Torus

If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere.

Real-world objects that approximate a solid torus include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels.

In topology, a ring torus is homeomorphic to the Cartesian product of two circles: S1 × S1, and the latter is taken to be the definition in that context.

An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is

of this torus is diffeomorphic (and, hence, homeomorphic) to a product of a Euclidean open disk and a circle.

As a torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used.

These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles".

[5] In modern use, toroidal and poloidal are more commonly used to discuss magnetic confinement fusion devices.

(This is the more typical meaning of the term "n-torus", the other referring to n holes or of genus n.[8]) Just as the ordinary torus is topologically the product space of two circles, the n-dimensional torus is topologically equivalent to the product of n circles.

And similar to the 2-torus, the n-torus, Tn can be described as a quotient of Rn under integral shifts in any coordinate.

Equivalently, the n-torus is obtained from the n-dimensional hypercube by gluing the opposite faces together.

Making them act on Rn in the usual way, one has the typical toral automorphism on the quotient.

, Z) can be identified with the exterior algebra over the Z-module Zn whose generators are the duals of the n nontrivial cycles.

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.

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.

A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows: where R and P are positive constants determining the aspect ratio.

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.)

In April 2012, an explicit C1 (continuously differentiable) isometric embedding of a flat torus into 3-dimensional Euclidean space R3 was found.

It is similar in structure to a fractal as it is constructed by repeatedly corrugating an ordinary torus at smaller scales.

[15] (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.

The Uniformization theorem guarantees that every Riemann surface is conformally equivalent to one that has constant Gaussian curvature.

Then one defines the "moduli space" of the torus to contain one point for each conformal equivalence class, with the appropriate topology.

The result is that this compactified moduli space is a sphere with three points each having less than 2π total angle around them.

(Such a point is termed a "cusp", and may be thought of as the vertex of a cone, also called a "conepoint".)

The homeomorphism group (or the subgroup of diffeomorphisms) of the torus is studied in geometric topology.

of invertible integer matrices, which can be realized as linear maps on the universal covering space

Thus the short exact sequence of the mapping class group splits (an identification of the torus as the quotient of

In combinatorial mathematics, a de Bruijn torus is an array of symbols from an alphabet (often just 0 and 1) that contains every m-by-n matrix exactly once.