Commensurability (mathematics)
Encyclopedia
In mathematics
, two non-zero real number
s a and b are said to be commensurable if a/b is a rational number
.
's notion of commensurability is anticipated in passing in the discussion between Socrates
and the slave boy in Plato's dialogue entitled Meno
, in which Socrates uses the boy's own inherent capabilities to solve a complex geometric problem through the Socratic Method. He develops a proof which is, for all intents and purposes, very Euclidean in nature and speaks to the concept of incommensurability.
The usage primarily comes to us from translations of Euclid
's Elements
, in which two line segments a and b are called commensurable precisely if there is some third segment c that can be laid end-to-end a whole number of times to produce a segment congruent to a, and also, with a different whole number, a segment congruent to b. Euclid did not use any concept of real number, but he used a notion of congruence of line segments, and of one such segment being longer or shorter than another.
That a/b is rational is a necessary and sufficient condition for the existence of some real number c, and integer
s m and n, such that
Assuming for simplicity that a and b are positive, one can say that a ruler
, marked off in units of length c, could be used to measure out both a line segment
of length a, and one of length b. That is, there is a common unit of length
in terms of which a and b can both be measured; this is the origin of the term. Otherwise the pair a and b are incommensurable.
, a generalisation to pairs of subgroup
s is obtained, by noticing that in the case given, the subgroups of the integers as an additive group
, generated respectively by a and by b, intersect in the subgroup generated by d, where d is the LCM
of a and b. This intersection has finite index
in the integers, and therefore in each of the subgroups. This gives rise to a general notion of commensurable subgroups: two subgroups A and B of a group are commensurable when their intersection
has finite index in each of them. That is, two subgroups H1 and H2 of a group G are commensurable if
for 
Sometimes in fact this relation is called commensurate, and to be commensurable requires only to be conjugate to a commensurate subgroup.
A relationship can similarly be defined on subspaces of a vector space
, in terms of projections
that have finite-dimension
al kernel and cokernel
.
In contrast, two subspace
s
and 
that are given by some moduli space
stacks over a Lie algebra
are not necessarily commensurable if they are described by infinite dimensional representations. In addition, if the completions
of
-type modules
corresponding to
and 
are not well-defined
, then
and 
are also not commensurable.
finite-sheeted covering spaces. Depending on the type of topological space under consideration one might want to use homotopy-equivalences
or diffeomorphisms
instead of homeomorphisms in the definition. Thus, if one uses homotopy-equivalences, commensurability of groups correspond to commensurability of spaces provided one associates the classifying space
to a discrete group. For smooth manifolds, the Gieseking manifold
is commensurate to the complement of the figure-eight knot.
, the terms commensurable and incommensurable are used in the same way as in mathematics. The two rational numbers a and b usually refer to periods of two distinct, but connected physical properties of the considered material, such as the crystal structure
and the magnetic superstructure
. The potential richness of physical phenomena related to this concept is exemplified in the devil's staircase
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, two non-zero real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s a and b are said to be commensurable if a/b is a rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
.
History of the concept
EuclidEuclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's notion of commensurability is anticipated in passing in the discussion between Socrates
Socrates
Socrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...
and the slave boy in Plato's dialogue entitled Meno
Meno
Meno is a Socratic dialogue written by Plato. It attempts to determine the definition of virtue, or arete, meaning virtue in general, rather than particular virtues, such as justice or temperance. The first part of the work is written in the Socratic dialectical style and Meno is reduced to...
, in which Socrates uses the boy's own inherent capabilities to solve a complex geometric problem through the Socratic Method. He develops a proof which is, for all intents and purposes, very Euclidean in nature and speaks to the concept of incommensurability.
The usage primarily comes to us from translations of Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's Elements
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
, in which two line segments a and b are called commensurable precisely if there is some third segment c that can be laid end-to-end a whole number of times to produce a segment congruent to a, and also, with a different whole number, a segment congruent to b. Euclid did not use any concept of real number, but he used a notion of congruence of line segments, and of one such segment being longer or shorter than another.
That a/b is rational is a necessary and sufficient condition for the existence of some real number c, and integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s m and n, such that
- a = mc and b = nc.
Assuming for simplicity that a and b are positive, one can say that a ruler
Ruler
A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines...
, marked off in units of length c, could be used to measure out both a line segment
Line segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
of length a, and one of length b. That is, there is a common unit of length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
in terms of which a and b can both be measured; this is the origin of the term. Otherwise the pair a and b are incommensurable.
Commensurability in group theory
In group theoryGroup theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a generalisation to pairs of subgroup
Subgroup
In group theory, 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 of G if the restriction of * to H x H is a group operation on H...
s is obtained, by noticing that in the case given, the subgroups of the integers as an additive group
Additive group
An additive group may refer to:*an abelian group, when it is written using the symbol + for its binary operation*a group scheme representing the underlying-additive-group functor...
, generated respectively by a and by b, intersect in the subgroup generated by d, where d is the LCM
Least common multiple
In arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...
of a and b. This intersection has finite index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
in the integers, and therefore in each of the subgroups. This gives rise to a general notion of commensurable subgroups: two subgroups A and B of a group are commensurable when their intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
has finite index in each of them. That is, two subgroups H1 and H2 of a group G are commensurable if
for Sometimes in fact this relation is called commensurate, and to be commensurable requires only to be conjugate to a commensurate subgroup.
A relationship can similarly be defined on subspaces of a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, in terms of projections
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....
that have finite-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al kernel and cokernel
Cokernel
In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
.
In contrast, two subspace
Subspace
-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...
s
and that are given by some moduli spaceModuli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
stacks over a Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
are not necessarily commensurable if they are described by infinite dimensional representations. In addition, if the completionsComplete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
of
-type modulesModule (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
corresponding to
and are not well-definedWell-defined
In mathematics, well-definition is a mathematical or logical definition of a certain concept or object which uses a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy...
, then
and are also not commensurable.In Topology
Two topological spaces are commensurable if they have homeomorphicHomeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
finite-sheeted covering spaces. Depending on the type of topological space under consideration one might want to use homotopy-equivalences
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
or diffeomorphisms
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
instead of homeomorphisms in the definition. Thus, if one uses homotopy-equivalences, commensurability of groups correspond to commensurability of spaces provided one associates the classifying space
Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...
to a discrete group. For smooth manifolds, the Gieseking manifold
Gieseking manifold
In mathematics, the Gieseking manifold, is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately 1.01494161...
is commensurate to the complement of the figure-eight knot.
In physics
In physicsPhysics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the terms commensurable and incommensurable are used in the same way as in mathematics. The two rational numbers a and b usually refer to periods of two distinct, but connected physical properties of the considered material, such as the crystal structure
Crystal structure
In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...
and the magnetic superstructure
ANNNI model
In statistical physics, the axial next-nearest-neighbor Ising model, usually known as the ANNNI model, is a variant of the Ising model in which competing ferromagnetic and...
. The potential richness of physical phenomena related to this concept is exemplified in the devil's staircase
Devil's staircase
In mathematics, a singular function is any function ƒ defined on the interval [a, b] that has the following properties:*ƒ is continuous on [a, b]...
.