In geometry, a torus (: tori or toruses) 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 tori include ring tori, horn tori, and spindle tori. A ring torus is sometimes colloquially referred to as a 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 (or self-crossing torus or self-intersecting 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 from 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 (compare Klein bottle).
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 (and, hence, homeomorphic) 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 (rectifying) 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 (so that and ), 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.
Differential geometry for the ring torus Using the parametrization of the torus given at the beginning of the section, it is straight forward to compute various common objects of differential calculus for the ring torus because it is not a self-intersecting surface. The partial angular velocity fields are given as,
,
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The transpose of the Jacobian matrix, , is given by,
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The Jacobian matrix determinant, , can then be computed as the Euclidean norm (magnitude) of the cross product of with as, . The result is . From this expression, one can then compute the surface area, and the volume by integrating over the coordinates of the ring torus's defined ranges: , , , and .
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The usual differential operators of vector calculus can be calculated using the same parametrization to obtain their ring toroidal forms. For example, the Laplace operator for the ring torus, with the same variable ranges as for the integrals above, is given by the expression below.
The metric tensor can be calculated as . The result is,
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One can check that the square root of the determinant of the metric tensor is equal to the determinant of the Jacobi matrix, i.e. .
Topology
Topologically, a torus is a closed surface defined as the product of two circles: . This can be viewed as lying in 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 (with two degenerate circles), a fact that is important in the study of as a fiber bundle over (the Hopf bundle).
The surface described above, given the relative topology from , 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: Padgett, Adele (2014). "Fundamental groups: motivation, computation methods, and applications", REA Program, Uchicago. https://math.uchicago.edu/~may/REU2014/REUPapers/Padgett.pdf
Intuitively speaking, this means that a closed path that circles the torus's "hole" (say, a circle that traces out a particular latitude) and then circles the torus's "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. 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 (see below), 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 the 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 (with the toe cut off). 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. (This is the more typical meaning of the term "-torus", the other referring to holes or of genus .) 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 of the integer lattice (with the action being taken as vector addition). 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 (when identified with the unit complex numbers with multiplication). 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 (like tori) 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•(, Z) can be identified with the exterior algebra over the -module whose generators are the duals of the nontrivial cycles.
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