aa r X i v : . [ m a t h . M G ] N ov LOCAL GROUPS IN DELONE SETS
NIKOLAY DOLBILIN
Abstract.
In the paper, we prove that in an arbitrary Delone set X in 3 D space, thesubset X of all points from X at which local groups has axes of the order not greater than6 is also a Delone set. Here, under the local group at point x ∈ X is meant the symmetrygroup S x (2 R ) of the cluster C x (2 R ) of x with radius 2 R , where R (according to Delone’stheory of the ’empty sphere’) is the radius of the largest ’empty’ ball, that is, the largestball free of points of X .The main result (Theorem 2.1) seems to be the first rigorously proved statement on absolutely generic Delone sets which implies substantial statements for Delone sets withstrong crystallographic restrictions. For instance, an important observation of Shtogrin onthe boundedness of local groups in Delone sets with equivalent 2 R -clusters (Theorem 1.7.)immediately follows from Theorem 2.1.In the paper, the ’crystalline kernel conjecture’ (Conjecture 1) and its two weaker versions(Conjectures 2 and 3) are suggested. According to Conjecture 1, in a quite arbitrary Deloneset, points with locally crystallographic axes (of order 2,3,4, or 6) only inevitably constituteessential part of the set. These conjectures significantly generalize the famous statement ofCrystallography on the impossibility of (global) 5-fold symmetry in a 3D lattice. Introduction and basic definitions
This paper grew out of the local theory for regular systems, i.e for Delone sets with verystrong requirements. On the other hand, here we consider an arbitrary Delone sets in R without any additional assumptions. For example, for a Delone set, we do not suppose atypical condition of the local theory such as the sameness of clusters of certain radius as wedid it in numerous papers (see e.g., [2], [6], [8]). Another feature of the paper is as follows,for a Delone set, we consider local groups operated over clusters of radius, namely 2 R (fordefinitions see below).In [9], A.L. Mackay says: ”In an infinite crystal there may be extra elements of symmetrywhich operate over a limited range. These may be seen by non-space-group extinctions indiffraction pattern. . . . The local operations need not to be ’crystallographic’.” Nevertheless,in this paper, we show that the last statement for regular systems (i.e. for sets with transitivegroups) should be significantly revised. If we fix for a Delone sets in 3D space, of type ( r, R )the range of action of local groups as 2 R (for details see below), then, it turns out, one canobtain interesting results on properties of such groups. To accurately formulate the resultsand open hypotheses we will need several definitions and notations.Euclidean distance between points x and x ′ in euclidean space R d is denoted by | x, x ′ | .Let d ( z, X ) denote distance from z ∈ R d to set X ⊂ R d , i.e. d ( z, X ) := inf x ∈ X | z, x | ]. Definition 1.1. [Delone set] Given positive real numbers r and R , a point subset X of R d is called a Delone set of type ( r, R ) if the following two conditions hold:(1) an open d -ball B oz ( r ) of radius r centered at any point z of space contains at most onepoint of X ; (2) a closed d -ball B z ( R ) of radius R centered at an arbitrary point z of space contains atleast one point of X . It is clear that a Delone set X of type ( r, R ) is obviously a Delone set of type ( r ′ , R ′ ) if r ′ ≤ r and R ′ ≥ R . Therefore, we can adopt the convention in designating the parameters( r, R ) to a Delone set X as follows. For a given Delone set X , we choose as r the largestpossible value satisfying Condition (1) of Definition 1.1, and choose as R the smallest valuesatisfying Condition (2).We will need also the following interpretations of the parameters r and R :inf x,x ′ ∈ X | xx ′ | = 2 r, sup z ∈ R d d ( z, X ) = R. (1)Thus, the value of r equals the half of the smallest (infimum) inter-point distance in X .The value of R is a distance from the most remote from X point of space R d to the set X .In the local theory of Delone sets, the key concepts are that of a cluster. Definition 1.2 ( ρ -cluster) . Let x be a point of a Delone set X of type ( r, R ) , ρ ≥ , and let B z ( ρ ) be a ball with radius ρ centered at point z ∈ R d . We call a point set C x ( ρ ) := X ∩ B x ( ρ ) the cluster of radius ρ at point x or simply the ρ -cluster at x . Definition 1.3 (equivalent clusters) . Two clusters C x ( ρ ) and C x ′ ( ρ ) of the same radius ρ at points x and x ′ are said to be equivalent if there is an isometry g ∈ Iso ( R ) such that g ( x ) = x ′ and g ( C x ( ρ )) = C x ′ ( ρ ) . (2) Definition 1.4 (cluster group) . Given a point x ∈ X and its ρ -cluster C x ( ρ ) , a group S x ( ρ ) of all isometries s ∈ Iso ( R d ) which leave the x fixed and the cluster C x ( ρ ) invariant is calledthe cluster group: S x ( ρ ) := { s ∈ Iso ( R d ) | s ( x ) = x, s ( C x ( ρ )) = C x ( ρ ) } . Groups of equivalent clusters C x ( ρ ) and C x ′ ( ρ ) are conjugate in the full group Iso ( d ) ofisometries: S x ( ρ ) = g − S x ′ ( ρ ) g , where g is determined by conditions of Definition 1.3 .It is clear that as the radius ρ increases, the cluster C x ( ρ ) expands but the cluster group S x ( ρ ) never increases and sometimes can contract only. It is clear that if 0 ≤ ρ < r group C x ( ρ ) = O x (3)? i.e. the full point group of all isometries that leave