Torus

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In geometry, a torus (pl. tori) is a doughnut-shaped surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle. Examples of tori include the surfaces of doughnuts and inner tubes. A circle rotated about a chord of the circle is called a torus in some contexts, but this is not a common usage in mathematics. The shape produced when a circle is rotated about a chord resembles a round cushion. Torus was the Latin word for a cushion of this shape.

1 Geometry
2 Topology
3 The n-torus
4 Colouring a torus
5 See also
6 External links

[edit] Geometry

A torus can be defined parametrically 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,

r is the radius of the tube.

An equation in Cartesian coordinates for a torus radially symmetric about the z-axis is

$\left(R - \sqrt{x^2 + y^2}\right)^2 + z^2 = r^2, \,\!$

and clearing the square root produces a quartic:

$(x^2+y^2+z^2 + R^2 - r^2)^2 = 4R^2(x^2+y^2) . \,\!$

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

$A = 4 \pi^2 R r = \left( 2\pi r \right) \left( 2 \pi R \right) \,$

$V = 2 \pi^2 R r^2 = \left( \pi r^2 \right) \left( 2\pi R \right). \,$

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.

[edit] Topology

A torus is the product of two circles.

Topologically, a torus is a closed surface defined as product of two circles: S¹ × S¹. This can be viewed as lying in C² and is a subset of the 3-sphere S³ of radius $\sqrt{2}$ . This topological torus is also often called the Clifford torus. In fact, S³ is filled out by a family of nested tori in this manner (with two degenerate cases, a circle and a straight line), a fact which is important in the study of S³ as a fiber bundle over S² (the Hopf bundle).

The surface described above, given the relative topology from R³, is homeomorphic to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by stereographically projecting the topological torus into R³ from the north pole of S³.

The torus can also be described as a quotient of the Cartesian 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, described as a fundamental polygon $A B A - 1 B - 1$ .

The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself:

$\pi_1(\mathbb{T}^2) = \pi_1(S^1) \times \pi_1(S^1) \cong \mathbb{Z} \times \mathbb{Z}$

Intuitively speaking, this means that a closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. This might be imagined as two shoelaces passing through each other, then unwinding, then rewinding.

The first homology group of the torus is isomorphic to the fundamental group (since the fundamental group is abelian).

[edit] The n-torus

There are two different definitions of the n-torus. Since the torus is the product space of two circles, we could define the n-torus as the product of n circles.

In this case we have:

$\mathbb{T}^n = \underbrace{S^1 \times S^1 \times \cdots \times S^1}_n$

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. That is, the n-torus is Rⁿ modulo the action of the integer lattice Zⁿ (with the action being taken as vector addition). Equivalently, the n-torus is obtained from the n-cube by gluing the opposite faces together.

An n-torus in this sense is an example of an n-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.

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.

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.

Algebraic topologists use the term n-torus with a different meaning. Instead of the product of n circles, they use the phrase to mean the connected sum of n torii. To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the circles that bound those disks. In this sense, an n-torus resembles the surface of n doughnuts stuck together side by side.

In this sense, an ordinary torus is a 1-torus, a 2-torus is called a double torus, a 3-torus a triple torus, and so on.

The classification theorem for surfaces states that every compact connected surface is either a sphere, an n-torus, or the connected sum of n projective planes.

[edit] Colouring a torus

If a torus is divided into regions, then it is always possible to colour the regions with at most seven colours so that neighbouring regions have different colours. (Contrast with four color theorem.) In the following example, the torus has been divided into seven regions, every one of which touches every other, illustrating why seven is the minimum necessary for a torus: