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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 Doughnut s and Inner Tube s. 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.


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

R

r


An equation in Cartesian Coordinates for a torus radially symmetric about the ''z-'' Axis is
:\left(R - \sqrt{x^2 + y^2} ight)^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 ight) \left( 2 \pi R ight) \,
:V = 2 \pi^2 R r^2 = \left( \pi r^2 ight) \left( 2\pi R ight). \,

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


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 ''S''3 as a Fiber Bundle over ''S''2 (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 ''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)
Or, equivalently, as the quotient of the Unit Square by pasting the opposite edges together, described as a Fundamental Polygon ABA^{-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) imes \pi_1(S^1) \cong \mathbb{Z} imes \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 ).


THE ''N''-DIMENSIONAL TORUS


The torus has a generalization to higher dimensions, the ''n''-dimensional torus, often called the ''n''-'''torus''' for short. (This is one of two different meanings of the term "''n''-torus".)
Recalling that the torus is the product space of two circles, the ''n''-dimensional torus is the product of ''n'' circles.
That is:
:\mathbb{T}^n = \underbrace{S^1 imes S^1 imes \cdots imes S^1}_n
The torus discussed above is the 2-dimensional torus. The 1-dimensional torus is just the circle. The 3-dimensional torus is rather difficult to visualize. Just as for the 2-torus, the ''n''-torus can be described as a quotient of R''n'' under integral shifts in any coordinate. That is, the ''n''-torus is R''n'' modulo the Action of the integer Lattice '''Z'''''n'' (with the action being taken as vector addition). Equivalently, the ''n''-torus is obtained from the ''n''-dimensional Hypercube 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 Number s 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 Group s. 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.


THE ''N''-FOLD TORUS


In the theory of Surface s 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 Plane s (that is, projective planes over the Real Numbers ) with ''n'' > 0.


COLOURING A TORUS



If a torus is divided into regions, then it is always possible to colour the regions with no more than seven colours so that neighbouring regions have different colours. (Contrast with the Four Color Theorem for the Plane .)


SEE ALSO




EXTERNAL LINKS



  • "Torus" From MathWorld by Eric W. Weisstein