point x fixed. On theother and, it is well-know that the S x (2 R )Since the main result grew up ideologically from the local theory of regular systems, here,we briefly recall basic concepts of this theory. Modern in form, the following definitions ofregular system and crystal are equivalent to those that go back to E.S. Fedorov. Definition 1.5 (regular system, crystal) . A Delone set X is called a regular system if it isan orbit of some point x with respect to a certain space group G ⊂ Iso ( d ) , i.e. X = G · x = { g ( x ) | g ∈ G } ; a Delone set X is a crystal if X is a union of several orbits: X = ∪ mi =1 G · x i . We emphasize that the notion of a regular system is an essential case of the crystal, i.e. amulti-regular system, and generalizes the lattice concept. In fact, a lattice is a a particularcase of a regular system when G is a group of translations generated by d linearly independent OCAL GROUPS IN DELONE SETS 3 translations. Moreover, due to a celebrated theorem by Schoenflies and Bieberbach, anyregular system is the union of congruent and mutually parallel lattices.The local theory of regular systems began with the Local criterion in [2].
Theorem 1.6 (Local Criterion, [2]) . A Delone set X is a regular system if and only if thereis some ρ > such that the following two conditions hold:1) all ρ + 2 R -clusters are mutually equivalent;2) S x ( ρ ) = S x ( ρ + 2 R ) for x ∈ X . In [3], [4], this criterion has been generalized for crystals, i.e multi-regular systems.From now on, we restrict ourselves only to the 3D case. One of central problems of thelocal theory of regular systems is to search for an upper (and lower) bound for the regularityradius, i.e. a minimum value ˆ ρ > ρ -clusters in a Delone set X implies the regularity of the set X ⊂ R . In [6],[8] is given a proof of the upper boundˆ ρ ≤ R . The long proof starts with selection of a special finite o of finite subgroups of O (3). Groups of this list have a chance to occur in Delone sets with equivalent 2 R -clustersas local groups S x (2 R ). The list of selected groups is provided by Theorem 1.7 found byShtogrin in the late 1970’s but published only in 2010 ([10]). Theorem 1.7 ([10]) . If in a Delone set X ∈ R all R -clusters are mutually equivalent,then the order of any rotational axis of S x (2 R ) does not exceed 6. Quite recently, [7], it was realized for the first time that an important statement aboutgroups in Delone sets with significant requirements on equivalent clusters may followfrom a certain statement true for pretty general
Delone sets.Namely, in [7], Theorem 1.8 is proved. Given an arbitrary Delone set X ⊂ R and x ∈ X ,let the maximal order of rotational axis in group S x (2 R ) be denoted by n x . Theorem 1.8.
In a Delone set X ⊂ R there is at least one point x with n x ≤ . It is obvious that Theorem 1.8 immediately implies Theorem 1.7 In fact, the subset X ofall points in a Delone set X with n x ≤ X is a Delone set itself.2. The main result and conjectures
Theorem 2.1 (Main result) . Given a Delone set X ⊂ R of type ( r, R ) , let X ⊆ X be thesubset of all points x ∈ X such that the maximal order n x of a rotation axes in S x (2 R ) doesnot exceed . Then X is a Delone set of a certain type ( r ′ , R ′ ) , where r ≤ r ′ ≤ R ′ = kR forsome k independent on X . At the moment, we do not care on the value kR of the upper bound for the parameter R ′ of the X . For us, so far it is more important to establish that the subset X is always aDelone subset.From now on, we will focus on clusters C x (2 R ) of radius 2 R and their groups S x (2 R ). Aswell-known, for a Delone set X , the 2 R -clusters all are full-dimensional (i.e. the dimensionof their convex hulls is d ). Hence, the cluster groups S x (2 R ) are necessarily finite. At thesame time, we emphasize that the value of 2 R is the smallest value of radius that guaranteesthe finiteness of the cluster group of radius 2 R for any Delone set with parameter R . Inother words, for an arbitrary ε > X with parameter R suchthat for some x ∈ X in S x (2 R − ε ) is infinite. N. DOLBILIN
By virtue of the above, we will single out group S x (2 R ) and call it a local group at x .Theorem 2.1 immediately implies Theorem 1.7 which concerns a Delone set X with mu-tually equivalent 2 R -clusters. Really, for such a Delone set X , the local groups at all pointsare pairwise conjugate and existence of points x with n x ≤ X ⊆ X we select points x with the condition n x = 5, i.e., all pointsof X whose local groups contain axes of only ’crystallographic’ orders 2, 3, 4, or 6. We callthe subset of all such points in X a crystalline kernel of X and denote by K . Conjecture 1 (Crystalline kernel conjecture) . The crystalline kernel K of a Delone set X is a Delone subset with some parameter R ′ ≤ kR , where k is some constant which does notdepend on X . Let Y denote the subset of all points x ∈ X at which local groups S x (2 R ) do not containthe pentagonal axis. It is clear that K ⊆ Y and if K is a Delone set then Y is a Delone settoo. Therefore Conjecture 1, if proven, immediately implies the following two Conjectures 2and 3. Conjecture 2 (5-gonal symmetry conjecture) . Given a Delone set X ⊂ R , the subset Y ofpoints x , whose groups S x (2 R ) are free of 5-fold axes, is also a Delone set. In its turn, Conjecture 2 enforces the following statement that seems to be much easierproved.
