In
geometry, the **Beckman–Quarles theorem** states that if a transformation of the
Euclidean plane or a higher-dimensional
Euclidean space preserves unit distances, then it preserves all
Euclidean distances. Equivalently, every
homomorphism from the
unit distance graph of the plane to itself must be an
isometry of the plane. The theorem is named after Frank S. Beckman and Donald A. Quarles Jr., who published this result in 1953; it was later rediscovered by other authors and re-proved in multiple ways. Analogous theorems for rational subsets of Euclidean spaces, or for
non-Euclidean geometry, are also known.

Formally, the result is as follows. Let be a
function 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.^{
[1]}

One way of rephrasing the BeckmanâQuarles theorem involves
graph homomorphisms, mappings between
undirected graphs that take vertices to vertices and edges to edges. 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, a homomorphism from this graph to itself is the same thing as a unit-distance-preserving transformation of the plane. Thus, the BeckmanâQuarles theorem states that the only homomorphisms from this graph to itself are the obvious ones coming from isometries of the plane.^{
[2]} For this graph, all homomorphisms are
symmetries of the graph, the defining property of a class of graphs called
cores.^{
[3]}

As well as the original proofs of Beckman and Quarles of the theorem,^{
[1]} and the proofs in later papers rediscovering the result,^{
[4]}^{
[5]}^{
[6]} several alternative proofs have been published.^{
[7]}^{
[8]}^{
[9]} If is the set of distances preserved by a mapping , then it follows from the
triangle inequality that certain comparisons of other distances with members of are preserved by . Therefore, if can be shown to be a
dense set, then all distances must be preserved. The main idea of several proofs of the BeckmanâQuarles theorem is to use the
structural rigidity of certain
unit distance graphs, such as the graph of a
regular simplex, to show that a mapping that preserves unit distances must preserve enough other distances to form a dense set.^{
[9]}

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 non-integers.^{
[1]}

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 numbers. "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 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 two discontinuous functions:

- The first function maps every point of the Hilbert space onto a nearby point in a countable dense subspace. For instance the dense subspace could be chosen as the subspace of sequences of rational numbers. As long as this transformation moves each point by a distance less than , 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 distance.^{
[1]}^{
[10]}

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
HadwigerâNelson problem, 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 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 HadwigerâNelson problem could improve this bound.^{
[11]}^{
[12]}

For transformations of the points with
rational number 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.^{
[13]}^{
[14]} Similar results hold also for mappings of the rational points that preserve other distances, such as the
square root of two, in addition to the unit distances.^{
[15]} For pairs of points whose distance is an
algebraic number , there is a finite version of this theorem: Maehara showed that, for every algebraic number , there is a finite
rigid unit distance graph in which some two vertices and must be at distance from each other. It follows from this that any transformation of the plane that preserves the unit distances in must also preserve the distance between and .^{
[16]}^{
[17]}^{
[18]}

A. D. Alexandrov asked which
metric spaces have the same property, that unit-distance-preserving mappings are isometries,^{
[19]} 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
quadratic form.^{
[20]} BeckmanâQuarles theorems have been proven for non-Euclidean spaces such as
Minkowski space,^{
[21]}
inversive distance in the
MĂ¶bius plane,^{
[22]} finite
Desarguesian planes,^{
[23]} and spaces defined over
fields with nonzero
characteristic.^{
[24]}^{
[25]} Additionally, theorems of this type have been used to characterize transformations other than the isometries, such as
Lorentz transformations.^{
[26]}

The BeckmanâQuarles theorem was first published by Frank S. Beckman and Donald A. Quarles Jr. in 1953.^{
[1]} It was already named as "a theorem of Beckman and Quarles" as early as 1960, by
Victor Klee.^{
[27]} It was later rediscovered by other authors, through the 1960s and 1970s.^{
[4]}^{
[5]}^{
[6]}

