On U-polygons of class c\geq 4 in planar point sets

On the Existence of $U$-Polygons of Class $c\geq 4$ in Planar Point Sets

Abstract.

For a finite set of directions in the Euclidean plane, a convex non-degenerate polygon is called a -polygon if every line parallel to a direction of that meets a vertex of also meets another vertex of . We characterize the numbers of edges of -polygons of class with all their vertices in certain subsets of the plane and derive explicit results in the case of cyclotomic model sets.

This work was supported by EPSRC via Grant EP/D058465/1. It is a pleasure to thank the CRC 701 at the University of Bielefeld for support during a stay in June 2007, where part of the manuscript was written.

1. Introduction

The (discrete parallel) -ray of a finite subset of the Euclidean plane in direction is the corresponding line sum function that gives the numbers of points of on each line parallel to . It was shown in [18, Proposition 4.6] that the convex subsets of an algebraic Delone set are determined by their discrete parallel -rays in the directions of a set of at least two pairwise non-parallel -directions (i.e., directions parallel to non-zero interpoint vectors of ) if and only if there is no -polygon with all its vertices in . By [18, Lemma 4.5], there always exists a -polygon with all its vertices in if is a set of at most three pairwise non-parallel -directions. This leads to the question which -polygons exist with all their vertices in for sets of four or more pairwise non-parallel -directions. We refer the reader to [12, 15, 16, 17, 18, 19] for more on discrete tomography and [11] for the role of -polygons in geometric tomography, where the -ray of a compact convex set in a direction gives the lengths of all chords of the set in this direction. Dulio and Peri have introduced the notion of class of a -polygon and demonstrated that for planar lattices the numbers of edges of -polygons of class with all their vertices in are precisely and ; cf. [10, Theorem 12]. As a first step beyond the case of planar lattices, this text provides a generalization of this result to planar sets that are non-degenerate in some sense and satisfy a certain affinity condition on finite scales (Theorem 3.1). It turns out that, for these sets , the existence of -polygons of class with all their vertices in is equivalent to the existence of certain affinely regular polygons with all their vertices in , a problem that was addressed in [19]. The obtained characterization of numbers of vertices of -polygons of class with all their vertices in can be expressed in terms of a simple inclusion of real field extensions of and particularly applies to algebraic Delone sets, thus including cyclotomic model sets, which form an important class of planar mathematical quasicrystals; cf. [2, 7]. For cyclotomic model sets , the numbers of vertices of -polygons of class with all their vertices in can be expressed by a simple divisibility condition (Corollary 4.1). In particular, the above result on lattice -polygons of class by Dulio and Peri is contained as a special case (Corollary 4.3(a)).

2. Definitions and preliminaries

Natural numbers are always assumed to be positive and the set of rational primes is denoted by . Primes for which the number is prime as well are called Sophie Germain primes. We denote by the set of Sophie Germain primes. The first few ones are

see entry A005384 of [20] for further details. The group of units of a given ring is denoted by . As usual, for a complex number , denotes the complex absolute value , where denotes the complex conjugation. Occasionally, we identify with . The unit circle in is denoted by . Moreover, the elements of are also called directions. For a direction , the angle between and the positive real axis is understood to be the unique angle with the property that a rotation of by in counter-clockwise order is a direction parallel to . For and , denotes the open ball of radius about . A subset of the plane is called uniformly discrete if there is a radius such that every ball with contains at most one point of . Further, is called relatively dense if there is a radius such that every ball with contains at least one point of . is called a Delone set if it is both uniformly discrete and relatively dense. A direction is called a -direction if it is parallel to a non-zero element of the difference set of . Further, a bounded subset of is called a convex subset of if its convex hull contains no new points of . A non-singular affine transformation of the Euclidean plane is given by , where and . Further, recall that a homothety of the Euclidean plane is given by , where is positive and . A convex polygon is the convex hull of a finite set of points in . For a subset , a polygon in is a convex polygon with all vertices in . A regular polygon is always assumed to be planar, non-degenerate and convex. An affinely regular polygon is a non-singular affine image of a regular polygon. In particular, it must have at least vertices. Let be a finite set of pairwise non-parallel directions. A non-degenerate convex polygon is called a -polygon if it has the property that whenever is a vertex of and , the line in the plane in direction which passes through also meets another vertex of . Then, every direction of is parallel to one of the edges of ; cf. [10, Lemma 5(i)]. Further, one can easily see that a -polygon has edges, where . For example, an affinely regular polygon with an even number of vertices is a -polygon if and only if each direction of is parallel to one of its edges. The following notion of class of a -polygon was introduced by Dulio and Peri; cf. [10, Definition 1]. For , a -polygon is said to be of class with respect to if is the maximal number of consecutive edges of whose directions belong to .

