Torus


In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a sphere. If the revolved curve is not a circle, the surface is a related shape, a toroid.
Real-world objects that approximate a torus of revolution include swim rings and inner tubes. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses.
A torus should not be confused with a solid torus, which is formed by rotating a disk, rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. 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. It is a compact 2-manifold of genus 1. 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 S1 in the plane with itself. This produces a geometric object called the Clifford torus, a surface in 4-space.
In the field of topology, a torus is any topological space that is topologically equivalent to a torus. A coffee cup and a doughnut are both topological tori.
An example of a torus can be constructed by taking a rectangular strip of flexible material, for example, a rubber sheet, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists.

Geometry

A torus can be defined parametrically by:
where
is known as the "major radius" and is known as the "minor radius". The ratio divided by is known as the "aspect ratio". The typical doughnut confectionery has an aspect ratio of about 3 to 2.
An implicit equation in Cartesian coordinates for a torus radially symmetric about the -axis is
or the solution of, where
Algebraically eliminating the square root gives a quartic equation,
The three classes of standard tori correspond to the three possible aspect ratios between and :
When, the interior
of this torus is diffeomorphic to a product of a Euclidean open disk and a circle. The volume of this solid torus and the surface area of its torus are easily computed using Pappus's centroid theorem, giving:
These formulas are the same as for a cylinder of length and radius, obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.
Expressing the surface area and the volume by the distance of an outermost point on the surface of the torus to the center, and the distance of an innermost point, yields
As a torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used.
In traditional spherical coordinates there are three measures,, the distance from the center of the coordinate system, and and, angles measured from the center point.
As a torus has, effectively, two center points, the centerpoints of the angles are moved; measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of is moved to the center of, and is known as the "poloidal" direction. 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".
In modern use, toroidal and poloidal are more commonly used to discuss magnetic confinement fusion devices.

Topology

, a torus is a closed surface defined as the product of two circles: S1 × S1. This can be viewed as lying in C2 and is a subset of the 3-sphere S3 of radius square root of 2|. 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, a fact which is important in the study of S3 as a fiber bundle over S2.
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
or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA−1B−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" and then circles the torus' "body" can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. An equivalent statement may 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. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock. Additionally, if the cylinder was made by gluing two opposite sides of a rectangle together, choosing the other two sides instead will cause the same reversal of orientation.
The first homology group of the torus is isomorphic to the fundamental group.

Two-sheeted cover

The 2-torus double-covers the 2-sphere, with four ramification points. Every conformal structure on the 2-torus can be represented as a two-sheeted cover of the 2-sphere. 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 cross-ratio of the four points.

''n''-dimensional torus

The torus has a generalization to higher dimensions, the n-dimensional torus, often called the n-torus or hypertorus for short. Recalling that the torus is the product space of two circles, the n-dimensional torus is the product of n circles. That is:
The 1-torus is just the circle: T1 = S1. The torus discussed above is the 2-torus, T2. And similar to the 2-torus, the n-torus, Tn 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. 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. 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. Toroidal groups are examples of protori, which are compact connected abelian groups, which are not required to be manifolds.
Automorphisms of T are easily constructed from automorphisms of the lattice Zn, which are classified by invertible integral matrices of size n with an integral inverse; these are just the integral matrices with determinant ±1. Making them 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• can be identified with the exterior algebra over the Z-module Zn whose generators are the duals of the n nontrivial cycles.

Configuration space

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 = n. The configuration space of unordered, not necessarily distinct points is accordingly the orbifold Tn/Sn, which is the quotient of the torus by the symmetric group on n letters.
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 : 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, being used to model musical triads.

Flat torus

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 : R2/Z2, which can also be described as the Cartesian plane under the identifications. This particular flat torus 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.
A simple 4-dimensional Euclidean embedding of a rectangular flat torus is as follows:
where R and P are constants determining the aspect ratio. It is diffeomorphic to a regular torus but not isometric. It can not be analytically 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
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 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
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.
It is known that there exists no C2 embedding of a flat torus into 3-space. On the other hand, according to the Nash-Kuiper theorem, proven in the 1950s, 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 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. 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. This is the first time that any such embedding was defined by explicit equations, or depicted by computer graphics.

Genus ''g'' surface

In the theory of surfaces there is another object, the "genus" g surface. Instead of the product of n circles, a genus g surface is the connected sum of g two-tori. To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the 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, a genus g surface resembles the surface of g doughnuts stuck together side by side, or a 2-sphere with g handles attached.
As examples, a genus zero surface is the two-sphere while a genus one surface is the ordinary torus. The surfaces of higher genus are sometimes called n-holed tori. The terms double torus and triple torus are also occasionally used.
The classification theorem for surfaces states that every compact connected surface is topologically equivalent to either the sphere or the connect sum of some number of tori, disks, and real projective planes.

genus two
genus three

Toroidal polyhedra

with the topological type of a torus are called toroidal polyhedra, and have Euler characteristic V − E + F = 0. For any number of 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.

Automorphisms

The homeomorphism group of the torus is studied in geometric topology. Its mapping class group is isomorphic to the group GL of invertible integer matrices, and can be realized as linear maps on the universal covering space Rn that preserve the standard lattice Zn 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 ; 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. More tersely, the map Homeo → SHE is 1-connected. This is a "homeomorphism reduces to homotopy reduces to algebra" result.
Thus the short exact sequence of the mapping class group splits :
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

The torus's Heawood number is seven, meaning every graph that can be embedded on the torus has a chromatic number of at most seven. Equivalently, in a torus divided into regions, it is always possible to color the regions using no more than seven colors so that no neighboring regions are the same color.

Cutting a torus

A solid torus of revolution can be cut by n planes into maximally
parts.
The first 11 numbers of parts, for 0 ≤ n ≤ 10, are as follows: