Beckman–Quarles theorem

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;[1] it was later rediscovered by other authors.[2][3]

Formal statement

Formally, the result is as follows. Let f be a function or multivalued function from a d-dimensional Euclidean space to itself, and suppose that, for every pair of points p and q that are at unit distance from each other, every pair of images f(p) and f(q) are also at unit distance from each other. Then f must be an isometry: it is a one-to-one function that preserves distances between all pairs of points.[1]

Counterexamples for other spaces

Beckman and Quarles observe that the theorem is not true for the real line (one-dimensional Euclidean space). For, the function f(x) that returns x + 1 if x is an integer and returns x otherwise obeys the preconditions of the theorem (it preserves unit distances) but is not an isometry.[1]

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.[1]

For transformations only of the subset of Euclidean space with Cartesian coordinates that are rational numbers, 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.[4][5] Similar results hold also for mappings of the rational points that preserve other distances, such as the square root of two.[6]

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 automorphisms are the obvious ones coming from isometries of the plane. For pairs of points whose distance is an algebraic number A, there is a finite version of this theorem: Maehara showed that there is a finite rigid unit distance graph G in which some two vertices p and q must be at distance A from each other, from which it follows that any transformation of the plane that preserves the unit distances in G must also preserve the distance between p and q.[7][8][9]

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.[10] Beckman–Quarles theorems have been proven for non-Euclidean spaces such as Minkowski space,[11] inversive distance in the Möbius plane,[12] finite Desarguesian planes,[13] and spaces defined over fields with nonzero characteristic.[14][15] Additionally, theorems of this type have been used to characterize transformations other than the isometries, such as Lorentz transformations.[16]

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References
  1. Beckman, F. S.; Quarles, D. A., Jr. (1953), "On isometries of Euclidean spaces", Proceedings of the American Mathematical Society, 4: 810–815, doi:10.2307/2032415, MR 0058193.
  2. Townsend, Carl G. (1970), "Congruence-preserving mappings", Mathematics Magazine, 43: 37–38, doi:10.2307/2688111, MR 0256252.
  3. Bishop, Richard L. (1973), "Characterizing motions by unit distance invariance", Mathematics Magazine, 46: 148–151, doi:10.2307/2687969, MR 0319026.
  4. Connelly, Robert; Zaks, Joseph (2003), "The Beckman-Quarles theorem for rational d-spaces, d even and d ≥ 6", Discrete geometry, Monogr. Textbooks Pure Appl. Math., 253, New York: Dekker, pp. 193–199, doi:10.1201/9780203911211.ch13, MR 2034715.
  5. Zaks, Joseph (2006), "The rational analogue of the Beckman-Quarles Theorem and the rational realization of some sets in $E^d$", Rendiconti di Matematica e delle sue Applicazioni. Serie VII, 26 (1): 87–94, MR 2215835.
  6. Zaks, Joseph (2005), "On mappings of Qd to Qd that preserve distances 1 and √2 and the Beckman-Quarles theorem", Journal of Geometry, 82 (1–2): 195–203, doi:10.1007/s00022-004-1660-3, MR 2161824.
  7. Maehara, Hiroshi (1991), "Distances in a rigid unit-distance graph in the plane", Discrete Applied Mathematics, 31 (2): 193–200, doi:10.1016/0166-218X(91)90070-D.
  8. Maehara, Hiroshi (1992), "Extending a flexible unit-bar framework to a rigid one", Discrete Mathematics, 108 (1–3): 167–174, doi:10.1016/0012-365X(92)90671-2, MR 1189840.
  9. Tyszka, Apoloniusz (2000), "Discrete versions of the Beckman-Quarles theorem", Aequationes Mathematicae, 59 (1–2): 124–133, arXiv:math/9904047, doi:10.1007/PL00000119, MR 1741475.
  10. Lester, June A. (1979), "Transformations of n-space which preserve a fixed square-distance", Canadian Journal of Mathematics, 31 (2): 392–395, doi:10.4153/CJM-1979-043-6, MR 0528819.
  11. Lester, June A. (1981), "The Beckman-Quarles theorem in Minkowski space for a spacelike square-distance", C. R. Math. Rep. Acad. Sci. Canada, 3 (2): 59–61, MR 0612389.
  12. Lester, June A. (1991), "A Beckman-Quarles type theorem for Coxeter's inversive distance", Canadian Mathematical Bulletin, 34 (4): 492–498, doi:10.4153/CMB-1991-079-6, MR 1136651.
  13. Benz, Walter (1982), "A Beckman-Quarles type theorem for finite Desarguesian planes", Journal of Geometry, 19 (1): 89–93, doi:10.1007/BF01930870, MR 0689123.
  14. Radó, Ferenc (1983), "A characterization of the semi-isometries of a Minkowski plane over a field K", Journal of Geometry, 21 (2): 164–183, doi:10.1007/BF01918141, MR 0745209.
  15. Radó, Ferenc (1986), "On mappings of the Galois space", Israel Journal of Mathematics, 53 (2): 217–230, doi:10.1007/BF02772860, MR 0845873.
  16. Benz, Walter (1980–1981), "A Beckman Quarles type theorem for plane Lorentz transformations", C. R. Math. Rep. Acad. Sci. Canada, 2 (1): 21–22, MR 0564486.
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