# Dictionary Definition

torus

### Noun

1 a ring-shaped surface generated by rotating a
circle around an axis that does not intersect the circle [syn:
toroid]

# User Contributed Dictionary

### Pronunciation

- /ˈtɔːrəs/
- Rhymes with: -ɔːrəs

#### Homophones

### Noun

- A three-dimensional shape consisting of a ring with a circular cross-section. The shape of an inner tube or hollow doughnut.
- A molding which projects at the base of a column and above the plinth.
- A botanical term for the end of the peduncle or flower stalk to which the floral parts (or in the Asteraceae, the florets of a flower head) are attached; see receptacle

#### Translations

three dimensional shape

- French: tore
- German: Torus
- Hungarian: tórusz
- Portuguese: toro
- Spanish: toro
- Swedish: torus

## Latin

### Noun

torus## Swedish

### Noun

- In the context of "topology|lang=sv": torus; a shape consisting of a ring, or an object of the same topology residing in a space of higher dimension; especially considered as a Cartesian product of two circles in a four-dimensional space

#### Related terms

# Extensive Definition

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.

## Geometry

A torus can be defined parametrically by:

- x(u, v) = (R + r \cos) \cos \,
- y(u, v) = (R + r \cos) \sin \,
- z(u, v) = r \sin \,

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\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( 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). \,

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.

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

Topologically, a
torus is a closed surface defined as the 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. 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).

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

- \pi_1(\mathbb^2) = \pi_1(S^1) \times \pi_1(S^1) \cong \mathbb \times \mathbb.

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).

## 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^n = \underbrace_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 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 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.

## Coloring a torus

If a torus is divided into regions, then it is
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.)

## See also

## External links

- 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.

torus in Bulgarian: Тор (геометрия)

torus in Catalan: Tor (figura geomètrica)

torus in Czech: Torus

torus in Danish: Torus

torus in German: Torus

torus in Spanish: Toro (matemática)

torus in Esperanto: Toro (Italio)

torus in Persian: چنبره

torus in French: Tore

torus in Ido: Toro

torus in Italian: Toro (geometria)

torus in Hebrew: טורוס

torus in Luxembourgish: Torus

torus in Lithuanian: Toras (geometrija)

torus in Hungarian: Tórusz

torus in Macedonian: Тор

torus in Dutch: Torus

torus in Japanese: トーラス

torus in Norwegian: Torus

torus in Polish: Torus (matematyka)

torus in Portuguese: Toro (topologia)

torus in Russian: Тор (поверхность)

torus in Slovak: Torus (geometria)

torus in Serbian: Торус

torus in Finnish: Torus

torus in Swedish: Torus

torus in Ukrainian: Тор (геометрія)

torus in Chinese: 环面