Everything about Torus totally explained
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 doesn't 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 isn't 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:
»
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
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-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 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
Zn, 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
Rn 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•(
Tn,
Z) can be identified with the
exterior algebra over the
Z-
module Zn whose generators are the duals of the
n nontrivial cycles.
The n-fold torus
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're 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.
Coloring a torus
If a torus is divided into regions, then it's always possible to color the regions with no more than seven colors so that neighboring regions have different colors. (Contrast with the
four color theorem for the
plane.)
Further Information
Get more info on 'Torus'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://torus.totallyexplained.com">Torus Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
