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 ...
, the Beckman–Quarles theorem, named after Frank S. Beckman and Donald A. Quarles Jr., states that if a transformation of the
Euclidean plane or a higher-dimensional
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 ...
preserves unit distances, then it preserves all
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
s. Equivalently, every
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from the
unit distance graph
In mathematics, 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 whenever the distance between them is exactly one. To distinguish these gra ...
of the plane to itself must be an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
of the plane.
Beckman and Quarles published this result in 1953; it was later rediscovered by other and re-proved in multiple Analogous theorems for rational subsets of Euclidean spaces, or for
non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...
, are also known.
Statement and proof idea
Formally, the result is as follows. Let
be a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
or
multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
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
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
: it is a
one-to-one function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
that preserves distances between all pairs of
One way of rephrasing the Beckman–Quarles theorem involves
graph homomorphism
In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs that maps adjacent vertices to adjacent verti ...
s, mappings between
undirected graph
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' ve ...
s that take vertices to vertices and edges to edges. For the
unit distance graph
In mathematics, 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 whenever the distance between them is exactly one. To distinguish these gra ...
whose vertices are all of the points in the plane, with an edge between any two points at unit distance, a homomorphism from this graph to itself is the same thing as a unit-distance-preserving transformation of the plane. Thus, the Beckman–Quarles theorem states that the only homomorphisms from this graph to itself are the obvious ones coming from isometries of the For this graph, all homomorphisms are
symmetries of the graph, the defining property of a class of graphs called
If
is the set of distances preserved by a then it follows from the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
that certain comparisons of other distances with members of
are preserved Therefore, if
can be shown to be a
dense set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
, then all distances must be preserved. The main idea of several proofs of the Beckman–Quarles theorem is to use the
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
Rigidity is the property of a struct ...
of certain
unit distance graph
In mathematics, 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 whenever the distance between them is exactly one. To distinguish these gra ...
s, such as the graph of a
regular simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
, to show that a mapping that preserves unit distances must preserve enough other distances to form a
Counterexamples for other spaces
Beckman and Quarles observe that the theorem is not true for the
real line (one-dimensional Euclidean space). As an example, consider the function
that returns
if
is an integer and returns
otherwise. This function obeys the preconditions of the theorem: it preserves unit distances. However, it does not preserve the distances between integers and
Beckman and Quarles provide another counterexample showing that their theorem cannot be generalized to an infinite-dimensional space, the
Hilbert space of square-summable sequences of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. "Square-summable" means that the sum of the squares of the values in a sequence from this space must be finite. The distance between any two such sequences can be defined in the same way as the
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
for finite-dimensional spaces, by summing the squares of the differences of coordinates and then taking the square root. To construct a function that preserves unit distances but not other distances, Beckman and Quarles
compose
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
two
discontinuous functions:
*The first function maps every point of the Hilbert space onto a nearby point in a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
dense subspace
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
. For instance the dense subspace could be chosen as the subspace of sequences of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s. As long as this transformation moves each point by a distance it will map points at unit distance from each other to distinct images.
*The second function maps this dense set onto a countable unit
simplex, an infinite set of points all at unit distance from each other. One example of a countable simplex in this space consists of the sequences of real numbers that take the value
in a single position and are zero everywhere else. There are infinitely many sequences of this form, and the distance between any two such sequences is one. This second function must be one-to-one but can otherwise be chosen arbitrarily.
When these two transformations are combined, they 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
Related results
![Hadwiger-Nelson](https://upload.wikimedia.org/wikipedia/commons/a/a3/Hadwiger-Nelson.svg)
Every Euclidean space can be mapped to a space of sufficiently higher dimension in a way that preserves unit distances but is not an isometry. To do so, following known results on the
Hadwiger–Nelson problem
In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color. ...
, color the points of the given space with a finite number of colors so that no two points at unit distance have the same color. Then, map each color to a vertex of a higher-dimensional
regular simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
with unit edge lengths. For instance, the Euclidean plane can be colored with seven colors, using a tiling by hexagons of slightly less than unit diameter, so that no two points of the same color are a unit distance apart. Then the points of the plane can be mapped by their colors to the seven vertices of a
six-dimensional regular simplex. It is not known whether six is the smallest dimension for which this is possible, and improved results on the Hadwiger–Nelson problem could improve this
For transformations of the points with
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
coordinates, the situation is more complicated than for the full Euclidean plane. There exist unit-distance-preserving maps of rational points to rational points that do not preserve other distances for 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
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princi ...
, in addition to the unit distances. For pairs of points whose distance is an there is a finite version of this theorem: Maehara showed that there is a finite
rigid unit distance in which some two vertices
must be at 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
A. D. Alexandrov asked which
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s have the same property, that unit-distance-preserving mappings are and following this question 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 Beckman–Quarles theorems have been proven for non-Euclidean spaces such as
inversive distance in the finite and spaces defined over
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
with nonzero Additionally, theorems of this type have been used to characterize transformations other than the isometries, such as
References
{{DEFAULTSORT:Beckman-Quarles theorem
Euclidean geometry
Metric geometry
Theorems in geometry
Mathematics of rigidity