Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut.
If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus. 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 double-covered sphere. If the revolved curve is not a circle, the surface is called a toroid, as in a square toroid.
Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings.
A torus is different than 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:, which is sometimes used as the definition. 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 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 homeomorphic to a torus. The surface of a coffee cup and a doughnut are both topological tori with genus one.
An example of a torus can be constructed by taking a rectangular strip of flexible material such as rubber, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists.

Etymology

Torus is a Latin word denoting something round, a swelling, an elevation, a protuberance.

Geometry

A torus of revolution in 3-space can be parametrized as:
using angular coordinates, representing rotation around the tube and rotation around the torus's axis of revolution, respectively, where the major radius is the distance from the center of the tube to the center of the torus and the minor radius is the radius of the tube.
The ratio is called the aspect ratio of the torus. 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 z-axis is
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 surface will be the familiar ring torus or anchor ring.
corresponds to the horn torus, which in effect is a torus with no "hole".
describes the self-intersecting spindle torus; its inner shell is a lemon and its outer shell is an apple.
When, the torus degenerates to the sphere radius.
When, the torus degenerates to the circle radius.

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 formulae 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 to the center, 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:. This can be viewed as lying in [complex coordinate space|] and is a subset of the 3-sphere of radius. This topological torus is also often called the Clifford torus. In fact, is filled out by a family of nested tori in this manner, a fact that is important in the study of as a fiber bundle over .
The surface described above, given the relative topology from [real coordinate space|], 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 from the north pole of.
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.
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's "hole" and then circles the torus's "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.
The fundamental group can also be derived from taking the torus as the quotient , so that may be taken as its universal cover, with deck transformation group.
Its higher homotopy groups are all trivial, since a universal cover projection always induces isomorphisms between the groups and for, and is contractible.
The torus has homology groups:
Thus, the first homology group of the torus is isomorphic to its fundamental group-- which in particular can be deduced from Hurewicz theorem since is abelian.
The cohomology groups with integer coefficients are isomorphic to the homology ones-- which can be seen either by direct computation, the universal coefficient theorem or even Poincaré duality.
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.

Two-sheeted cover

The 2-torus is a twofold branched cover of the 2-sphere, with four ramification points. Every conformal structure on the 2-torus can be represented as such 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, often called the or for short. Just as the ordinary torus is topologically the product space of two circles, the -dimensional torus is topologically equivalent to the product of circles. That is:
The standard 1-torus is just the circle:. The torus discussed above is the standard 2-torus,. And similar to the 2-torus, the -torus, can be described as a quotient of under integral shifts in any coordinate. That is, the n-torus is modulo the action (mathematics)|action] of the integer lattice . Equivalently, the -torus is obtained from the -dimensional hypercube by gluing the opposite faces together.
An -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 one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension. Such maximal tori have a controlling role to play in theory of connected. Toroidal groups are examples of protori, which are compact connected abelian groups, which are not required to be manifolds.
Automorphisms of are easily constructed from automorphisms of the lattice, which are classified by invertible integral matrices of size with an integral inverse; these are just the integral matrices with determinant. Making them act on 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 . The th homology group of an -torus is a free abelian group of rank n choose . It follows that the Euler characteristic of the -torus is for all . The cohomology ring H^• can be identified with the exterior algebra over the -module whose generators are the duals of the nontrivial cycles.

Configuration space

As the -torus is the -fold product of the circle, the -torus is the configuration space of ordered, not necessarily distinct points on the circle. Symbolically,. The configuration space of unordered, not necessarily distinct points is accordingly the orbifold, which is the quotient of the torus by the symmetric group on letters.
For, the quotient is the Möbius strip, the edge corresponding to the orbifold points where the two coordinates coincide. For 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

