Beckman–Quarles theorem

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.

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 Beckman–Quarles 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 Beckman–Quarles theorem states that the only homomorphisms from this graph to itself are the obvious ones coming from isometries of the plane.

[9] Beckman and Quarles observe that the theorem is not true for the real line (one-dimensional Euclidean space).

This function obeys the preconditions of the theorem: it preserves unit distances.

[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.

Then, map each color to a vertex of a higher-dimensional regular simplex with unit edge lengths.

Then the points of the plane can be mapped by their colors to the seven vertices of a six-dimensional regular simplex.

[11][12] For transformations of the points with rational number coordinates, the situation is more complicated than for the full Euclidean plane.

, there is a finite version of this theorem: Maehara showed that, for every algebraic number

[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.

[26] The Beckman–Quarles 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.

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.

[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.