In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis 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.
Real-world examples of toroidal objects include inner tubes. A torus should not be confused with a solid torus, which is formed by rotating a disc, rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. Real-world approximations include doughnuts, non-inflatable lifebuoys, and O-rings.
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.
R is known as the “major radius” and r is known as the “minor radius”. The ratio R divided by r is known as the “aspect ratio”. The typical doughnut confectionery has an aspect ratio of about 3 to 2.
or the solution of f(x, y, z) = 0, where
The three different classes of standard tori correspond to the three possible aspect ratios between R and r:
When R ≥ r, the interior
of this torus is diffeomorphic (and, hence, homeomorphic) to a product of a Euclidean open disc 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 2πR and radius r, 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 p of an outermost point on the surface of the torus to the center, and the distance q of an innermost point (so that R = p + q/ and r = p − q/), 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, R, the distance from the center of the coordinate system, and θ and φ, angles measured from the center point.
As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in the spherical system, but is known as the “toroidal” direction. The center point of θ is moved to the center of r, 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 these terms are more commonly used to discuss magnetic confinement fusion devices.
Topologically, 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 √. 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 (with two degenerate circles), a fact which is important in the study of S3 as a fiber bundle over S2 (the Hopf bundle).
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.
Intuitively speaking, this means that a closed path that circles the torus’ “hole” (say, a circle that traces out a particular latitude) and then circles the torus’ “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.
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 different 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.
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.
The torus has a generalization to higher dimensions, the n-dimensional torus, often called the n-torus or hypertorus for short. (This is one of two different meanings of the term “n-torus”.) 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 (with the action being taken as vector addition). 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 (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 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 (like tori) 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•(Tn, Z) can be identified with the exterior algebra over the Z-module Zn whose generators are the duals of the n nontrivial cycles.
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/Sn, 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.
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: 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) 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:
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 (smooth of class Ck, 2 ≤ k ≤ ∞) 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 (R, P) = (cos(η), sin(η)) 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
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) embedding of a 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, 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 (continuously differentiable) 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 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.
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 (without boundary) is the two-sphere while a genus one surface (without boundary) is the ordinary torus. The surfaces of higher genus are sometimes called n-holed tori (or, rarely, n-fold 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.
Polyhedra with the topological type of a torus are called toroidal polyhedra, and have 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.
The homeomorphism group (or the subgroup of diffeomorphisms) of the torus is studied in geometric topology. Its mapping class group ( the connected components of the homeomorphism group) 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.
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. (Since the complete graph can be embedded on the torus, and , the upper bound is tight.) 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. (Contrast with the four color theorem for the plane.)
A solid torus of revolution can be cut by n (> 0) planes into maximally
The first 11 numbers of parts, for 0 ≤ n ≤ 10 (including the case of n = 0, not covered by the above formulas), are as follows:
The brow ridge, or supraorbital ridge known as superciliary arch in medicine, refers to a bony ridge located above the eye sockets of all primates. In Homo sapiens sapiens (modern humans) the eyebrows are located on their lower margin.Bumpy torus
The bumpy torus is a class of magnetic fusion energy devices that consist of a series of magnetic mirrors connected end-to-end to form a closed torus. Such an arrangement is not stable on its own, and most bumpy torus designs use secondary fields or relativistic electrons to create a stable field inside the reactor. The main disadvantage of magnetic mirror confinement, that of excessive plasma leakage, is circumvented by the arrangement of multiple mirrors end-to-end in a ring. It is described as "bumpy" because the fuel ions comprising the plasma tend to concentrate inside the mirrors at greater density than the leakage currents between mirror cells.
