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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a torus (plural tori, colloquially donut or doughnut) is a
surface of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
generated by revolving a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
about an axis that is
coplanar In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. How ...
with the circle. If the
axis of revolution In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the '' axis of revolution'') that lies on the same plane. The surface created by this revolution and which bounds the solid is th ...
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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
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 A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
. If the revolved curve is not a circle, the surface is called a ''
toroid In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...
'', as in a square toroid. Real-world objects that approximate a torus of revolution include swim rings,
inner tube An inner tube is an inflatable ring that forms the interior of some pneumatic tires. The tube is inflated with a valve stem, and fits inside of the casing of the tire. The inflated inner tube provides structural support and suspension, while the ...
s and ringette rings. Eyeglass lenses that combine spherical and cylindrical correction are
toric lens A toric lens is a lens with different optical power and focal length in two orientations perpendicular to each other. One of the lens surfaces is shaped like a "cap" from a torus (see figure at right), and the other one is usually spherical. Such ...
es. 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 Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
inside the torus. Real-world objects that approximate a ''solid torus'' include
O-ring An O-ring, also known as a packing or a toric joint, is a mechanical gasket in the shape of a torus; it is a loop of elastomer with a round cross-section, designed to be seated in a groove and compressed during assembly between two or more par ...
s, non-inflatable
lifebuoy A lifebuoy is a life-saving buoy designed to be thrown to a person in water, to provide buoyancy and prevent drowning. Some modern lifebuoys are fitted with one or more seawater-activated lights, to aid rescue at night. Other names Other nam ...
s, ring doughnuts, and
bagel A bagel ( yi, בײגל, translit=beygl; pl, bajgiel; also spelled beigel) is a bread roll originating in the Jewish communities of Poland. It is traditionally shaped by hand into a roughly hand-sized ring from yeasted wheat dough that is first ...
s. In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a ring torus is homeomorphic to the Cartesian product of two
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
s: S^\times S^, 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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, but another way to do this is the Cartesian product of the
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is g ...
of S^ in the plane with itself. This produces a geometric object called the Clifford torus, a surface in 4-space. In the field of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, 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 Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
one. 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 (compare Möbius strip).


Geometry

A torus can be defined
parametrically A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
by: \begin x(\theta, \varphi) &= (R + r \cos \theta) \cos\\ y(\theta, \varphi) &= (R + r \cos \theta) \sin\\ z(\theta, \varphi) &= r \sin \theta\\ \text~ \theta, \varphi \in [0,2\pi) \end where *, are angles which make a full circle, so their values start and end at the same point, * is the distance from the center of the tube to the center of the torus, * is the radius of the tube. Angle represents rotation around the tube, whereas represents rotation around the torus' axis of revolution. 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 function, implicit equation in Cartesian coordinates for a torus radially symmetric about the -coordinate axis, axis is \left(\sqrt - R\right)^2 + z^2 = r^2, or the solution of , where f(x,y,z) = \left(\sqrt - R\right)^2 + z^2 - r^2. Algebraically eliminating the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
gives a
quartic equation In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomi ...
, \left(x^2 + y^2 + z^2 + R^2 - r^2\right)^2 = 4R^2\left(x^2+y^2\right). 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 The lemon (''Citrus limon'') is a species of small evergreen trees in the flowering plant family Rutaceae, native to Asia, primarily Northeast India (Assam), Northern Myanmar or China. The tree's ellipsoidal yellow fruit is used for culin ...
'' and its outer shell is an ''
apple An apple is an edible fruit produced by an apple tree (''Malus domestica''). Apple trees are cultivated worldwide and are the most widely grown species in the genus ''Malus''. The tree originated in Central Asia, where its wild ancestor, ' ...
'' *When , the torus degenerates to the sphere. When , the interior \left(\sqrt - R\right)^2 + z^2 < r^2 of this torus is
diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two man ...
(and, hence, homeomorphic) to a
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of a Euclidean open disk and a circle. The
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
of this solid torus and the surface area of its torus are easily computed using
Pappus's centroid theorem In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The ...
, giving: \begin A &= \left( 2\pi r \right) \left(2 \pi R \right) = 4 \pi^2 R r \\ V &= \left ( \pi r^2 \right ) \left( 2 \pi R \right) = 2 \pi^2 R r^2. \end 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 (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 \begin A &= 4 \pi^2 \left(\frac\right) \left(\frac\right) = \pi^2 (p+q) (p-q) \\ V &= 2 \pi^2 \left(\frac\right) \left(\frac\right)^2 = \tfrac14 \pi^2 (p+q) (p-q)^2 \end 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 Toroidal describes something which resembles or relates to a torus or toroid: Mathematics *Torus *Toroid, a surface of revolution which resembles a torus *Toroidal polyhedron *Toroidal coordinates, a three-dimensional orthogonal coordinate system ...
are more commonly used to discuss
magnetic confinement fusion Magnetic confinement fusion is an approach to generate thermonuclear fusion power that uses magnetic fields to confine fusion fuel in the form of a plasma. Magnetic confinement is one of two major branches of fusion energy research, along with ...
devices.


