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Encyclopedia > Torus
A torus
A torus

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 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. Image File history File links Emblem-important. ... Image File history File links Torus. ... Image File history File links Torus. ... For other uses, see Geometry (disambiguation). ... The parabola y=x2 rotated about the z-axis A surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of rotation) that lies on the same plane. ... Circle illustration This article is about the shape and mathematical concept of circle. ... A set of points is said to be coplanar if and only if they lie on the same geometric plane. ... For other uses, see Doughnut (disambiguation). ... Firestone tire A tire (US spelling) or tyre (UK spelling) is a roughly toroidal piece of (usually) rubber placed on a wheel to cushion it. ... A chord of a curve is a geometric line segment whose endpoints both lie on the curve. ... For other uses, see Latins and Latin (disambiguation). ... Cushion: makes a lounge or rug softer A cushion (from Old French coisson, coussin; from Latin culcita, a quilt), is a soft bag of some ornamental material, stuffed with wool, hair, feathers, polyester staple fiber, non-woven material, or even paper torn into fragments. ...

Contents

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-axis is Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... A coordinate axis is one of a set of vectors that defines a coordinate system. ...

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 Area is the measure of how much exposed area any two dimensional object has. ... For other uses, see Volume (disambiguation). ...

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.


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. In Abstract Algebra, a generator is defined as follows: Let G be a group and , then a is called a generator and G is a cyclic group. ... For other uses, see Ellipse (disambiguation). ... Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ...


Topology

A torus is the product of two circles.
A torus is the product of two circles.

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{2}. 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). homology cycles on a torus. ... homology cycles on a torus. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... An open surface with X-, Y-, and Z-contours shown. ... Circle illustration This article is about the shape and mathematical concept of circle. ... In mathematics, a foliation is a geometric device used to study manifolds. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, the Hopf bundle (or Hopf fibration), named after Heinz Hopf, is an important example of a fiber 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. In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology). ... This word should not be confused with homomorphism. ... Stereographic projection of a circle of radius R onto the x axis. ...


The torus can also be described as a quotient of the Cartesian plane under the identifications In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

(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 − 1B − 1. The unit square in a Cartesian coordinate system with coordinates (x,y) is defined as the square consisting of the points where both x and y lie in the unit interval from 0 to 1. ... In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges. ...

Turning a torus inside-out (animated version)
Turning a torus inside-out (animated version)

The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself: In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ...

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 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. In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X f : I → X. The initial point of the path is f(0) and the terminal point is f(1). ...


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). In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...


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 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. In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. ... A square A projection of a cube (into a two-dimensional image) A projection of a hypercube (into a two-dimensional image) In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ...


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. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... Illustration of a unit circle. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...


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. In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. ... In the theory of Lie groups in mathematics, especially those that are compact, a special role is played by the torus groups. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...


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. In linear algebra, an n-by-n (square) matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...


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. In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In abstract algebra, a free abelian group is an abelian group that has a basis in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, particularly in combinatorics, a binomial coefficient is a coefficient of any of the terms in the expansion of the binomial (x+1)n. ... In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological spaces shape or structure. ... In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. ... In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...


The n-fold torus

A triple torus
A triple 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. Image File history File links Size of this preview: 792 × 600 pixelsFull resolution (1320 × 1000 pixels, file size: 366 KB, MIME type: image/png) % illustration of a triple torus. ... Image File history File links Size of this preview: 792 × 600 pixelsFull resolution (1320 × 1000 pixels, file size: 366 KB, MIME type: image/png) % illustration of a triple torus. ... An open surface with X-, Y-, and Z-contours shown. ... In geometric topology, a connected sum of two connected -dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres. ... For other uses, see Sphere (disambiguation). ...


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. In mathematics, a double torus is a topological object formed by the connected sum of two torii. ... The torus is an orientable surface. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... For other uses, see Sphere (disambiguation). ...


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. There exist several classification theorems in mathematics: Classification theorem of surfaces Classification of two-dimensional closed manifolds Enriques-Kodaira classification of algebraic surfaces (complex dimension two, real dimension four) Nielsen-Thurston classification which characterizes homeomorphisms of a compact surface Classification of finite simple groups Artin–Wedderburn theorem — a classification theorem... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... Projective plane - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... Please refer to Real vs. ...


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.) Example of a four-colored map The four color theorem (also known as the four color map theorem) states that given any plane separated into regions, such as a political map of the states of a country, the regions may be colored using no more than four colors in such... This article is about the mathematical construct. ...

This construction shows the torus divided into the maximum of seven regions, every one of which touches every other.

Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

See also

As the distance to the axis of revolution decreases, the ring torus becomes a spindle torus and then degenerates into a sphere. ... In mathematics, an algebraic torus over a field K is an algebraic group which is isomorphic over the algebraic closure of K to (GL1)r for some integer r, the rank of the torus. ... In geometry, given an arbitrary point on a torus, four circles can be drawn through it. ... An annulus In mathematics, an annulus (the Latin word for little ring, with plural annuli) is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. ... For other uses, see Doughnut (disambiguation). ... A catalog of elliptic curves. ... In differential geometry, Loewners torus inequality applies to general Riemannian metrics on the 2-torus. ... In the theory of Lie groups in mathematics, especially those that are compact, a special role is played by the torus groups. ... In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ... For other uses, see Sphere (disambiguation). ... An open surface with X-, Y-, and Z-contours shown. ... A toroid is a doughnut-shaped object whose surface is a torus. ... In nuclear physics a torus is a large fusion reactor which is shaped like an elliptical or circular torus. ... These are an example of large mandibular tori. ... This is an example of palatal torus. ...

External links

Wikimedia Commons has media related to:
Torus
  • Creation of a torus at cut-the-knot
  • Eric W. Weisstein, Torus at MathWorld.
  • "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.
cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

  Results from FactBites:
 
Torus - Wikipedia, the free encyclopedia (649 words)
In geometry, a torus (pl. tori) is a doughnut-shaped surface of revolution generated by revolving a circle about an axis coplanar with the circle.
The sphere is a special case of the torus obtained when the axis of rotation is a diameter of the circle.
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.
  More results at FactBites »

 
 

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