Definition 2.1.

For a subset , we denote by the intermediate field of that is given by

Further, we set , the maximal real subfield of .

For , we always let , as a specific choice for a primitive th root of unity in . Denoting by Euler’s totient function, one has the following standard result for the th cyclotomic field .

Fact 2.2 (Gauß).

[21, Theorem 2.5] . The field extension is a Galois extension with Abelian Galois group , where corresponds to the automorphism given by .

It is well known that is the maximal real subfield of and is of degree over ; see [21]. Throughout this text, we shall use the notation

Note that that and are the rings of integers in and , respectively; cf. [21, Theorem 2.6 and Proposition 2.16]. For odd, one has by the multiplicativity of the arithmetic function and thus ; cf. Fact 2.2.

Definition 2.3.

For a set , we define the following properties:

(Alg)
(Aff) For all finite subsets of , there is a non-singular affine
transformation of the plane such that  .
(Hom) For all finite subsets of , there is a homothety of the
plane such that  .

Moreover, we call degenerate if and only if is a subfield of .

Remark 2.4.

For any non-degenerate , the field is a complex extension of . Trivially, property (Hom) implies property (Aff). If satisfies property (Alg), then one has , meaning that is a real algebraic number field.

We need the following result of Darboux [9] on second mid-point polygons, where the midpoint polygon of a convex polygon is the convex polygon whose vertices are the midpoints of the edges of ; compare also [13, Lemma 5] or [11, Lemma 1.2.9].

Fact 2.5.

Let be a convex -gon in with centroid at the origin. For each , define . Then the sequence converges in the Hausdorff metric to an affinely regular polygon.

If, in the situation of Fact 2.5, is a -polygon of class , then, for all , is a -polygon of class , whence also is a -polygon of class . This proves the next

Lemma 2.6.

Let be a finite set of directions and let . Then, there exists a -polygon of class if and only if there is an affinely regular -polygon of class .

Let be an ordered tuple of four distinct elements of the set . Then, its cross ratio is defined by

with the usual conventions if one of the equals , thus . The following property of cross ratios of slopes of elements is standard.

Fact 2.7.

Let , , be four pairwise non-parallel elements of the Euclidean plane and let . Then, one has

Lemma 2.8.

[18, Fact 4.7] For a set , the cross ratio of slopes of four pairwise non-parallel -directions is an element of the field .

3. The characterization

Theorem 3.1.

For a non-degenerate subset of the plane with property (Aff) and an even number , the following statements are equivalent:

  • There is a -polygon of class in with edges.

  • There is an affinely regular -polygon of class with edges for a set of -directions.

  • .

  • There is an affinely regular polygon in with edges.

If additionally fulfils property (Alg), then the above assertions only hold for finitely many values of .

Proof.

Direction (i) (ii) immediately follows from Lemma 2.6. For direction (ii) (iii), let be an affinely regular -polygon of class with edges for a set of -directions. There is then a non-singular affine transformation of the plane such that is a regular -gon. Since is a -polygon of class for a set of -directions and since, by Fact 2.7, the cross ratio of slopes of directions of edges is preserved by non-singular affine transformations, there are four consecutive edges of whose cross ratio of slopes of their directions, say arranged in order of increasing angle with the positive real axis, is an element of ; cf. Lemma 2.8. Applying a suitable rotation, if necessary, we may assume that these directions are given in complex form by and ; cf. Fact 2.7 again. Using , one easily calculates that

This implies that

the latter being equivalent to (iii). Direction (iii) (iv) is an immediate consequence of [19, Theorem 3.3] in conjunction with the identity . For direction (iv) (i), assume first that is odd. Here, we are done since every affinely regular polygon in with edges is a -polygon of class with respect to any set of directions parallel to consecutive of its edges. If is even, there is an affinely regular polygon in with edges. Attach translates of edge-to-edge to in the obvious way and consider the convex hull of the resulting point set. Clearly, is a -polygon in of class with edges, where consists of the pairwise non-parallel -directions given by the edges and diagonals of . By property (Aff), there is a non-singular affine transformation of the plane such that is a polygon in . Then, is a -polygon of class in with edges, where is a set of pairwise non-parallel -directions parallel to the elements of . Assertion (i) follows. If additionally has property (Alg), then is an algebraic number field by Remark 2.4. Thus, the field extension has only finitely many intermediate fields and the assertion follows from condition (iii) in conjunction with [19, Corollary 2.7, Remark 2.8 and Lemma 2.9].∎

Corollary 3.2.