Conjecture 3 (Weak 5-gonal symmetry conjecture) . Given a Delone set X ⊂ R withmutually equivalent R -clusters, the local group S x (2 R ) contains no 5-fold axis. It is obvious that these hypotheses relate to a celebrated crystallographic theorem on theimpossibility of the global 5-fold symmetry in a three-dimensional lattice. Conjectures 1–3significantly reinforce a classical statement on famous crystallographic restrictions.It is well-known that for a 3-dimensional lattice even in the group S x ( r ), there are no the5-fold symmetry, where r = 2 r ≤ R is the minimum inter-point distance in the lattice.In contrast to lattices, in regular systems in 3 D space, the pentagonal symmetry can locally manifest itself on clusters of a certain radius less than 2 R . So, for instance, there are regularsystems such that even group S x ( r ) contains the 5-fold axis, but the local group S x (2 R ) doesnot. Here, in the systems, r ( r < r < r < R ) is the 3rd inter-point distance in X . But,it is still unknown whether there are regular systems with 5-fold symmetrical 2 R -clusters.Since the regularity radius for dimension 3 is not less than 6 R (see [1]), among Delone sets X with mutually equivalent 2 R -clusters, there are non-regular and even non-crystallographicsets. Thus, even the weakest Conjecture 3 concerns also a wide class of those non-regularsets. 3. Proof of Theorem 2.1
Proof.
Generally speaking, in the local group S x (2 R ) ⊂ O (3), x ∈ X , there are several axesof maximal order n x . Bearing in mind the well-known list of all finite subgroups of O (3),we see that more than one axes of the maximal order n x in S x (2 R ) cannot happen provided n x >
5. Let ℓ x be one of those axes. Since rk( C x (2 R )) = 3, i.e. the convex hull of the2 R -cluster is 3-dimensional, in C x (2 R ), there are necessarily points off the ℓ x . OCAL GROUPS IN DELONE SETS 5
Since X is a subset of X , the minimal inter-point distance 2˜ r in X (in fact, the infimumof such distances) is not less than 2 r = inf x,x ′ ∈ X | x, x ′ | . In order to prove that X is a Deloneset with a certain parameter ˜ R , we will prove that the distance from a given point z ∈ R to the nearest point of X does not exceed ˜ R : min x ∈ X ′ | z, x | ≤ ˜ R (due to interpretations (1)for the parameters r and R in Section 1. We will be looking for the point x ∈ X nearest to z ∈ R by walking along a special finite point sequence in X . Definition 3.1.
A sequence of points [ x , x , . . . x m , . . . ] ∈ X (finite or infinite, no matter‘)is termed an off-axial chain if the following condition holds for any i = 1 , , . . . :the point x i +1 ∈ X is the nearest point to x i among all points of X which are off the axis ℓ x i , where the axis ℓ x i means an axis of the local group S x i (2 R ) of the maximal order n x i .In case the subgroup of all (orientation-preserving) rotations of S x i (2 R ) is trivial (that is, inthe local group at x i no axes through x ), any nearest to x i point of X can be chosen as x i +1 . Note that for any point x ∈ X , there are off-axial sequences [ x (= x ) , x , x , . . . ]. Lemma 3.2.
