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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. A circle rotated about a
chord (geometry) 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,
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-Coordinate_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 (mathematics) of a torus need not be a circle but could also be an ellipse or any other conic section.
Topology
Topology, a
torus is a closed
surface defined as product topology 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 Foliation 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 Stereographic projection the topological torus into
R3 from the north pole of
S3.
The torus can also be described as a quotient space 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) \times \pi_1(S^1) \cong \mathbb{Z} \times \mathbb{Z}.
Intuitively speaking, this means that a closed path (topology) 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 group).
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 \times S^1 \times \cdots \times 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
Rn under integral shifts in any coordinate. That is, the
n-torus is
Rn modulo the group action of the integer
lattice (group) 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 space
manifold. It is also an example of a compact
abelian group 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 matrix 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 binomial coefficient 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 (mathematics)
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 (mathematics)"
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 space connected space 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.
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 colour theorem for the plane (mathematics).)
See also
External links
- Creation of a torus at cut-the-knot
- "Torus" From MathWorld by Eric W. Weisstein
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. A circle rotated about a
chord (geometry) 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,
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-Coordinate_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 (mathematics) of a torus need not be a circle but could also be an
ellipse or any other conic section.
Topology
Topology, a
torus is a closed
surface defined as product topology 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 Foliation 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
Stereographic projection the topological torus into
R3 from the north pole of
S3.
The torus can also be described as a
quotient space 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) \times \pi_1(S^1) \cong \mathbb{Z} \times \mathbb{Z}.
Intuitively speaking, this means that a closed path (topology) 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 group).
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 \times S^1 \times \cdots \times 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
Rn under integral shifts in any coordinate. That is, the
n-torus is
Rn modulo the
group action of the integer lattice (group)
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 space
manifold. It is also an example of a compact
abelian group 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 matrix 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 binomial coefficient 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 (mathematics)
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 (mathematics)"
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 space connected space 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.
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 colour theorem for the
plane (mathematics).)
See also
External links
- Creation of a torus at cut-the-knot
- "Torus" From MathWorld by Eric W. Weisstein
Torus - Wikipedia, the free encyclopedia
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 ...
Definition: torus from Online Medical Dictionary
The Online Medical Dictionary is a searchable dictionary of definitions from medicine, science and technology.
Torus Games
Developers of Max Steel and Hello Kitty.
The TORUS 3D Radiative Transfer Code
TORUS is a Monte-Carlo radiative-transfer code that is designed to compute polarization images and spectra from a three-dimensional opacity grid.
Torus
Torus Facts; Notice these interesting things: It can be made by revolving a small circle along a line made by another circle. It has no edges or vertices
Torus -- from Wolfram MathWorld
An (ordinary) torus is a surface having genus one, and therefore possessing a single "hole" (left figure). The single-holed "ring" torus is known in older literature as an "anchor ...
Veneered Torus Skirting
White Oak, Ash or Beech Veneered Torus Skirting available either sanded and trimmed or pre-finished.
Digital Chemistry - Torus
Torus™- Introducing Markush Structure Searching in Oracle® A Chemistry Cartridge with a Difference! Torus is a major new development in cheminformatics software, able to ...
torus - definition of torus in the Medical dictionary - by the Free ...
Definition of torus in the Medical Dictionary. torus explanation. Information about torus in Free online English dictionary. What is torus? Meaning of torus medical term. What does ...
AskOxford: torus
torus / tor ss/ • noun (pl. tori / tor i/ or toruses) 1 Geometry a surface or solid resembling a ring doughnut, formed by rotating a closed curve about a line which lies in the ...