Quarles was the son of communications engineer and defense executive
Donald A. Quarles. He was educated at the
Phillips Academy,
Yale University, and the
United States Naval Academy. He served as a meteorologist in the US Navy during
World War II, and became an engineer for IBM. His work there included projects for tracking
Sputnik, the development of a
supercomputer,
inkjet printing, and
magnetic resonance imaging;^{
[28]} he completed a Ph.D. in 1964 at the
Courant Institute of Mathematical Sciences on the
computer simulation of
shock waves, jointly supervised by
Robert D. Richtmyer and
Peter Lax.^{
[29]}

Beckman studied at the
City College of New York and served in the US Army during the war. Like Quarles, he worked for IBM, beginning in 1951.^{
[30]} He earned a Ph.D. in 1965, under the supervision of
Louis Nirenberg at
Columbia University, on
partial differential equations.^{
[31]} In 1971, he left IBM to become the founding chair of the Computer and Information Science Department at
Brooklyn College, and he later directed the graduate program in computer science at the
Graduate Center, CUNY.^{
[30]}

- ^
^{a}^{b}^{c}^{d}^{e}Beckman, F. S.; Quarles, D. A., Jr. (1953), "On isometries of Euclidean spaces",*Proceedings of the American Mathematical Society*,**4**: 810â815, doi: 10.2307/2032415, MR 0058193 **^**Kuzminykh, Alexandr (2009), "On diversity and stability of unit bases for the Euclidean metric",*Journal of Geometry*,**94**(1â2): 143â150, doi: 10.1007/s00022-009-0010-x, MR 2534414**^**NeĆĄetĆil, Jaroslav; Ossona de Mendez, Patrice (2012),*Sparsity: Graphs, Structures, and Algorithms*, Algorithms and Combinatorics, vol. 28, Heidelberg: Springer, p. 43, doi: 10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058- ^
^{a}^{b}Zvengrowski, P. (1965), "Appendix to Chapter II", in Modenov, P. S.; Parkhomenko, A. S. (eds.),*Geometric Transformations*, vol. 1, New York: Academic Press; as cited by Rassias 2001 - ^
^{a}^{b}Townsend, Carl G. (1970), "Congruence-preserving mappings",*Mathematics Magazine*,**43**: 37â38, doi: 10.2307/2688111, JSTOR 2688111, MR 0256252 - ^
^{a}^{b}Bishop, Richard L. (1973), "Characterizing motions by unit distance invariance",*Mathematics Magazine*,**46**: 148â151, doi: 10.2307/2687969, JSTOR 2687969, MR 0319026 **^**Benz, Walter (1987), "An elementary proof of the theorem of Beckman and Quarles",*Elemente der Mathematik*,**42**(1): 4â9, MR 0881889**^**JuhĂĄsz, RozĂĄlia (2015), "Another proof of the BeckmanâQuarles theorem",*Advances in Geometry*,**15**(4): 519â521, doi: 10.1515/advgeom-2015-0027, MR 3406479- ^
^{a}^{b}Totik, Vilmos (2021), "The BeckmanâQuarles theorem via the triangle inequality" (PDF),*Advances in Geometry*,**21**(4): 541â543, doi: 10.1515/advgeom-2020-0024, MR 4323350 **^**Greenwell, Donald; Johnson, Peter D. (1976), "Functions that preserve unit distance",*Mathematics Magazine*,**49**(2): 74â79, doi: 10.1080/0025570X.1976.11976543, JSTOR 2689433, MR 0394445**^**Rassias, Themistocles M. (1987), "Some remarks on isometric mappings",*Facta Universitatis, Series Mathematics and Informatics*,**2**: 49â52, MR 0963783; as cited by Rassias 2001**^**Rassias, Themistocles M. (2001), "Isometric mappings and the problem of A. D. Aleksandrov for conservative distances", in Florian, H.; Ortner, N.; Schnitzer, F. J.; Tutschke, W. (eds.),*Functional-Analytic and Complex