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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, which does not touch the circle. Examples of tori include the surfaces of doughnuts and inner tubes. The solid contained by the surface is known as a toroid. 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.

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 are in the interval [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

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

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

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

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

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

These formulas are the same as for a cylinder of length 2πR and radius r, created by cutting the tube and unrolling it by straightening out the line running around the centre of the tube. The losses in surface area and volume on the inner side of the tube happen to exactly cancel out the gains on the outer side.

Topologically, a torus is a closed surface defined as the product of two circles: S^{1} × S^{1}. This can be viewed as lying in C^{2} and is a subset of the 3-sphere S^{3} of radius $sqrt\{2\}$. This topological torus is also often called the Clifford torus. In fact, S^{3} is filled out by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of S^{3} as a fiber bundle over S^{2} (the Hopf bundle).

The surface described above, given the relative topology from R^{3}, 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 R^{3} from the north pole of S^{3}.

The torus can also be described as a quotient of the Cartesian plane under the identifications

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

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

- $pi\_1(mathbb\{T\}^2)\; =\; pi\_1(S^1)\; times\; pi\_1(S^1)\; cong\; mathbb\{Z\}\; times\; mathbb\{Z\}.$

If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged.

The first homology group of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian).

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

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 G one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension. Such maximal tori T have a controlling role to play in theory of connected G.

Automorphisms of T are easily constructed from automorphisms of the lattice Z^{n}, which are classified by integral matrices M of size n×n which are invertible with integral inverse; these are just the integral M of determinant +1 or −1. Making M act on R^{n} in the usual way, one has the typical toral automorphism on the quotient.

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^{•}(T^{n},Z) can be identified with the exterior algebra over the Z-module Z^{n} whose generators are the duals of the n nontrivial cycles.

A simple 4-d Euclidean embedding is as follows:

In the theory of surfaces the term n-torus has a different meaning. Instead of the product of n circles, they use the phrase to mean the connected sum of n 2-dimensional tori. To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the disks' boundary circles. To form the connected sum of more than two surfaces, sum two of them at a time until they are all connected together. In this sense, an n-torus resembles the surface of n doughnuts stuck together side by side, or a 2-dimensional sphere with n handles attached.

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 n-torus is said to be an "orientable surface" of "genus" n, the genus being the number of handles. The 0-torus is the 2-dimensional sphere.

The classification theorem for surfaces states that every compact connected surface is either a sphere, an n-torus with n > 0, or the connected sum of n projective planes (that is, projective planes over the real numbers) with n > 0.

- Creation of a torus at cut-the-knot
- "4D torus" Fly-through cross-sections of a four dimensional torus.
- "Relational Perspective Map" An algorithm that uses flat torus to visualize high dimensional data.
- "Torus Games" Several games that highlight the topology of a torus.

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Last updated on Saturday October 11, 2008 at 15:31:19 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday October 11, 2008 at 15:31:19 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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