Beckman–Quarles theorem
Encyclopedia
In geometry
, the Beckman–Quarles theorem, named after F. S. Beckman and D. A. Quarles, Jr., states that if a transformation of the Euclidean plane or a higher dimensional Euclidean space
preserves unit distances, then it preserves all distances. Equivalently, every automorphism
of the unit distance graph
of the plane must be an isometry
of the plane.
Beckman and Quarles published this result in 1953; it was later rediscovered by other authors.
or multivalued function
from a -dimensional Euclidean space to itself, and suppose that, for every pair of points and that are at unit distance from each other, every pair of images and are also at unit distance from each other. Then must be an isometry
: it is a one-to-one function that preserves distances between all pairs of points.
(one-dimensional Euclidean space). For, the function that returns if is an integer and returns otherwise obeys the preconditions of the theorem (it preserves unit distances) but is not an isometry.
Beckman and Quarles also provide a counterexample for Hilbert space
, the space of square-summable sequences of real numbers. This example involves the composition
of two discontinuous functions
: one that maps every point of the Hilbert space onto a nearby point in a countable
dense subspace
, and a second that maps this dense set into a countable unit simplex
(an infinite set of points all at unit distance from each other). These two transformations map any two points at unit distance from each other to two different points in the dense subspace, and from there map them to two different points of the simplex, which are necessarily at unit distance apart. Therefore, their composition preserves unit distances. However, it is not an isometry, because it maps every pair of points, no matter their original distance, either to the same point or to a unit distance.
Cartesian coordinates, the situation is more complicated than for the full Euclidean plane: in this case, there exist unit-distance-preserving non-isometries of dimensions up to four, but none for dimensions five and above. Similar results hold also for mappings of the rational points that preserve other distances, such as the square root of two.
One way of rephrasing the Beckman–Quarles theorem is that, for the unit distance graph
whose vertices are all of the points in the plane, with an edge between any two points at unit distance, the only graph automorphism
s are the obvious ones coming from isometries of the plane. For pairs of points whose distance is an algebraic number
, there is a finite version of this theorem: Maehara showed that there is a finite rigid
unit distance graph in which some two vertices and must be at distance from each other, from which it follows that any transformation of the plane that preserves the unit distances in must also preserve the distance between and .
Several authors have studied analogous results for other types of geometries. For instance, it is possible to replace Euclidean distance by the value of a quadratic form
.
Beckman–Quarles theorems have been proven for non-Euclidean spaces such as Minkowski space
the Möbius plane
, finite Desarguesian plane
s, and spaces defined over fields
with nonzero characteristic
.
Additionally, theorems of this type have been used to characterize transformations other than the isometries, such as Lorentz transformation
s.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the Beckman–Quarles theorem, named after F. S. Beckman and D. A. Quarles, Jr., states that if a transformation of the Euclidean plane or a higher dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
preserves unit distances, then it preserves all distances. Equivalently, every automorphism
Graph automorphism
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity....
of the unit distance graph
Unit distance graph
In mathematics, and particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points by an edge whenever the distance between the two points is exactly one...
of the plane must be an isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
of the plane.
Beckman and Quarles published this result in 1953; it was later rediscovered by other authors.
Formal statement
Formally, the result is as follows. Let be a functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
or multivalued function
Multivalued function
In mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...
from a -dimensional Euclidean space to itself, and suppose that, for every pair of points and that are at unit distance from each other, every pair of images and are also at unit distance from each other. Then must be an isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
: it is a one-to-one function that preserves distances between all pairs of points.
Counterexamples for other spaces
Beckman and Quarles observe that the theorem is not true for the real lineReal line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
(one-dimensional Euclidean space). For, the function that returns if is an integer and returns otherwise obeys the preconditions of the theorem (it preserves unit distances) but is not an isometry.
Beckman and Quarles also provide a counterexample for Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, the space of square-summable sequences of real numbers. This example involves the composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
of two discontinuous functions
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
: one that maps every point of the Hilbert space onto a nearby point in a countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
dense subspace
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
, and a second that maps this dense set into a countable unit simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
(an infinite set of points all at unit distance from each other). These two transformations map any two points at unit distance from each other to two different points in the dense subspace, and from there map them to two different points of the simplex, which are necessarily at unit distance apart. Therefore, their composition preserves unit distances. However, it is not an isometry, because it maps every pair of points, no matter their original distance, either to the same point or to a unit distance.
Related results
For transformations only of the subset of Euclidean space with rational numberRational 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...
Cartesian coordinates, the situation is more complicated than for the full Euclidean plane: in this case, there exist unit-distance-preserving non-isometries of dimensions up to four, but none for dimensions five and above. Similar results hold also for mappings of the rational points that preserve other distances, such as the square root of two.
One way of rephrasing the Beckman–Quarles theorem is that, for the unit distance graph
Unit distance graph
In mathematics, and particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points by an edge whenever the distance between the two points is exactly one...
whose vertices are all of the points in the plane, with an edge between any two points at unit distance, the only graph automorphism
Graph automorphism
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity....
s are the obvious ones coming from isometries of the plane. For pairs of points whose distance is an algebraic number
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
, there is a finite version of this theorem: Maehara showed that there is a finite rigid
Structural rigidity
In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges.-Definitions:...
unit distance graph in which some two vertices and must be at distance from each other, from which it follows that any transformation of the plane that preserves the unit distances in must also preserve the distance between and .
Several authors have studied analogous results for other types of geometries. For instance, it is possible to replace Euclidean distance by the value of a quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
.
Beckman–Quarles theorems have been proven for non-Euclidean spaces such as Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
the Möbius plane
Möbius plane
A Möbius plane or inversive plane is a particular kind of plane geometry, built upon some affine planes by adding one point, called the ideal point or point at infinity. In a Möbius plane straight lines are a special case of circles; they are the circles that pass through the ideal point. Möbius...
, finite Desarguesian plane
Desarguesian plane
In projective geometry a Desarguesian plane, named after Gérard Desargues, is a plane in which Desargues' theorem holds. The ordinary real projective plane is a Desarguesian plane. More generally any projective plane over a division ring is Desarguesian, and conversely Hilbert showed that any...
s, and spaces defined over fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
with nonzero characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
.
Additionally, theorems of this type have been used to characterize transformations other than the isometries, such as Lorentz transformation
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
s.