Beckman and Quarles theorem

The Beckman-Quarles theorem is a theorem about geometric transformations . It was first published in 1953 by Frank S. Beckman and Donald A. Quarles Jr. and has been independently verified by several other authors.

The proposition says that any self-mapping of the n-dimensional Euclidean space ( ), which converts all point pairs with distance 1 into such, is already an isometry , i.e. leaves all distances unchanged. This is equivalent to saying that every automorphism of the unit distance graph is an isometry. ${\displaystyle n>1}$

Formal statement

The statement of the theorem is still correct even if the distance 1 is replaced by any fixed distance and ambiguous functions are allowed . ${\displaystyle r>0}$

Be for an ambiguous function in itself with the following property: ${\displaystyle \phi \colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle n>1}$

There is , such that for all with for all pairs of images applies ${\displaystyle r>0}$ ${\displaystyle x,y\in \mathbb {R} ^{n}}$ ${\displaystyle |x-y|=r}$ ${\displaystyle |\phi (x)-\phi (y)|=r}$ .

Then there is an unambiguous, bijective function and it applies to all that . ${\displaystyle \phi }$ ${\displaystyle x,y\in \mathbb {R} ^{n}}$ ${\displaystyle |x-y|=|\phi (x)-\phi (y)|}$

Counterexamples in real spaces

Using a simple counterexample, you can see that the requirement for the dimension of the room is really necessary. Consider the function that maps all integers to and records all other numbers. The figure obviously gets distance 1, but no other positive distances. In graph theory , this counterexample exists because the unit distance graph breaks down into individual connected components to which different graph automorphisms are applied. In contrast, the unit distance graph is connected in all dimensions . ${\displaystyle n\neq 1}$ ${\displaystyle f\colon \mathbb {R} ^{1}\to \mathbb {R} ^{1}}$ ${\displaystyle x}$ ${\displaystyle x+1}$ ${\displaystyle f}$ ${\displaystyle \mathbb {R} ^{1}}$ ${\displaystyle n>1}$

Seven-coloring of the plane using hexagonal tiling.

The requirement that the dimensions of the original image and the target area of ​​the image match is also necessary. In the event that the archetype space is the Euclidean plane and the target space is the space , one finds a function that holds the distance 1, but is not an isometry. To do this, parquet the level with hexagons of diameter 1. These can be colored in seven different colors (see illustration), which corresponds to a 7- color of the unit distance graph. In the target area, determine a 6- simplex with edge length 1. The mapping that maps all points of a color class to one point of the simplex is obviously a mapping that fixes the distance 1, but is not an isometry. ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{6}}$ ${\displaystyle \mathbb {R} ^{6}}$

Furthermore, the finiteness of the dimension of space is necessary for the application of the theorem: Beckman and Quarles provided a counterexample in the Hilbert space of quadratic summable sequences . ${\displaystyle \ell ^{2}}$

A finite variant of the Beckman and Quarles theorem

For any algebraic number A , a unit distance graph can G in which some pairs of nodes are found, the distance A -representations unit distance of all G have. This implies a finite variant of the Beckman and Quarles theorem: for every two points p and q with distance A, there is a finite, rigid unit distance graph that contains p and q and the transformation of the plane that preserves the unit distance also the distance between p and q receives.

Generalizations and further results

If one looks at self-images in space and using the Euclidean metric, the situation is more complicated than in real space. For dimensions , all self-images that contain the unit distance are isometrics. Counterexamples can be found in the dimensions , since in these spaces the unit distance graph breaks down into individual connected components. Even if you also assume that the distance is maintained, nothing changes in the statement. The finite variant of the theorem is known for the rational space only for certain special cases. ${\displaystyle \mathbb {Q} ^{n}}$ ${\displaystyle n\geq 5}$ ${\displaystyle \mathbb {Q} ^{n}}$ ${\displaystyle n<5}$ ${\displaystyle {\sqrt {2}}}$

