the center of the tube to the center of the torus, and r is the radius of the tube.

The surface area and interior volume of this torus are given by

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.

Topology

A torus is the product of two circles.

Topologically, a torus is defined as product of two circles: S¹ × S¹. One can show that the surface described above, together with its induced topology from R³, is homeomorphic to a topologocial torus. The torus can also be desribed as a quotient of the Euclidean plane under the identifications

(x,y) ~ (x+1,y) ~ (x,y+1)

Or, equivalently, as the quotient of the unit square by pasting the opposite edges together.

One can easily generalize the torus to arbitrary dimensions. An n-torus is defined as a product of n circles:

<math>\mathbb{T}^n = S^1 \times S^1 \times \cdots \times S^1<math>

The torus discussed above is the 2-torus. The 1-torus is just the circle. The 3-torus is rather difficult to visualize. Just as for the 2-torus, the n-torus can be described as a quotient of Rⁿ under integral shifts in any coordinate. Or, equivalently, as the quotient of the n-cube obtained by glueing the opposite faces together.

An n-torus is an example of an n-dimensional compact manifold.

The fundamental group of an n-torus is a free abelian group of rank n. The k-th homology group of an n-torus is a free abelian group of rank n choose k. It follows that the Euler characteristic of the n-torus is 0 for all n. The cohomology ring H^•(Tⁿ,Z) can be identified with the exterior algebra over the Z-module Zⁿ whose generators are the duals of the n nontrivial cycles.

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.