A flat torus is a torus with the metric inherited from its representation as the quotient,, where is a discrete subgroup of isomorphic to. This gives the quotient the structure of a Riemannian manifold, as well as the structure of an abelian Lie group. Perhaps the simplest example of this is when :, 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. It 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 this cylinder cannot be bent 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 positive 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 one to stretch it, in which case it looks like a regular torus. For example, in the following map:
If and in the above flat torus parametrization form a unit vector then u, v, and parameterize the unit 3-sphere as Hopf coordinates. 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 defined by
Other tori in having this partitioning property include the square tori of the form, where is a rotation of 4-dimensional space, or in other words is a member of the Lie group.
It is known that there exists no embedding of a flat torus into 3-space. On the other hand, according to the Nash-Kuiper theorem, which was 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 isometric embedding of a flat torus into 3-dimensional Euclidean space was found. It is a flat torus in the sense that, as a metric space, it is isometric to a flat square torus. It is similar in structure to a fractal as it is constructed by repeatedly corrugating an ordinary torus at smaller scales. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined surface normals, yielding a so-called "smooth fractal". The key to obtaining the smoothness of this corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their "wavelengths". This is the first time that any such embedding was defined by explicit equations or depicted by computer graphics.

Conformal classification of flat tori

In the study of Riemann surfaces, one says that any two smooth compact geometric surfaces are "conformally equivalent" when there exists a smooth homeomorphism between them that is both angle-preserving and orientation-preserving. The Uniformization theorem guarantees that every Riemann surface is conformally equivalent to one that has constant Gaussian curvature. In the case of a torus, the constant curvature must be zero. Then one defines the "moduli space" of the torus to contain one point for each conformal equivalence class, with the appropriate topology. It turns out that this moduli space M may be identified with a punctured sphere that is smooth except for two points that have less angle than 2π around them: One has total angle π and the other has total angle 2π/3.
M may be turned into a compact space M* – topologically equivalent to a sphere – by adding one additional point that represents the limiting case as a rectangular torus approaches an aspect ratio of 0 in the limit. The result is that this compactified moduli space is a sphere with three points each having less than 2π total angle around them. This third conepoint will have zero total angle around it. Due to symmetry, M* may be constructed by glueing together two congruent geodesic triangles in the hyperbolic plane along their boundaries, where each triangle has angles of,, and. As a result, the Gauss–Bonnet theorem shows that the area of each triangle can be calculated as, so it follows that the compactified moduli space M* has area equal to.
The other two cusps occur at the points corresponding in M* to the square torus and the hexagonal torus. These are the only conformal equivalence classes of flat tori that have any conformal automorphisms other than those generated by translations and negation.

Genus ''g'' surface

In the theory of surfaces there is a more general family of objects, the "genus" surfaces. A genus surface is the connected sum of 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, successively take the connected sum of two of them at a time until they are all connected. In this sense, a genus surface resembles the surface of doughnuts stuck together side by side, or a 2-sphere with 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 -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. For any number of holes, the formula generalizes to, where is the topological genus.
Toroidal polyhedra have been used to show that the maximum number of colors to color a map on a torus is seven. The Szilassi polyhedron is one example of a toroidal polyhedron with this property.
The Szilassi polyhedron's dual, the Császár polyhedron, is the only polyhedron other than the tetrahedron which has the property that every possible edge connecting two vertices is an edge of the polyhedron.
The term "toroidal polyhedron" is also used for higher-genus polyhedra and for immersions of toroidal polyhedra, although some authors only include those with genus 1.
Self-crossing toroidal polyhedra are determined by the topology of their abstract manifold. One subset of the self-crossing toroidal polyhedra are the crown polyhedra, which are the only toroidal polyhedra that are also noble.

Automorphisms

The homeomorphism group of the torus is studied in geometric topology. Its mapping class group is surjective onto the group of invertible integer matrices, which can be realized as linear maps on the universal covering space that preserve the standard lattice 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 ; all homotopy equivalences of the torus can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism.
Thus the short exact sequence of the mapping class group splits :
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.

de Bruijn torus

In combinatorial mathematics, a de Bruijn torus is an array of symbols from an alphabet that contains every -by- matrix exactly once. It is a torus because the edges are considered wraparound for the purpose of finding matrices. Its name comes from the De Bruijn sequence, which can be considered a special case where is 1.

Cutting a torus

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