geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a Danzer set is a set of points that touches every

convex body In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...

of unit volume. Ludwig Danzer asked whether it is possible for such a set to have bounded

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

. Several variations of this problem remain unsolved.

Density

One way to define the problem more formally is to consider the growth rate of a set

S

in Euclidean space, defined as the function that maps a real number

r

to the number of points of

S

that are within distance

r

of the

origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...

. Danzer's question is whether it is possible for a Danzer set to have growth expressed in

big O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...

. If so, this would equal the growth rate of well-spaced point sets like the

integer lattice In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid l ...

(which is not a Danzer set). It is possible to construct a Danzer set of growth rate that is within a polylogarithmic factor For instance, overlaying rectangular grids whose cells have constant volume but differing aspect ratios can achieve a growth rate Constructions for Danzer sets are known with a somewhat slower growth rate, but the answer to Danzer's question remains unknown.

Bounded coverage

Another variation of the problem, posed by

Timothy Gowers Sir William Timothy Gowers, (; born 20 November 1963) is a British mathematician. He is Professeur titulaire of the Combinatorics chair at the Collège de France, and director of research at the University of Cambridge and Fellow of Trinity Col ...

, asks whether there exists a Danzer set

S

for which there is a finite bound

C

on the number of points of intersection between

S

and any convex body of unit volume. This version has been solved: it is impossible for a Danzer set with this property to exist.

Separation

A third variation of the problem, still unsolved, is Conway's dead fly problem.

John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

recalled that, as a child, he slept in a room with wallpaper whose flower pattern resembled an array of dead flies, and that he would try to find convex regions that did not have a dead fly in them. In Conway's formulation, the question is whether there exists a Danzer set in which the points of the set (the dead flies) are separated at a bounded distance from each other. Such a set would necessarily also have an upper bound on the distance from each point of the plane to a dead fly (in order to touch all circles of unit area), so it would form a

Delone set In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points, a ...

, a set with both lower and upper bounds on the spacing of the points. It would also necessarily have growth so if it exists then it would also solve the original version of Danzer's problem. Conway offered a $1000 prize for a solution to his problem, as part of a set of problems also including

Conway's 99-graph problem In graph theory, Conway's 99-graph problem is an unsolved problem asking whether there exists an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices ...

, the analysis of

sylver coinage Sylver coinage is a mathematical game for two players, invented by John H. Conway. It is discussed in chapter 18 of '' Winning Ways for Your Mathematical Plays''. This article summarizes that chapter. The two players take turns naming positive ...

, and the thrackle conjecture.

Additional properties

It is also possible to restrict the classes of point sets that may be Danzer sets in other ways than by their densities. In particular, they cannot be the union of finitely many lattices, they cannot be generated by choosing a point in each tile of a

substitution tiling In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The ...

(in the same position for each tile of the same type), and they cannot be generated by the cut-and-project method for constructing

aperiodic tiling An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- period ...

s. Therefore, the vertices of the

pinwheel tiling In geometry, pinwheel tilings are non-periodic tilings defined by Charles Radin and based on a construction due to John Conway. They are the first known non-periodic tilings to each have the property that their tiles appear in infinitely many or ...

and

Penrose tiling A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any finite distance, without r ...

are not Danzer sets.

References

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On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...

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\mathbb{R}^d

, year = 2017, arxiv = 1510.07179 {{citation , last1 = Solomon , first1 = Yaar , last2 = Weiss , first2 = Barak , arxiv = 1406.3807 , doi = 10.24033/asens.2303 , issue = 5 , journal = Annales Scientifiques de l'École Normale Supérieure , mr = 3581810 , pages = 1053–1074 , title = Dense forests and Danzer sets , volume = 49 , year = 2016, s2cid = 672315 Convex analysis Metric geometry