# 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.

## 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

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

A torus is the product of two circles.
Topologically, a torus is a closed surface defined as product of two circles: S1 Ã— S1. 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 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).
Or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon .

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. 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).

## 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). 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

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.

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.

## Colouring a torus

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

Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
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.

Examples of surfaces generated by a straight line are the cylindrical and conical surfaces.
circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the centre.

Circles are simple closed curves which divide the plane into an interior and exterior.
In geometry, a set of points in space is coplanar if the points all lie in the same geometric plane. For example, three points are always coplanar; but four points in space are usually not coplanar.
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A chord of a curve is a geometric line segment whose endpoints both lie on the curve. A secant or a secant line is the line extension of a chord.

## Chords of a circle

Further information: Chord properties

Latin}}}
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Official language of: Vatican City
Used for official purposes, but not spoken in everyday speech
Regulated by: Opus Fundatum Latinitas
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ISO 639-2: lat
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.
Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
Area is the measure of how much exposed area any two dimensional object has. It is expressed in square units, and is calculated by adding together the areas of all the faces of the object.

## Area formulas

Note: For 2D figures, the surface area and the area are the same.
The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses.
ellipse (from the Greek ἔλλειψις, literally absence) is the locus of points on a plane where the sum of the distances from any point on the curve to two fixed points is constant.
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. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their
Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure.
surface is a two-dimensional manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space, EÂ³.
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology.
circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the centre.

Circles are simple closed curves which divide the plane into an interior and exterior.
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, cut from a striped fabric.
fiber bundle (or fibre bundle) is a space which looks locally like a product space. It may have a different global topological structure in that the space as a whole may not be homeomorphic to a product space.
Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle.
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, or the
In the mathematical field of topology, a homeomorphism or topological isomorphism
stereographic projection is a certain mapping (function) that projects a sphere onto a plane. Intuitively, it gives a planar picture of the sphere.

The projection is defined on the entire sphere, except at one point — the projection point.
quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation.
Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.