Configuration space[edit]

The configuration space of 2 not necessarily distinct points on the circle is the orbifold quotient of the 2-torus, T²/S₂, which is the Möbius strip.

The Tonnetz is an example of a torus in music theory.
The Tonnetz is only truly a torus if enharmonic equivalence is assumed, so that the (F♯-A♯) segment of the right edge of the repeated parallelogram is identified with the (G♭-B♭) segment of the left edge. If not, it is merely a cylinder.

As the n-torus is the n-fold product of the circle, the n-torus is the configuration space of n ordered, not necessarily distinct points on the circle. Symbolically, Tⁿ = (S¹)ⁿ. The configuration space of unordered, not necessarily distinct points is accordingly the orbifold Tⁿ/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 cross-section an equilateral triangle, with a twist; equivalently, as a triangular prism whose top and bottom faces are connected with a 1/3 twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge corresponds to points with all 3 coordinates identical.

These orbifolds have found significant applications to music theory in the work of Dmitri Tymoczko and collaborators (Felipe Posada and Michael Kolinas, et al.), being used to model musical triads.^[6][7]

Flat torus[edit]

In three dimensions, one can bend a rectangle into a torus, but doing this typically stretches the surface, as seen by the distortion of the checkered pattern

Seen in stereographic projection a 4D flat torus can be projected into 3-dimensions and rotated on a fixed axis

The flat torus is a torus with the metric inherited from its representation as the quotient, R²/L, where L is a discrete subgroup of R² isomorphic to Z². This gives the quotient the structure of a Riemannian manifold. Perhaps the simplest example of this is when L = Z² itself: R²/Z², which can also be described as the Cartesian plane under the identifications (x, y) ~ (x+1, y) ~ (x, y+1). This particular flat torus (and any uniformly scaled version of it as well) is known as the "square" flat torus.

This metric of the square flat torus can also be realised by specific embeddings of the familiar 2-torus into Euclidean 4-space 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 one 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 (unless you give up some regularity and differentiability conditions, see below).

A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows:

(x, y, z, w) = (R cosu, R sinu, P cosv, P sinv)

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 3-space. Mapping it into 3-space requires you to stretch 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)).

If R and P in the above flat torus form a unit vector = then u, v, and η can be used to parameterize the unit 3-sphere in a parameterization associated with the Hopf map. In particular, for certain very specific choices of a square flat torus in the 3-sphere S³, where η = π/4 above, the torus will partition the 3-sphere into two congruent solid tori subsets with the aforesaid flat torus surface as their common boundary. One example is the torus T defined by

          T  =  { (x,y,z,w) ∈ S3  |  x2 + y2 = 1/2,     z2 + w2 = 1/2 }.

Other tori in S³ having this partitioning property include the square tori of the form Q⋅T, where Q is a rotation of 4-dimensional space R⁴, or in other words Q is a member of the Lie group SO(4).

It is known that there exists no C² (twice continuously differentiable) flat torus into 3-space. (The idea of the proof is to take a large sphere containing such a flat torus in its interior, and shrink the radius of the sphere until it just touches the torus for the first time. Such a point of contact must be a tangency. But that would imply that part of the torus, since it has zero curvature everywhere, must lie strictly outside the sphere, which is a contradiction.) On the other hand, according to the Nash-Kuiper theorem showed in the 1950s that an isometric C¹ embedding exists. This is solely an existence proof, and does not provide explicit equations for such an embedding.

In April 2012, an explicit C¹ (continuously differentiable) embedding of a flat torus into 3-dimensional Euclidean space R³ was found.^{[8][9][10][11]} It is similar in structure to a fractal as it is constructed by repeatedly corrugating a normal torus. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined surface normals. It "is" a flat torus in the sense that as metric spaces, it is isometric to a flat square torus. (These infinitely recursive corrugations are used only for embedding into three dimensions; they are not an intrinsic feature of the flat torus.) This is the first time that any such embedding was defined by explicit equations, or depicted by computer graphics.

n-fold torus[edit]

In the theory of surfaces there is another object, the n-fold torus, often called the n-holed torus. Instead of the product of n circles, an n-fold torus is the connected sum of n 2-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. In this sense, an n-torus resembles the surface of n doughnuts stuck together side by side, or a 2-sphere with n handles attached.

An ordinary torus is a 1-fold torus, a 2-fold torus is called a double torus, a 3-fold torus a triple torus, and so on. The n-fold torus is said to be an "orientable surface" of "genus" n, the genus being the number of handles. The 0-fold torus is the 2-sphere.

The classification theorem for surfaces states that every compact connected surface is topologically equivalent to a) the sphere, or b) the n-fold torus with n > 0, or c) the connected sum of n projective planes (that is, projective planes over the real numbers) with n > 0.

double torus

triple torus

Toroidal polyhedra[edit]

A toroidal polyhedron with 6 × 4 = 24 quadrilateral faces.

Polyhedra with the topological type of a torus are called toroidal polyhedra, and satisfy a modified version of the Euler characteristic: V − E + F = 0. For any number holes, the formula generalizes to V − E + F = 2 − 2N, where N is the number of holes.

The term "toroidal polyhedron" is also used for higher-genus polyhedra and for immersions of toroidal polyhedra.

This section requires expansion. (April 2010)

Automorphisms[edit]

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 Rⁿ that preserve the standard lattice Zⁿ (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 the fundamental group, as these are all naturally isomorphic; also the first cohomology group generates the cohomology algebra):

Since the torus is an Eilenberg–MacLane 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 Homeo(Tⁿ) → SHE(Tⁿ) is 1-connected (isomorphic on path-components, 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 Rⁿ 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.

Coloring a torus[edit]

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.)

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

Cutting a torus[edit]

A standard torus (specifically, a ring torus) can be cut with n planes into at most

parts.^[12]

The initial terms of this sequence for n starting from 1 are:

2, 6, 13, 24, 40, … (sequence A003600 in OEIS).

See also[edit]

Algebraic torus Annulus (mathematics) Bagel Clifford torus Complex torus Doughnut Dupin cyclide Elliptic curve Irrational cable on a torus Joint European Torus Klein Bottle Loewner's torus inequality Maximal torus

Mathematics portal Period lattice Real projective plane Sphere Spiric section Surface Toric lens Toric section Toric variety Toroid Toroidal and poloidal Torus knot Umbilic torus Villarceau circles

Hechos 10