Topology

Topologically, a torus is a
closed surface In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as g ...
defined as the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of two
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
s: ''S''1 × ''S''1. This can be viewed as lying in C2 and is a subset of the
3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
''S''3 of radius √2. This topological torus is also often called the Clifford torus. In fact, ''S''3 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 ''S''3 as a fiber bundle over ''S''2 (the Hopf bundle). The surface described above, given the relative topology from \mathbb^, 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 \mathbb^ from the north pole of ''S''3. The torus can also be described as a
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
under the identifications :(x,y) \sim (x+1,y) \sim (x,y+1), \, or, equivalently, as the quotient of the
unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordin ...
by pasting the opposite edges together, described as a
fundamental polygon In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal eq ...
''ABA''−1''B''−1. The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself: :\pi_1(\mathbb^2) = \pi_1(\mathbb^1) \times \pi_1(\mathbb^1) \cong \mathbb \times \mathbb. 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 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 first
homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian).


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 In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, th ...
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 "''n''-torus", the other referring to ''n'' holes or of genus ''n''.) Recalling that the torus is the product space of two circles, the ''n''-dimensional torus is the product of ''n'' circles. That is: :\mathbb^n = \underbrace_n. The standard 1-torus is just the circle: \mathbb^=\mathbb^ . The torus discussed above is the standard 2-torus, \mathbb^2. And similar to the 2-torus, the ''n''-torus, \mathbb^ can be described as a quotient of \mathbb^ under integral shifts in any coordinate. That is, the ''n''-torus is \mathbb^ modulo the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of the integer
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
\mathbb^ (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 In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
is a compact abelian Lie group (when identified with the unit
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s 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 group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
s. This is due in part to the fact that in any compact Lie group ''G'' one can always find a
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefor ...
; that is, a closed
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a 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 \mathbb^, which are classified by
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
integral matrices of size ''n'' with an integral inverse; these are just the integral matrices with determinant ±1. Making them act on \mathbb^ 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 In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
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''(\mathbb^, Z) can be identified with the
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
over the Z- module \mathbb^ 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, \mathbb^= (\mathbb^)^. The configuration space of ''unordered'', not necessarily distinct points is accordingly the
orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...
\mathbb^ / \mathbb_, which is the quotient of the torus by the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite 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 In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
, with a twist; equivalently, as a
triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A ...
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, Michael Kolinas, et al.), being used to model musical triads.


Flat torus

A flat torus is a torus with the metric inherited from its representation as the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
, \mathbb^/L, where L is a discrete subgroup of \mathbb^ isomorphic to \mathbb^. This gives the quotient the structure of a Riemannian manifold. Perhaps the simplest example of this is when : \mathbb^/\mathbb^, which can also be described as the
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
under the identifications . 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 this cylinder cannot be bent into a torus without stretching the paper (unless some regularity and differentiability conditions are given up, see below). A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows: :(x,y,z,w) = (R\cos u, R\sin u, P\cos v, P\sin v) where ''R'' and ''P'' are positive constants determining the aspect ratio. It is
diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two man ...
to a regular torus but not isometric. It can not be analytically embedded (
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
of class ) 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: :(x,y,z) = ((R+P\sin v)\cos u, (R+P\sin v)\sin u, P\cos v). If ''R'' and ''P'' in the above flat torus parametrization form a unit vector then ''u'', ''v'', and 0 < ''η'' < /2 parameterize the unit 3-sphere as Hopf coordinates. In particular, for certain very specific choices of a square flat torus in the
3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
''S''3, where above, the torus will partition the 3-sphere into two
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
solid tori subsets with the aforesaid flat torus surface as their common
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
. One example is the torus T defined by :T = \left\. Other tori in ''S''3 having this partitioning property include the square tori of the form ''Q''⋅T, where ''Q'' is a rotation of 4-dimensional space \mathbb^, or in other words ''Q'' is a member of the Lie group SO(4). It is known that there exists no ''C''2 (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, which was proven in the 1950s, an isometric ''C''1 embedding exists. This is solely an existence proof and does not provide explicit equations for such an embedding. In April 2012, an explicit ''C''1 (continuously differentiable) embedding of a flat torus into 3-dimensional Euclidean space \mathbb^ was found. It is a flat torus in the sense that as metric spaces, 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. 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 obtain the smoothness of this corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their "wavelengths". (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.