Let be a complex algebraic number field with and let be the ring of integers in . Let be a translate of or a translate of . Further, let be an even number. Denoting the maximal real subfield of by , the following statements are equivalent:

  • There is a -polygon of class in with edges.

  • There is an affinely regular -polygon of class with edges for a set of -directions.

  • .

  • There is an affinely regular polygon in with edges.

Additionally, the above assertions only hold for finitely many values of .

Proof.

This follows immediately from Theorem 3.1 in conjunction with the fact that has properties (Aff) and (Alg) with ; cf. [19, Section 3]. ∎

Remark 3.3.

In particular, Corollary 3.2 applies to translates of complex cyclotomic fields and their rings of integers, respectively, with for a suitable ; cf. Fact 2.2 and also compare the equivalences of Corollary 4.1 below.

4. Application to cyclotomic model sets

Delone subsets of the plane satisfying properties (Alg) and (Hom) were introduced as algebraic Delone sets in [18, Definition 4.1]. Note that algebraic Delone sets are always non-degenerate, since this is true for all relatively dense subsets of the plane. Examples of algebraic Delone sets are the so-called cyclotomic model sets ; cf. [18, Proposition 4.31]. By definition, any cyclotomic model set is contained in a translate of , where , in which case the -module is called the underlying -module of . More precisely, for , let be any map of the form

where the set arises from by choosing exactly one automorphism from each pair of complex conjugate ones; cf. Fact 2.2. Then, for any such choice, each translate of the set , where is a sufficiently ‘nice’ set with non-empty interior and compact closure, is a cyclotomic model set with underlying -module ; cf. [16, 17, 18, 19] for more details and properties of (cyclotomic) model sets. Since for odd , we might restrict ourselves to values when dealing with cyclotomic model sets with underlying -module . With the exception of the crystallographic cases of translates of the square lattice and translates of the triangular lattice , cyclotomic model sets are aperiodic (they have no non-zero translational symmetries) and have long-range order; cf. [18, Remark 4.23]. Well-known examples of cyclotomic model sets with underlying -module are the vertex sets of aperiodic tilings of the plane like the Ammann-Beenker tiling [1, 4, 14] (), the Tübingen triangle tiling [5, 6] () and the shield tiling [14] (); cf. Figure 1 for an illustration. For definitions of the above vertex sets of aperiodic tilings of the plane in algebraic terms, we refer the reader to [17, Section 1.2.3.2] or [16]. As an immediate consequence of Theorem 3.1 in conjunction with [19, Corollary 4.1] and the identity , one obtains the following

Corollary 4.1.

Let with an even number and . Further, let be a cyclotomic model set with underlying -module . The following statements are equivalent:

  • There is a -polygon of class in with edges.

  • There is an affinely regular -polygon of class with edges for a set of -directions.

  • .

  • There is an affinely regular polygon in with edges.

  • .

  • , or .

  • , or , or with an odd divisor of .

  • , or .

  • .

Remark 4.2.

Combining Corollary 4.1 and Fact 2.7, one sees that the cross ratios of slopes of directions of edges of -polygons of class in cyclotomic model sets , say arranged in order of increasing angle with the positive real axis, easily follow from a direct computation with a finite number of regular polygons; cf. [12, 8] for deep insights into this in the case of planar lattices.

The following consequence follows immediately from Corollary 4.1 in conjunction with [19, Corollary 4.2]. Restricted to values , it deals with the two cases where the degree of over is either or a prime number ; cf. [19, Lemma 2.10].

Corollary 4.3.

Let with an even number and . Further, let be a cyclotomic model set with underlying -module . Then, one has:

  • If , there is a -polygon of class in with edges if and only if .

  • If , there is a -polygon of class in with edges if and only if

Figure 1. An -icosagon of class in the vertex set of the Tübingen triangle tiling with respect to the set of -directions given by the edges and diagonals of the central regular decagon.
Example 4.4.

As mentioned above, the vertex set of the Tübingen triangle tiling is a cyclotomic model set with underlying -module . By Corollary 4.3 there is a -polygon of class in with edges if and only if ; see Figure 1 for an -icosagon of class in .

Acknowledgements

It is a pleasure to thank Michael Baake, Richard J. Gardner and Uwe Grimm for their cooperation and for useful hints on the manuscript.

References

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