Bumpy torus designs were an area of active research starting in the 1960s and continued until 1986 with the ELMO (ELectro Magnetic Orbit) Bumpy Torus at the Oak Ridge National Laboratory. One in particular has been described: "Imagine a series of magnetic mirror machines placed end to end and twisted into a torus. An ion or electron that leaks out of one mirror cavity finds itself in another mirror cell. This constitutes a bumpy torus." These demonstrated problems and most research on the concept has ended.Carrick mat
The carrick mat is a flat woven decorative knot which can be used as a mat or pad. Its name is based on the mat's decorative-type carrick bend with the ends connected together, forming an endless knot. A larger form, called the prolong knot, is made by expanding the basic carrick mat by extending, twisting, and overlapping its outer bights, then weaving the free ends through them. This process may be repeated to produce an arbitrarily long mat.In its basic form it is the same as a 3-lead, 4-bight Turk's head knot. The basic carrick mat, made with two passes of rope, also forms the central motif in the logo of the International Guild of Knot Tyers.When tied to form a cylinder around the central opening, instead of lying flat, it can be used as a woggle.Cylinder chess
Cylinder chess (or cylindrical chess) is a chess variant with an unusual board. The game is played as if the board were a cylinder, with the left side of the board joined to the right side. According to Bill Wall, in 947 in a history of chess in India and Persia, the Arabic historian Ali al-Masudi described six different variants of chess, including astrological chess, circular chess and cylinder chess.Cylindrical board is also used in chess problems.Double torus knot
In knot theory, a double torus knot is a closed curve drawn on the surface called a double torus (think of the surface of two doughnuts stuck together). More technically, a double torus knot is the homeomorphic image of a circle in S³ which can be realized as a subset of a genus two handlebody in S³. If a link is a subset of a genus two handlebody, it is a double torus link.The simplest example of a double torus knot that is not a torus knot is the figure-eight knot.
While torus knots and links are well understood and completely classified, there are many open questions about double torus knots.
Two different notations exist for describing double torus knots. The T/I notation is given in F. Norwood, "Curves on Surfaces", and a different notation is given in P. Hill, "On double-torus knots (I)". The big problem, solved in the case of the torus, still open in the case of the double torus, is: when do two different notations describe the same knot?Greenstick fracture
A greenstick fracture is a fracture in a young, soft bone in which the bone bends and breaks. Greenstick fractures usually occur most often during infancy and childhood when bones are soft. The name is by analogy with green (i.e., fresh) wood which similarly breaks on the outside when bent. It was discovered by British-American orthopedist, John Insall, and Polish-American orthopedist, Michael Slupecki.Joint European Torus
JET, the Joint European Torus, is the world's largest operational magnetically confined plasma physics experiment, located at Culham Centre for Fusion Energy in Oxfordshire, UK. Based on a tokamak design, the fusion research facility is a joint European project with a main purpose of opening the way to future nuclear fusion grid energy.
JET was one of a number of tokamak reactors built in the early 1980s that tested new design concepts. It was one of only two designed to work with a real deuterium-tritium fuel mix, the other being the US-built TFTR. Both were built with the hope of reaching breakeven, the point where the energy created by the fusion reactions is greater than the energy being fed into it to keep it hot.
JET began operation in 1983 and spent most of the next decade increasing its performance in a lengthy series of experiments and upgrades. In 1991 the first experiments including tritium were made, making JET the first reactor in the world to run on the production fuel of a 50-50 mix of tritium and deuterium. In 1997, using this fuel, JET set the current world record for fusion output at 16 MW from an input of 24 MW of heating and a total input of 700-800 MW of electrical power. This is also the world record for Q, at 0.67. A Q of 1 is scientific breakeven, a point JET was not ultimately able to reach.
JET was built by an international consortium, which formed the nucleus for the European Union's contribution to the International Thermonuclear Experimental Reactor. JET has been described as a "little ITER" as the designs are very similar in many ways. In recent years, JET has been used to test a number of features from the ITER design.KK Torus
KK Torus (Macedonian: КК Торус) is a defunct basketball club based in Skopje, North Macedonia. They played in the Macedonian First League and the Balkan League until the season 2011/2012.Klein bottle
In topology, a branch of mathematics, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary (for comparison, a sphere is an orientable surface with no boundary).
The Klein bottle was first described in 1882 by the German mathematician Felix Klein. It may have been originally named the Kleinsche Fläche ("Klein surface") and then misinterpreted as Kleinsche Flasche ("Klein bottle"), which ultimately may have led to the adoption of this term in the German language as well.List of moments of inertia
In physics and applied mathematics, the mass moment of inertia, usually denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. Mass moments of inertia have units of dimension ML2([mass] × [length]2). It should not be confused with the second moment of area, which is used in beam calculations. The mass moment of inertia is often also known as the rotational inertia, and sometimes as the angular mass.
For simple objects with geometric symmetry, one can often determine the moment of inertia in an exact closed-form expression. Typically this occurs when the mass density is constant, but in some cases the density can vary throughout the object as well. In general, it may not be straightforward to symbolically express the moment of inertia of shapes with more complicated mass distributions and lacking symmetry. When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems.
