Locally optimal 2-periodic sphere packings
LLocally optimal 2-periodic sphere packings
Alexei Andreanov ∗ Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS),Daejeon 34051, Republic of Korea
Yoav Kallus
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
Abstract
The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell ( m -periodic packings). For the case m = 1 (latticepackings), Voronoi presented an algorithm to enumerate all local optima in afinite computation, which has been implemented in up to d = 8 dimensions.We generalize Voronoi’s algorithm to m > d = 3 ,
4, and 5.In particular, we show that no 2-periodic packing surpasses the density of theoptimal lattice in these dimensions. A partial enumeration is performed in d = 6. Keywords: sphere packing, periodic point set, quadratic form, Ryshkovpolyhedron
1. Introduction
The sphere packing problem asks for the highest possible density achieved byan arrangement of nonoverlapping spheres in a Euclidean space of d dimensions.The exact solutions are known in d = 2 [1], d = 3 [2], d = 8 [3] and d = 24 [4]. ∗ Corresponding author
Email addresses: [email protected] (Alexei Andreanov), [email protected] (YoavKallus)
Preprint submitted to Journal of L A TEX Templates July 9, 2018 a r X i v : . [ m a t h . M G ] A p r his natural geometric problem is useful as a model of material systems andtheir phase transitions, and, even in nonphysical dimensions, it is related tofundamental questions about crystallization and the glass transition [5]. Thesphere packing problem also arises in the problem of designing an optimal error-correcting code for a continuous noisy communication channel [6]. Perhaps evenmore than the obvious applications of the sphere packing problem, contributingto its importance is the unexpected wealth of remarkable structures that appearas possible solutions and merit study in their own right [7, 8].There does not appear to be a systematic solution or construction thatachieves the optimal packing in every dimension, and every dimension seemsto have its own quirks, unearthing new surprises [6]. In some dimensions, suchas d = 8 and d = 24, the solution is unique and given by exceptionally sym-metric lattices. In others, such as d = 3, there is a lattice that achieves thehighest density, but this density can also be achieved by other periodic packingswith larger fundamental unit cells and even by aperiodic packings. For d = 10it seems that the highest density, achieved by a periodic arrangement with 40spheres per unit cell, cannot be achieved by a lattice. In any given dimension,the densest packing known is periodic, but it is not known whether there issome dimension where the densest packing is not periodic. In fact, frighteninglylittle is known in general as we go up in dimensions. It is possible that in alldimensions the densest packing is achieved by a periodic packing with a uni-versally bounded number of spheres in the unit cell, that in all dimensions it isachieved by a periodic packing, but only with unboundedly many spheres in theunit cell, or that in some dimension it is not achieved by a periodic packing atall. In any case, periodic packings with arbitrarily many spheres in the unit cellcan approximate the optimal density in any dimension to arbitrary precision.Therefore, it is reasonable to ask for the optimal density achieved by aperiodic packing in d dimensions with m spheres per unit cell, which we denote φ d,m . In the limit m → ∞ , this density approaches (and possibly equals forsome finite m ) the optimal packing density φ d . Apart from the dimensionswhere φ d is known, there are no cases with m > φ d,m is known. In this2aper, we describe a general method for calculating φ d, , and use it to obtain φ , = φ , and φ , = φ , . Our method could also be generalized to larger m ,but becomes more complicated.For the case m = 1, corresponding to lattices, Voronoi gave an algorithmto enumerate all the locally optimal solutions to the problem. In geometricalterms, Voronoi’s algorithm uses the fact that the space of lattices up to isometrycan be parameterized (redundantly) by positive definite quadratic forms. Thesubset in the linear space of quadratic forms that corresponds to lattices withno two points less than a certain distance apart is a polyhedron. The additionalfact that the density of lattice points is a quasiconvex function implies that localmaxima can only occur at the vertices of this polyhedron. Because there are onlyfinitely many vertices that correspond to distinct lattices, the local optima ofthe lattice sphere packing problem can be fully enumerated. This interpretationof Voronoi’s algorithm was suggested by Ryshkov [9] and so the polyhedron isknown as the Ryshkov polyhedron. The lattices corresponding to vertices ofthe polyhedron are known as perfect lattices , but not every perfect lattice islocally optimal, or an extreme lattice . This algorithm has been implementedand executed to determine φ d, for d ≤