Methods, their Interactions, and Applications to Partial Differential Equations: Proceedings of the International Workshop held at Graz University Of Technology, Graz, February 12â16, 2001*, River Edge, New Jersey: World Scientific Publishing Co., Inc., pp. 118â125, doi: 10.1142/4822, MR 1893253**^**Connelly, Robert; Zaks, Joseph (2003), "The BeckmanâQuarles theorem for rational d-spaces, d even and*d*â„ 6", in Bezdek, AndrĂĄs (ed.),*Discrete Geometry: In honor of W. Kuperberg's 60th birthday*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 253, New York: Dekker, pp. 193â199, doi: 10.1201/9780203911211, MR 2034715**^**Zaks, Joseph (2006), "The rational analogue of the BeckmanâQuarles Theorem and the rational realization of some sets in*E*^{d}" (PDF),*Rendiconti di Matematica e delle sue Applicazioni*, Serie VII,**26**(1): 87â94, MR 2215835, archived from the original (PDF) on 2006-05-12**^**Zaks, Joseph (2005), "On mappings of to that preserve distances and and the BeckmanâQuarles theorem",*Journal of Geometry*,**82**(1â2): 195â203, doi: 10.1007/s00022-004-1660-3, MR 2161824**^**Maehara, Hiroshi (1991), "Distances in a rigid unit-distance graph in the plane",*Discrete Applied Mathematics*,**31**(2): 193â200, doi: 10.1016/0166-218X(91)90070-D**^**Maehara, Hiroshi (1992), "Extending a flexible unit-bar framework to a rigid one",*Discrete Mathematics*,**108**(1â3): 167â174, doi: 10.1016/0012-365X(92)90671-2, MR 1189840**^**Tyszka, Apoloniusz (2000), "Discrete versions of the BeckmanâQuarles theorem",*Aequationes Mathematicae*,**59**(1â2): 124â133, arXiv: math/9904047, doi: 10.1007/PL00000119, MR 1741475**^**Aleksandrov, A. D. (1970), "Mappings of families of sets",*Doklady Akademii Nauk SSSR*,**190**: 502â505, MR 0256256**^**Lester, June A. (1979), "Transformations of -space which preserve a fixed square-distance",*Canadian Journal of Mathematics*,**31**(2): 392â395, doi: 10.4153/CJM-1979-043-6, MR 0528819**^**Lester, June A. (1981), "The BeckmanâQuarles theorem in Minkowski space for a spacelike square-distance",*Archiv der Mathematik*,**37**(6): 561â568, doi: 10.1007/BF01234395, MR 0646516**^**Lester, June A. (1991), "A BeckmanâQuarles type theorem for Coxeter's inversive distance",*Canadian Mathematical Bulletin*,**34**(4): 492â498, doi: 10.4153/CMB-1991-079-6, MR 1136651**^**Benz, Walter (1982), "A BeckmanâQuarles type theorem for finite Desarguesian planes",*Journal of Geometry*,**19**(1): 89â93, doi: 10.1007/BF01930870, MR 0689123**^**RadĂł, Ferenc (1983), "A characterization of the semi-isometries of a Minkowski plane over a field ",*Journal of Geometry*,**21**(2): 164â183, doi: 10.1007/BF01918141, MR 0745209**^**RadĂł, Ferenc (1986), "On mappings of the Galois space",*Israel Journal of Mathematics*,**53**(2): 217â230, doi: 10.1007/BF02772860, MR 0845873**^**Benz, Walter (1981), "A Beckman Quarles type theorem for plane Lorentz transformations",*Mathematische Zeitschrift*,**177**(1): 101â106, doi: 10.1007/BF01214341, MR 0611472**^**Klee, Victor (1960),*Unsolved Problems in Intuitive Geometry*, p. 41, hdl: 1773/16201**^**"Donald Quarles Jr.",*The Cape Codder*, June 14, 2014 – via Legacy.com**^**Donald Aubrey Quarles, Jr. at the Mathematics Genealogy Project- ^
^{a}^{b}"Frank Samuel Beckman",*Richmond Times-Dispatch*, October 23, 2009 – via Legacy.com **^**Frank S. Beckman at the Mathematics Genealogy Project