Genus ''g'' surface

In the theory of
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s there is another object, the "
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
" ''g'' surface. Instead of the product of ''n'' circles, a genus ''g'' surface is the
connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
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 A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
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 In mathematics, a genus ''g'' surface (also known as a ''g''-torus or ''g''-holed torus) is a surface formed by the connected sum of ''g'' many tori: the interior of a disk is removed from each of ''g'' many tori and the boundaries of the ''g'' ...
and triple torus are also occasionally used. The
classification theorem In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few issues relate ...
for surfaces states that every compact
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
surface is topologically equivalent to either the sphere or the connect sum of some number of tori, disks, and real
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
s.


Toroidal polyhedra

Polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...
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 − 2''N'', 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 (or the subgroup of diffeomorphisms) of the torus is studied in geometric topology. Its
mapping class group In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space. Mo ...
(the connected components of the homeomorphism group) is surjective onto the group \operatorname(n,\mathbb) of invertible integer matrices, which can be realized as linear maps on the universal covering space \mathbb^ that preserve the standard lattice \mathbb^ (this corresponds to integer coefficients) and thus descend to the quotient. At the level of
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
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 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
generates the
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
algebra: :\operatorname_(\mathbb^n) = \operatorname(\pi_1(X)) = \operatorname(\mathbb^n) = \operatorname(n,\mathbb). Since the torus is an
Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
''K''(''G'', 1), its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); all homotopy equivalences of the torus can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism. Thus the
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
of the mapping class group splits (an identification of the torus as the quotient of \mathbb^ gives a splitting, via the linear maps, as above): :1 \to \operatorname_0(\mathbb^n) \to \operatorname(\mathbb^n) \to \operatorname_(\mathbb^n) \to 1. The mapping class group of higher genus surfaces is much more complicated, and an area of active research.


Coloring a torus

The torus's
chromatic number In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices ...
is seven, meaning every graph that can be embedded on the torus has a
chromatic number In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices ...
of at most seven. (Since the
complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is ...
\mathsf can be embedded on the torus, and \chi (\mathsf) = 7, 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.)


de Bruijn torus

In
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
mathematics, a ''de Bruijn torus'' is an
array An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...
of symbols from an alphabet (often just 0 and 1) that contains every ''m''-by-''n'' 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 In combinatorial mathematics, a de Bruijn sequence of order ''n'' on a size-''k'' alphabet ''A'' is a cyclic sequence in which every possible length-''n'' string on ''A'' occurs exactly once as a substring (i.e., as a ''contiguous'' subseq ...
, which can be considered a special case where ''n'' is 1 (one dimension).


Cutting a torus

A solid torus of revolution can be cut by ''n'' (> 0) planes into maximally :\beginn+2 \\ n-1\end +\beginn \\ n-1\end = \tfrac(n^3 + 3n^2 + 8n) parts. 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: :1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, ... .


See also


Notes

*''Nociones de Geometría Analítica y Álgebra Lineal'', , Author: Kozak Ana Maria, Pompeya Pastorelli Sonia, Verdanega Pedro Emilio, Editorial: McGraw-Hill, Edition 2007, 744 pages, language: Spanish *Allen Hatcher
''Algebraic Topology''
Cambridge University Press, 2002. . *V. V. Nikulin, I. R. Shafarevich. ''Geometries and Groups''. Springer, 1987. , .
"Tore (notion géométrique)" at ''Encyclopédie des Formes Mathématiques Remarquables''


References


External links


Creation of a torus
at cut-the-knot
"4D torus"
Fly-through cross-sections of a four-dimensional torus
"Relational Perspective Map"
Visualizing high dimensional data with flat torus

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