Beckman–Quarles theorem

The theorem of Beckman and Quarles is a theorem about geometric transformations. He was first published in 1953 by Frank S. Beckman and Donald A. Quarles Jr. and proved it independently by several other authors.

The theorem says that any self-map of n-dimensional Euclidean space (), by far the one transferred all pairs of points in precisely those already is an isometry, ie all distances leaves unchanged. This is equivalent to saying that every automorphism of the unit distance graph is an isometry.

Formal statement

The statement of the theorem is true even then, if the distance 1 is replaced by an arbitrary fixed distance and allow ambiguous features.

Be an ambiguous function of in with the following property:

There are, such that for all with for all image pairs.

Then is a unique bijective function, and it applies to all that.

Counter-examples in real rooms

Using a simple counter-example can be seen that really is the prerequisite to the dimension of the space needed. Consider this the function that maps x to x 1 all integers and holds all the other numbers. The map f is replaced by the apparent distance 1, but no other positive distances. Spoken graph theory exists this counter-example, since the unit distance graph of disintegrates into individual connected components, on which different graph automorphisms are applied. In all dimensions of the unit - distance graph, however, is contiguous.

The condition that the dimensions of original image and the target area of the image match, is also necessary. In the event that the original image space is the Euclidean plane and the target space of the room you will find a function that have the distance 1 holds, but no isometry is .. To do parquetted the plane with hexagons of diameter 1 These can be different in seven colors are colored (see figure), which corresponds to a 7- coloring of the unit distance graph. At the finish area, determine a 6- simplex with edge length 1 Those mapping that maps all points of a color class in each case on a point of the simplex, is obviously a figure that holds the distance 1, but no isometry.

Furthermore, the finiteness of the dimension of space is necessary for the application of the theorem: From Beckman and Quarles comes a counter-example in the Hilbert space of square summable sequences.

A finite variant of the theorem of Beckman and Quarles

For any algebraic number A is a unit distance graph G can be found in which some pairs of nodes are at a distance A in all unit distance representations of G. So that is a finite variant the set of Beckman and Quarles implies: for each pair of points P and Q at a distance A, there exists a finite rigid unit distance graph, the p and q containing and at each unit distance - preserving transform to the level and the distance between p and q receives.

Generalizations and further results

Regarding self-images of the space, and using the Euclidean metric, the situation is more complicated than in the real space. For all dimensions of self-images are on receiving the unit distance, isometrics. In the dimensions can be found counterexamples, since decays in these spaces the unit distance graph into individual connected components. Even if one additionally assumes that the distance is maintained, does not change the statement. The finite variant of the theorem is known for the rational space only for certain special cases.

There are various other geometries phrases corresponding to the set of Beckman and Quarles. June Lester showed for example, that are preserved under a self-image that gets a fixed value of a quadratic form, all values ​​of the quadratic form. From various authors analog sets for Minkowski spaces that Möbius plane projective plane and metric spaces over field with characteristic not equal to 0 were proved.