A torus

In geometry, a torus (pl. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position 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. A surface of revolution is a Surface created by rotating a Curve lying on some plane (the Generatrix) around a Straight line (the Axis Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Geometry, a set of points in space is coplanar if the points all lie in the same geometric plane. Examples of tori include the surfaces of doughnuts and inner tubes. A doughnut (also spelled "donut" is a sweet Deep-fried piece of Dough or batter. This article is about tires used on road Vehicles including pneumatic tires and solid tires. 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. A chord of a Curve is a geometric Line segment whose endpoints both lie on the curve 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. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. A cushion (from Old French coisson, coussin; from Latin culcita, a quilt is a soft bag of some ornamental material stuffed with

Geometry

A torus can be defined parametrically by:

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

and clearing the square root produces a quartic:

The surface area and interior volume of this torus are given by

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. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point Surface area is the measure of how much exposed Area an object has The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically 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 Mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface

Topology

A torus is the product of two circles.

Topologically, a torus is a closed surface defined as the product of two circles: S¹ × S¹. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the This can be viewed as lying in C² and is a subset of the 3-sphere S³ of radius . This topological torus is also often called the Clifford torus. In Geometric topology, the Clifford torus is a special kind of Torus sitting inside R 4 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). In Mathematics, a foliation is a geometric device used to study manifolds Informally speaking a foliation is a kind of "clothing" worn on a manifold In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. In the mathematical field of Topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a Hypersphere

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. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is Topological equivalence redirects here see also Topological equivalence (dynamical systems. A particular homeomorphism is given by stereographically projecting the topological torus into R³ from the north pole of S³. In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane

The torus can also be described as a quotient of the Cartesian plane under the identifications

(x,y) ~ (x+1,y) ~ (x,y+1). In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

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 unit square is a square with all of the side lengths equalling 1 In Mathematics, each closed Surface in the sense of Geometric topology can be constructed from an even-sided oriented Polygon, called a fundamental

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:

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. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics, one can often define a direct product of objectsalready known giving a new one In Mathematics, a path in a Topological space X is a continuous map f from the Unit interval I = to So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. This might be imagined as two shoelaces passing through each other, then unwinding, then rewinding.

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 ὁμός homos "identical" is In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, the Hurewicz theorem is a basic result of Algebraic topology, connecting Homotopy theory with Homology theory via a map known An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

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:

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). In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of Equivalently, the n-torus is obtained from the n-dimensional hypercube by gluing the opposite faces together. 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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be It is also an example of a compact abelian Lie group. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the 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). In Mathematics, a unit circle is Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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. In Mathematics, a compact ( topological, often understood group is a Topological group whose Topology is Compact. 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. In the mathematical theory of Compact Lie groups a special role is played by Torus subgroups in particular by the maximal torus subgroups In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of 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. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- 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. 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 The k-th homology group of an n-torus is a free abelian group of rank n choose k. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x   k term in the Polynomial It follows that the Euler characteristic of the n-torus is 0 for all n. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant 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, specifically Algebraic topology, the cohomology ring of a Topological space X is a ring formed from the Cohomology 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

The n-fold torus

A triple torus

In the theory of surfaces the term n-torus has a different meaning. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. Instead of the product of n circles, they use the phrase to mean the connected sum of n 2-dimensional tori. In Mathematics, specifically in Topology, the operation of connected sum is a geometric modification on Manifolds Its effect is to join two given manifolds 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. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

An ordinary torus is a 1-torus, a 2-torus is called a double torus, a 3-torus a triple torus, and so on. In mathematics a double torus is a topological object formed by the Connected sum of two torii. The n-torus is said to be an "orientable surface" of "genus" n, the genus being the number of handles. A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back In Mathematics, genus has a few different but closely related meanings Topology Orientable surface The 0-torus is the 2-dimensional sphere. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

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. In Mathematics, a classification theorem answers the classification problem "What are the objects of a given type up to some equivalence?" In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics In Mathematics, the real numbers may be described informally in several different ways

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. 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 )

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

See also

External links

• Creation of a torus at cut-the-knot
• Eric W. Weisstein, Torus at MathWorld. In Mathematics, a standard torus is a circular Torus of revolution that is any Surface of revolution generated by rotating a Circle In Mathematics, an algebraic torus is a type of commutative affine Algebraic group. In Geometry, Villarceau circles (viːlɑrˈsoʊ are a pair of Circles produced by cutting a Torus diagonally through the center at the correct angle In Mathematics, an annulus (the Latin word for "little ring" with plural annuli) is a ring-shaped geometric figure or more generally a term A doughnut (also spelled "donut" is a sweet Deep-fried piece of Dough or batter. In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O In Differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner for the systole of an arbitrary Riemannian metric In the mathematical theory of Compact Lie groups a special role is played by Torus subgroups in particular by the maximal torus subgroups In Mathematics, a fundamental pair of periods is an Ordered pair of Complex numbers that define a lattice in the Complex plane. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. A tokamak is a machine producing a toroidal Magnetic field for confining a plasma. Torus mandibularis (pl mandibular tori) is a bony growth in the Mandible along the surface nearest to the Tongue. Torus palatinus (pl palatal tori) is a bony growth on the Palate. Created by Helaman Ferguson, the umbilic torus is a single-edged 3-dimensional figure In Mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of any Standard torus. Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• "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.