Given a Delone set X and an off-axial sequence [ x , x , . . . , x m , . . . ] ⊂ X ,assume that x i / ∈ X , ∀ i ∈ , m . Then for i ∈ , m the following holds: | x i , x i +1 | < . i − · R and | x , x m | < . · R = 15 . R, for all m. (3) Proof. (of Lemma 3.2). Let [ x , x , , . . . , x m , . . . ] be an off-axial chain and assume that itbelongs X \ X . Recall that, by construction, in this chain, the length r ∗ i of each link x i , x i +1 is less than 2 R . Hence x i +1 ∈ C x i (2 R ). Therefore, the rotation g x i ∈ S x i (2 R ) can be appliedto point x i +1 too.By assumption, x / ∈ X , that is, n x ≥
7. Let x be the nearest to x point x which isoff the axis l x , also let g x be a rotation around axis ℓ x by angle 2 π/n x . Since r ∗ ≤ R the cluster C x (2 R ) necessarily contains vertices of a regular n x -gon P which is generatedby the rotation g x ∈ S x (2 R ) applied to the point x . The polygon P is located in a planeorthogonal to the ℓ x . The center of P is on l x (Figure 1).Denote the side-length of P by a . Since the circumradius of P does not exceed r ∗ x and n x ≥ a and r ∗ x the following estimate r ∗ x ≤ a ≤ r ∗ x sin πn x ≤ r ∗ x sin π < . r ∗ x < . · R. (4) P z x x y y ′ < R r ∗ x a Figure 1.
Polygon P and the beginning of an off-axial chain [ x , x , . . . ] N. DOLBILIN
Assuming now that x / ∈ X , i.e. n x ≥
7, we will construct the next point x in the off-axialchain [ x , x , . . . ] and obtain upper estimates for r x and a (see inequalities (5) below).The rotation g x about the axis ℓ x is assumed to belong to the local group S x (2 R ). Since x ∈ C x (2 R ), the g x can be applied to the point x . Hence the cluster C x (2 R ) necessarilycontains vertices of a regular n x -gon P generated by rotation g x applied to the point x .Denote the side-length of P by a and note that a ≤ r ∗ ≤ R .Point x , like a vertex of the regular n x -gon P , has two adjacent vertices in P at distance a from vertex x . Vertex x and two adjacent vertices of P form a non-collinear triple.Therefore, no matter how the axis ℓ x passes through the point x , anyway, at least one oftwo neighboring points is off the axis ℓ x . It follows that the distance r ∗ x from x to thenearest point x ∈ X \ ℓ x does not exceed a . Since we bear in mind that in the set X \ ℓ x ,there may be points nearer to the x than distance a we have r ∗ x ≤ a .Since n x ≥
7, by the same argument as above, we have r ∗ x < a ≤ r ∗ x sin πn x ≤ r ∗ x sin π < . r ∗ x < . · R. (5)This reasoning can be repeated over and over again. Under condition n x i ≥ ∀ i ∈ , m , weget the off-axial chain [ x , x , . . . ] for which the sequence of inter-point distances r ∗ x i = | x i x i +1 | is dominated by a geometric progression r ∗ x i +1 < . r ∗ x i < (0 . i r ∗ x < (0 . i R. (6)Thus, we obtain the required inequalities (3). Lemma 3.2 is proved. (cid:3) Now we are going to complete the proof of Theorem 2.1. Given a Delone set X , fromLemma 3.2 it follows that any off-axial chain provided n x i ≥ m can be bounded from above: m < M := log( Rr ) / log . . This implies that in the chain[ x , x , . . . ] there are points x m . with n x m ≤
6, i.e. x m ∈ X . By Lemma 3.2 the segment-length | x , x, x m | < . z of space tothe nearest point of the subset X . Let the nearest to z point of X be x (see Figure 1) and x m ∈ X (by Lemma 3.2). Then | z, x | ≤ R and we getmin x ∈ X | z, x | ≤ | z, x m | ≤ | z, x | + | x , x m | = 16 . R. (7)In other words, we proved that the subset X is a Delone set with some parameter ˜ R ≤ . R. Theorem 2.1 is proved. (cid:3) Concluding remarks
We emphasize that the upper bound for the parameter ˜ R , established here, is far fromoptimal one. We believe that we will soon be able to present a sharper bound [11] . Thepurpose of this paper was to present the‘ result, in our opinion, of a new type. The resultsuggests a few conjectures which should be interesting both in itself and in context of thetheory of quasicrystals. So, for instance, in Penrose patterns, in structures of real Shechtmanquasicrystals, the centers of 2 R -clusters with local 5-fold symmetry constitute a rich Delonesubset. At the same time, in these known quasicrystalline structures, there are also Delonesubsets of points with local crystallographic axes (including identical) ‘ However, according OCAL GROUPS IN DELONE SETS 7 to Conjecture 1 , not only in in these structures but in any other possible Delone sets, pointswith local crystallographic axes inevitably constitute an essential part of the structure.5.
Acknowledgments
This work was supported by the Russian Science Foundation under grant 20-11-20141 andperformed in Steklov Mathematical Institute of Russian Academy of Sciences.
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