In
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 c ...
, a torus (plural tori, colloquially donut or doughnut) is a
surface of revolution 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 const ...
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 (geometry), position of an element (i.e., Point (m ...
about an axis that is
coplanar with the circle.
If the
axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is
tangent
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. More ...
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 Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. 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 ring
A swim ring (also known as a swimming ring, swim tube, rubber ring, water donut, floatie, inner tube, or, in the United States, a lifesaver) is a toroid-shaped (hence the name "ring" or "doughnut") inflatable water toy. The swim ring was derived ...
s,
inner tubes and
ringette ring
Ringette is a non-contact winter team sport played on ice hockey rinks using ice hockey skates, straight sticks with drag-tips, and a blue, rubber, pneumatic ring designed for use on ice surfaces. The sport is among a small number of organi ...
s. Eyeglass lenses that combine spherical and cylindrical correction are
toric lenses.
A torus should not be confused with a ''
solid torus
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle, endowed with the product topology.
A standard way to visuali ...
'', which is formed by rotating a
disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
, 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). The de ...
inside the torus. Real-world objects that approximate a ''solid torus'' include
O-rings, non-inflatable
lifebuoys, ring
doughnut
A doughnut or donut () is a type of food made from leavened fried dough. It is popular in many countries and is prepared in various forms as a sweet snack that can be homemade or purchased in bakeries, supermarkets, food stalls, and franc ...
s, 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 language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, a ring torus is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
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 const ...
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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, 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
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdo ...
, a surface in
4-space
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
.
In the field of
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, 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 extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
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
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Augu ...
).
Geometry
A torus can be defined
parametrically by:
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
or the solution of , where
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 .
E ...
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
An apple is an edible fruit produced by an apple tree (''Malus domestica''). Apple fruit tree, trees are agriculture, cultivated worldwide and are the most widely grown species in the genus ''Malus''. The tree originated in Central Asia, wh ...
''
*When , the torus degenerates to the sphere.
When , the
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
of this torus is
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
(and, hence, homeomorphic) to a
product 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). The de ...
of this solid torus and the
surface area
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
of its torus are easily computed using
Pappus's centroid theorem, giving:
These formulas are the same as for a cylinder of length and radius , obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (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
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
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
Topologically
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 ho ...
, a torus is a
closed surface defined as the
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 const ...
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 dimensi ...
''S''
3 of radius √2. This topological torus is also often called the
Clifford torus
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdo ...
. 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
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
over ''S''
2 (the
Hopf bundle
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...
).
The surface described above, given the
relative topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
from
, is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
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 ''S''
3.
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
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 coordinate ...
by pasting the opposite edges together, described as a
fundamental polygon ''ABA''
−1''B''
−1.
The
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of the torus is just the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of the fundamental group of the circle with itself:
:
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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the fundamental group (this follows from
Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results ...
since the fundamental group is
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
).
Two-sheeted cover
The 2-torus double-covers the 2-sphere, with four
ramification point
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
s. Every
conformal structure
In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two d ...
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 point In mathematics, a Weierstrass point P on a nonsingular algebraic curve C defined over the complex numbers is a point such that there are more functions on C, with their poles restricted to P only, than would be predicted by the Riemann–Roch theore ...
s. 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, the ...
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:
:
The standard 1-torus is just the circle:
. The torus discussed above is the standard 2-torus,
. And similar to the 2-torus, the ''n''-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 ''n''-torus is obtained from the ''n''-dimensional
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
by gluing the opposite faces together.
An ''n''-torus in this sense is an example of an ''n-''dimensional
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. It is also an example of a compact
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
. 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 Eucl ...
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 form ...
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 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
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s.
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s of ''T'' are easily constructed from automorphisms of the lattice
, which are classified by
invertible integral matrices In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Int ...
of size ''n'' with an integral inverse; these are just the integral matrices with determinant ±1. Making them act on
in the usual way, one has the typical ''toral automorphism'' on the quotient.
The
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of an ''n''-torus is a
free abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
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
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
of the ''n''-torus is 0 for all ''n''. The
cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually und ...
''H''
•(
, Z) can be identified with the
exterior algebra over the Z-
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
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,
. 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 ''n'' letters (by permuting the coordinates).
For ''n'' = 2, the quotient is the
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Augu ...
, 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
Twist may refer to:
In arts and entertainment Film, television, and stage
* ''Twist'' (2003 film), a 2003 independent film loosely based on Charles Dickens's novel ''Oliver Twist''
* ''Twist'' (2021 film), a 2021 modern rendition of ''Olive ...
; 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 unif ...
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,
/L, where L is a discrete subgroup of
isomorphic to
. This gives the quotient the structure of a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
. Perhaps the simplest example of this is when :
, which can also be described as the
Cartesian plane 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
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
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:
:
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 Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
to a regular torus but not
isometric
The term ''isometric'' comes from the Greek for "having equal measurement".
isometric may mean:
* Cubic crystal system, also called isometric crystal system
* Isometre, a rhythmic technique in music.
* "Isometric (Intro)", a song by Madeon from ...
. It can not be
analytically embedded (
smooth 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:
:
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 dimensi ...
''S''
3, where above, the torus will partition the 3-sphere into two
congruent solid tori subsets with the aforesaid flat torus surface as their common
boundary. One example is the torus T defined by
:
Other tori in ''S''
3 having this partitioning property include the square tori of the form ''Q''⋅T, where ''Q'' is a rotation of 4-dimensional space
, 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
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instan ...
, 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
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
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
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
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
, 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
surfaces there is another object, the "
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
" ''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
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
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 ...
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
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'' ...
are also occasionally used.
The
classification theorem for surfaces states that every
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
connected surface is topologically equivalent to either the sphere or the connect sum of some number of tori, disks, and real
projective planes.
Toroidal polyhedra
Polyhedra with the topological type of a torus are called toroidal polyhedra, and have
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
''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 In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important in ...
(or the subgroup of diffeomorphisms) of the torus is studied in
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated i ...
. Its
mapping class group (the connected components of the homeomorphism 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
(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 deforma ...
and
homology
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
* Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chrom ...
, the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
, as these are all naturally isomorphic; also the first
cohomology group 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:
:
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); 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 (an identification of the torus as the quotient of
gives a splitting, via the linear maps, as above):
:
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 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
In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions sh ...
for the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
.)
de Bruijn torus
In
combinatorial mathematics, a ''de Bruijn torus'' is an
array of symbols from an alphabet (often just 0 and 1) that contains every ''m''-by-''n''
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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 ''n'' is 1 (one dimension).
Cutting a torus
A solid torus of revolution can be cut by ''n'' (> 0) planes into maximally
:
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 torusat
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
"4D torus"Fly-through cross-sections of a four-dimensional torus
"Relational Perspective Map"Visualizing high dimensional data with flat torus
*Archived a
Ghostarchiveand th
Wayback Machine
*
{{Authority control
Surfaces