This article mainly considers symmetric mass distributions, with constant density throughout the object, and the axis of rotation is taken to be through the center of mass unless otherwise specified.National Spherical Torus Experiment
The National Spherical Torus Experiment (NSTX) is a magnetic fusion device based on the spherical tokamak concept. It was constructed by the Princeton Plasma Physics Laboratory (PPPL) in collaboration with the Oak Ridge National Laboratory, Columbia University, and the University of Washington at Seattle.
The spherical tokamak (ST) is an offshoot of the conventional tokamak design. Proponents claim that it has a number of practical advantages over these devices, some of them dramatic. For this reason the ST has seen considerable interest since it was proposed in the late 1980s. However, development remains effectively one generation behind mainline efforts such as JET. Other major experiments in the field include the pioneering START and MAST at Culham in the UK.
NSTX studies the physics principles of spherically shaped plasmas—hot ionized gases in which nuclear fusion will occur under the appropriate conditions of temperature and density, which are produced by confinement in a magnetic field.Stanford torus
The Stanford torus is a proposed NASA design for a space habitat capable of housing 10,000 to 140,000 permanent residents.The Stanford torus was proposed during the 1975 NASA Summer Study, conducted at Stanford University, with the purpose of exploring and speculating on designs for future space colonies (Gerard O'Neill later proposed his Island One or Bernal sphere as an alternative to the torus). "Stanford torus" refers only to this particular version of the design, as the concept of a ring-shaped rotating space station was previously proposed by Wernher von Braun and Herman Potočnik.It consists of a torus, or doughnut-shaped ring, that is 1.8 km in diameter (for the proposed 10,000 person habitat described in the 1975 Summer Study) and rotates once per minute to provide between 0.9g and 1.0g of artificial gravity on the inside of the outer ring via centrifugal force.Sunlight is provided to the interior of the torus by a system of mirrors, including a large non-rotating primary solar mirror.
The ring is connected to a hub via a number of "spokes", which serve as conduits for people and materials travelling to and from the hub. Since the hub is at the rotational axis of the station, it experiences the least artificial gravity and is the easiest location for spacecraft to dock. Zero-gravity industry is performed in a non-rotating module attached to the hub's axis.The interior space of the torus itself is used as living space, and is large enough that a "natural" environment can be simulated; the torus appears similar to a long, narrow, straight glacial valley whose ends curve upward and eventually meet overhead to form a complete circle. The population density is similar to a dense suburb, with part of the ring dedicated to agriculture and part to housing.Tokamak
A tokamak (Russian: Токамáк) is a device which uses a powerful magnetic field to confine a hot plasma in the shape of a torus. The tokamak is one of several types of magnetic confinement devices being developed to produce controlled thermonuclear fusion power. As of 2016, it is the leading candidate for a practical fusion reactor.Tokamaks were initially conceptualized in the 1950s by Soviet physicists Igor Tamm and Andrei Sakharov, inspired by a letter by Oleg Lavrentiev. Meanwhile, the first working tokamak was attributed to the work of Natan Yavlinskii on the T-1. It had been demonstrated that a stable plasma equilibrium requires magnetic field lines that wind around the torus in a helix. Devices like the z-pinch and stellarator had attempted this, but demonstrated serious instabilities. It was the development of the concept now known as the safety factor (labelled q in mathematical notation) that guided tokamak development; by arranging the reactor so this critical factor q was always greater than 1, the tokamaks strongly suppressed the instabilities which plagued earlier designs.
The first tokamak, the T-1, began operation in 1958. By the mid-1960s, the tokamak designs began to show greatly improved performance. Initial results were released in 1965, but were ignored; Lyman Spitzer dismissed them out of hand after noting potential problems in their system for measuring temperatures. A second set of results was published in 1968, this time claiming performance far in advance of any other machine, and was likewise considered unreliable. This led to the invitation of a delegation from the United Kingdom to make their own measurements. These confirmed the Soviet results, and their 1969 publication resulted in a stampede of tokamak construction.
By the mid-1970s, dozens of tokamaks were in use around the world. By the late 1970s, these machines had reached all of the conditions needed for practical fusion, although not at the same time nor in a single reactor. With the goal of breakeven now in sight, a new series of machines were designed that would run on a fusion fuel of deuterium and tritium. These machines, notably the Joint European Torus (JET), Tokamak Fusion Test Reactor (TFTR) and JT-60, had the explicit goal of reaching breakeven.