There are theorems for various other geometries that correspond to the Beckman and Quarles theorem. June Lester showed, for example, that under a self-mapping that receives a fixed value of a square shape , all values ​​of the square shape are preserved. Analogous theorems for Minkowski spaces , the Möbius plane , the projective plane and metric spaces over bodies with characteristics other than 0 have been proven by various other authors . ${\displaystyle \phi \colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}\ (n>2)}$

literature

• FS Beckman, DA Quarles: On isometries of Euclidean spaces . In: Proc. Amer. Math. Soc. tape 4 , 1953, pp. 810-815 ( MR0058193 ).
• W. Benz: Geometric transformations with special consideration of the Lorentz transformation . BI-Wiss.-Verl., Mannheim (inter alia) 1992, ISBN 3-411-15071-8 , pp. 15-31 ( MR1183223 ).
• H. Lenz: Comments on the Beckman-Quarles problem . In: Mitt. Math. Ges. Hamburg . tape 12 , no. 2 , 1991, p. 429-446 ( MR1144794 ).

Individual evidence

1. ↑ CG Townsend: Congruence-preserving mappings . In: Math. Mag. Band 43 , 1970, pp. 37-38 ( MR0256252 ).
2. ^ RL Bishop: Characterizing motions by unit distance invariance . In: Math. Mag. Band 46 , 1970, pp. 148-151 ( MR0319026 ).
3. ↑ BV Dekster: Nonisometric distance one preserving mapping ${\displaystyle E_{2}\to E_{6}}$ . In: Arch. Math. Band 45 , 1985, pp. 283-283 ( MR0807663 ).
4. ^ H. Maehara: Distances in a rigid unit-distance graph in the plane . In: Discrete Appl. Math. Band 31 , no. 2 , 1991, p. 193-200 ( MR1106700 ).
5. ^ H. Maehara: Extending a flexible unit-bar framework to a rigid one . In: Discrete Math . tape 108 , no. 1-3 , 1992, pp. 167-174 ( MR1189840 ).
6. ^ A. Tyszka: Discrete versions of the Beckman-Quarles theorem . In: Aequationes Math . tape 59 , no. 1-2 , 2000, pp. 124-133 ( MR1741475 ).
7. ↑ J. Zaks: The rational analogue of the Beckman-Quarles Theorem and the rational realization of some sets in ${\displaystyle E_{d}}$ . In: Rend. Mat. Appl. (7) . tape 26 , no. 1 , 2006, p. 87-94 ( MR2215835 ).
8. ^ R. Connelly, J. Zaks: The Beckman-Quarles theorem for rational d-spaces, d even and d ≥ 6 . In: Monogr. Textbooks Pure Appl. Math. No. 253 , 2003, p. 193-199 ( MR2034715 ).
9. ^ J. Zaks: On mappings of to that preserve distances 1 and and the Beckman-Quarles theorem ${\displaystyle \mathbb {Q} ^{d}}$ ${\displaystyle \mathbb {Q} ^{d}}$ ${\displaystyle {\sqrt {2}}}$ . In: J. Geom. Volume 82 , no. 1-2 , 2005, pp. 195-203 ( MR2034715 ).
10. ^ J. Zaks: A discrete form of the Beckman-Quarles theorem for rational spaces . In: J. Geom. Volume 72 , no. 1-2 , 2001, pp. 199-205 ( MR1891467 ).
11. ^ JA Lester: Transformations of n-space which preserve a fixed square-distance . In: Canad. J. Math. Band 31 , no. 2 , 1979, p. 392-395 ( MR0528819 ).
12. ^ JA Lester: The Beckman-Quarles theorem in Minkowski space for a spacelike square-distance . In: CR Math. Rep. Acad. Sci. Canada . tape 3 , no. 2 , 1981, p. 59-61 ( MR0612389 ).
13. ^ JA Lester: A Beckman-Quarles type theorem for Coxeter's inversive distance . In: Canad. Math. Bull. Volume 34 , no. 4 , 1991, pp. 492-498 ( MR1136651 ).
14. ^ W. Benz: A Beckman-Quarles type theorem for finite Desarguesian planes . In: J. Geom. Volume 19 , no. 1 , 1982, pp. 89-93 ( MR0689123 ).
15. ↑ F. Radó: A characterization of the semi-isometries of a Minkowski plane over a field . In: KJ Geom. Volume 21 , no. 2 , 1983, p. 164-183 ( MR0745209 ).
16. ^ F. Radó: On mappings of the Galois space . In: Israel J. Math. Band 53 , no. 2 , 1986, p. 217-230 ( MR0845873 ).