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. In most contexts it is assumed that the axis does not touch the circle and in this case the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit. Other types of torus include the horn torus, which is generated when the axis is tangent to the circle, and the spindle torus, which is generated when the axis is a chord of the circle. A degenerate case is when the axis is a diameter of the circle and surface is a sphere. The ring torus bounds a solid known as a toroid. The adjective toroidal can be applied to tori, toroids or, more generally, any ring shape as in toroidal inductors and transformers. Real world examples of (approximately) toroidal objects include doughnuts, inner tubes, many lifebuoys, Orings and Vortex rings.
In topology, a ring torus is homeomorphic to the Cartesian product of two circles: S^{1} × S^{1}, and the latter is taken to be the definition in that context. It is a compact 2manifold of genus 2. The ring torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian product of the embedding of S^{1} in the plane. This produces a geometric object called the Clifford torus, surface in 4space.
The word torus comes from the Latin word meaning cushion.^{[1]}
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A torus can be defined parametrically by:^{[2]}
where
An implicit equation in Cartesian coordinates for a torus radially symmetric about the zaxis is
clearing the square root and rotating 2π (equivalent to replacing x^{2} by (r / 2R)^{2} − y^{2} → ) produces a quartic:
The surface area and interior volume of this torus are easily computed using Pappus's centroid theorem giving^{[3]}
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.
Topologically, a torus is a closed surface defined as the product of two circles: S^{1} × S^{1}. This can be viewed as lying in C^{2} and is a subset of the 3sphere S^{3} of radius . This topological torus is also often called the Clifford torus. In fact, S^{3} is filled out by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of S^{3} as a fiber bundle over S^{2} (the Hopf bundle).
The surface described above, given the relative topology from R^{3}, 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 R^{3} from the north pole of S^{3}.
The torus can also be described as a quotient of the Cartesian plane under the identifications
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:
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.
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 2torus doublecovers the 2sphere, with four ramification points. Every conformal structure on the 2torus can be represented as a twosheeted cover of the 2sphere. The points on the torus corresponding to the ramification points are the Weierstrass points. In fact, the conformal type of the torus is determined by the crossratio of the four points.
The torus has a generalization to higher dimensions, the ndimensional torus, often called the ntorus for short. (This is one of two different meanings of the term "ntorus".) Recalling that the torus is the product space of two circles, the ndimensional torus is the product of n circles. That is:
The torus discussed above is the 2dimensional torus. The 1dimensional torus is just the circle. The 3dimensional torus is rather difficult to visualize. Just as for the 2torus, the ntorus can be described as a quotient of R^{n} under integral shifts in any coordinate. That is, the ntorus is R^{n} modulo the action of the integer lattice Z^{n} (with the action being taken as vector addition). Equivalently, the ntorus is obtained from the ndimensional hypercube by gluing the opposite faces together.
An ntorus in this sense is an example of an ndimensional 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 coordinatewise 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 Z^{n}, 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 R^{n} in the usual way, one has the typical toral automorphism on the quotient.
The fundamental group of an ntorus is a free abelian group of rank n. The kth homology group of an ntorus is a free abelian group of rank n choose k. It follows that the Euler characteristic of the ntorus is 0 for all n. The cohomology ring H^{•}(T^{n},Z) can be identified with the exterior algebra over the Zmodule Z^{n} whose generators are the duals of the n nontrivial cycles.
As the ntorus is the nfold product of the circle, the ntorus is the configuration space of n ordered, not necessarily distinct points on the circle. Symbolically, T^{n} = (S^{1})^{n}. The configuration space of unordered, not necessarily distinct points is accordingly the orbifold T^{n} / S_{n}, which is the quotient of the torus by the symmetric group on n letters (by permuting the coordinates).
For n = 2, the quotient is the Möbius strip, the edge corresponding to the orbifold points where the two coordinates coincide. For n = 3 this quotient may be described as a solid torus with crosssection an equilateral triangle, with a twist; equivalently, as a triangular prism whose top and bottom faces are connected with a ⅓ twist (120°): the 3dimensional interior corresponds to the points on the 3torus where all 3 coordinates are distinct, the 2dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1dimensional edge corresponds to points with all 3 coordinates different.
These orbifolds have found significant applications to music theory in the work of Dmitri Tymoczko and collaborators, being used to model musical triads.^{[4]}^{[5]}
The flat torus is a specific embedding of the familiar 2torus into Euclidean 4space or higher dimensions. Its surface has zero Gaussian curvature everywhere. Its surface is "flat" in the same sense that the surface of a cylinder is "flat". In 3 dimensions you can bend a flat sheet of paper into a cylinder without stretching the paper, but you cannot then bend this cylinder into a torus without stretching the paper. In 4 dimensions you can (mathematically).
A simple 4d Euclidean embedding is as follows: <x,y,z,w> = <Rcos u, Rsin u, Pcos v, Psin v> where R and P are constants determining the aspect ratio. It is diffeomorphic to a regular torus but not isometric. It can not be isometrically embedded into Euclidean 3space. Mapping it into 3space requires you to "bend" it, in which case it looks like a regular torus, for example, the following map <x,y,z> = <(R + Psin v)cos u, (R + Psin v)sin u, Pcos v>.
In the theory of surfaces the term ntorus has a different meaning. Instead of the product of n circles, they use the phrase to mean the connected sum of n 2dimensional 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 ntorus resembles the surface of n doughnuts stuck together side by side, or a 2dimensional sphere with n handles attached.
An ordinary torus is a 1torus, a 2torus is called a double torus, a 3torus a triple torus, and so on. The ntorus is said to be an "orientable surface" of "genus" n, the genus being the number of handles. The 0torus is the 2dimensional sphere.
The classification theorem for surfaces states that every compact connected surface is either a sphere, an ntorus with n > 0, or the connected sum of n projective planes (that is, projective planes over the real numbers) with n > 0.
The homeomorphism group (or the subgroup of diffeomorphisms) of the torus is studied in geometric topology. Its mapping class group (the group of connected components) is isomorphic to the group GL(n, Z) of invertible integer matrices, and can be realized as linear maps on the universal covering space that preserve the standard lattice (this corresponds to integer coefficients) and thus descend to the quotient.
At the level of homotopy and homology, the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on on fundamental group, as these are all naturally isomorphic; note also that the first cohomology group generates the cohomology algebra):
Since the torus is an EilenbergMacLane space K(G, 1), its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); that this agrees with the mapping class group reflects that all homotopy equivalences can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism – and that homotopic homeomorphisms are in fact isotopic (connected through homeomorphisms, not just through homotopy equivalences). More tersely, the map is 1connected (isomorphic on pathcomponents, onto fundamental group). This is a "homeomorphism reduces to homotopy reduces to algebra" result.
Thus the short exact sequence of the mapping class group splits (an identification of the torus as the quotient of gives a splitting, via the linear maps, as above):
so the homeomorphism group of the torus is a semidirect product,
The mapping class group of higher genus surfaces is much more complicated, and an area of active research.
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.)
A standard torus (specifically, a ring torus) can be cut with n planes into at most parts, integer sequence A003600.^{[6]} The initial terms of this sequence are 1, 2, 6, 13, for n starting from 0.
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Torus m. (genitive Torus, plural Tori)