Instead, these machines demonstrated new problems that limited their performance. Solving these would require a much larger and more expensive machine, beyond the abilities of any one country. After an initial agreement between Ronald Reagan and Mikhail Gorbachev in November 1985, the International Thermonuclear Experimental Reactor (ITER) effort emerged and remains the primary international effort to develop practical fusion power. Many smaller designs, and offshoots like the spherical tokamak, continue to be used to investigate performance parameters and other issues.Torus (Merzbow EP)
Torus is an EP by the Japanese noise musician Merzbow. It was released by Jezgro on June 12, 2017. The track "Torus 2" was premiered by The Brvtalist on June 10, 2017.Torus knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q are not coprime (in which case the number of components is gcd(p, q)). A torus knot is trivial (equivalent to the unknot) if and only if either p or q is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.Torus mandibularis
Torus mandibularis is a bony growth in the mandible along the surface nearest to the tongue. Mandibular tori are usually present near the premolars and above the location of the mylohyoid muscle's attachment to the mandible. In 90% of cases, there is a torus on both the left and right sides, making this finding a predominantly bilateral condition.
The prevalence of mandibular tori ranges from 5% - 40%. It is less common than bony growths occurring on the palate, known as torus palatinus. Mandibular tori are more common in Asian and Inuit populations, and slightly more common in males. In the United States, the prevalence is 7% - 10% of the population.
It is believed that mandibular tori are caused by several factors. They are more common in early adult life and are associated with bruxism. The size of the tori may fluctuate throughout life, and in some cases the tori can be large enough to touch each other in the midline of mouth. Consequently, it is believed that mandibular tori are the result of local stresses and not due solely to genetic influences.
Mandibular tori are usually a clinical finding with no treatment necessary. It is possible for ulcers to form in the area of the tori due to trauma. The tori may also complicate the fabrication of dentures. If removal of the tori is needed, surgery can be done to reduce the amount of bone, but the tori may reform in cases where nearby teeth still receive local stresses.Torus palatinus
Torus palatinus (pl. tori palatini) [palatinus torus (pl. palatal tori) in English] is a bony protrusion on the palate. Palatal tori are usually present on the midline of the hard palate. Most palatal tori are less than 2 cm in diameter, but their size can change throughout life.
Prevalence of palatal tori ranges from 9–60% and are more common than bony growths occurring on the mandible, known as torus mandibularis. Palatal tori are more common in Asian, Native American and Inuit populations, and are twice as common in females. In the United States, the prevalence is 20% - 35% of the population, with similar findings between black and white people.
Although some research suggest palatal tori to be an autosomal dominant trait, it is generally believed that palatal tori are caused by several factors. They are more common in early adult life and can increase in size. In some older people, the size of the tori may decrease due to bone resorption. It is believed that tori of the lower jaw are the result of local stresses and not due solely to genetic influences.
Sometimes, the tori are categorized by their appearance. Arising as a broad base and a smooth surface, flat tori are located on the midline of the palate and extend symmetrically to either side. Spindle tori have a ridge located at their midline. Nodular tori have multiple bony growths that each have their own base. Lobular tori have multiple bony growths with a common base.
Palatal tori are usually a clinical finding with no treatment necessary. It is possible for ulcers to form on the area of the tori due to repeated trauma. Also, the tori may complicate the fabrication of dentures. If removal of the tori is needed, surgery can be done to reduce the amount of bone present.Torus tubarius
The base of the cartilaginous portion of the auditory tube (eustachian tube, pharyngotympanic tube) lies directly under the mucous membrane of the nasal part of the pharynx, where it forms an elevation, the torus tubarius, the torus of the auditory tube, or cushion, behind the pharyngeal orifice of the tube. The torus tubarius is very close to the tubal tonsil, which is sometimes also called the tonsil of (the) torus tubarius. Equating the torus with its tonsil however might be seen as incorrect or imprecise.
Two folds run posteriorly and anteriorly:
posteriorly, the vertical fold of mucous membrane, the salpingopharyngeal fold, stretches from the lower part of the torus tubarius; it contains the Salpingopharyngeus muscle which originates from the superior border of the medial lamina of the cartilage of the auditory tube, and passes downward and blends with the posterior fasciculus of the palatopharyngeus muscle.
anteriorly, the second and smaller fold, the salpingopalatine fold, smaller than the salpingopharyngeal fold, contains some fibers of muscle, called salpingopalatine muscle by Simkins (1943), it stretches from the superior border of lateral lamina of the cartilage, anteroinferiorly, to the back of the hard palate. The tensor veli palatini does not contribute to the fold, since the origin is deep to the cartilaginous opening.Trefoil knot
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory.
The trefoil knot is named after the three-leaf clover (or trefoil) plant.