8, as well as to enumerate all perfectlattices in these dimensions [10]. However, the exploding number of perfectlattices as d increases make the execution of this algorithm currently unfeasibleeven for d = 9 [11].Sch¨urmann transported some of the concepts from Voronoi’s theory to thecase of periodic packings and was able to show that any extreme lattice, whenviewed as a periodic packing, is still locally optimal [12, 13]. However thenonlinearities that arise in the case m > m . These elements behave nicely enough for m = 2to allow us to provide an algorithm for the enumeration of all locally optimalpackings, and we execute this algorithm for d ≤ . Theoretical preparation For a set of points, Ξ ⊂ R d , the packing radius is the largest radius, such thatif balls were centered at each of the points, they would not overlap. Namely,the packing radius is half the infimum distance between any two points, ρ (Ξ) = inf x,y ∈ Ξ (cid:107) x − y (cid:107) . A set of points is periodic if there are d linearly independentvectors a , . . . , a d , such that the set is invariant under translation by any ofthese vectors. The group generated by a , . . . , a d is a lattice, Λ = { (cid:80) di =1 n i a i : n , . . . , n d ∈ Z } . If all translations that fix the set are in Λ, we say that Λ is aprimitive lattice for the periodic set. The primitive lattice is unique. A set thatis periodic under translation by a lattice Λ and has m orbits under translationsby Λ called m -periodic. We will only deal with sets of positive packing radius,and therefore m is necessarily finite.The number density of a periodic set Ξ is given by δ (Ξ) = m det Λ , (1)where det Λ is the volume of a parallelotope generated by the generators of Λ, { (cid:80) di =1 x i a i : 0 ≤ x , . . . , x d ≤ } . This parallelotope and its translates underΛ tile R d , and its volume is the same as the volume of any other fundamentalcell of Λ, that is, a polytope whose Λ-translates tile R d . Therefore, this is aproperty of Λ, and not of the particular basis. Also, since Λ itself is not uniquelydetermined from Ξ, but can be any sublattice of the primitive lattice of Ξ, it isimportant to note that the formula for δ (Ξ) above is independent of the choiceof Λ. The largest density that can be achieved by a packing of equal-sized ballscentered at the points of Ξ is φ (Ξ) = V d ρ (Ξ) d δ (Ξ). where V d is the volume ofa unit ball in R d . In any family of point sets closed under homothety, such asperiodic sets or m -periodic sets, maximizing the packing density φ is equivalentto maximizing the number density under the constraint ρ (Ξ) ≥ ρ for somefixed ρ . 4 .2. Voronoi’s algorithm A 1-periodic packing is a translate of a lattice. The packing radius of alattice is given by ρ (Λ) = min l , l (cid:48) ∈ Λ (cid:107) l − l (cid:48) (cid:107) = min l ∈ Λ (cid:107) l (cid:107) . (2)The packing radius of a lattice and the volume of its fundamental cell are bothinvariant under rotations of the lattice. Therefore, to find the densest latticepacking, we only need to consider lattices up to rotation. Consider a lattice Λ = A Z d generated by a , . . . , a d . The quadratic form Q : Z d → R , Q ( n , . . . , n d ) = (cid:107) n a + . . . + n d a d (cid:107) determines Λ up to rotations. However, since a latticecan have different generating vectors, different quadratic forms can correspondto the same lattice. Q and Q (cid:48) correspond to the same lattice if and only if Q = Q ◦ U , where U ∈ GL d ( Z ). This is precisely the group of linear mapsthat map Z d to itself. The packing radius of a lattice Λ corresponding to thequadratic form Q is ρ ( λ ) = (min Q ) / , where min Q is the minimum of Q overnonzero vectors. The determinant of Λ is given by (det Q ) / .The linear space of quadratic forms can be identified with the linear space ofsymmetric matrices S d using the standard basis of Z d , so that Q ( n ) = n T Q n .And the natural inner product in this space is given by (cid:104) Q, Q (cid:48) (cid:105) = tr QQ (cid:48) . Thecondition min Q ≥ λ can be written as the intersection of infinitely many linearinequalities: Q ( n ) = (cid:104) nn T , Q (cid:105) ≥ λ for all n ∈ Z d (3)It can be easily checked that for λ >
0, Eq. (3) implies that Q is a positive defi-nite matrix, and a lattice corresponding to it can be recovered, e.g. by Choleskydecomposition. Ryshkov observed that the set { Q : min Q ≥ λ } , albeit an infi-nite intersection of linear inequalities, behaves locally like a finite intersection,namely a polyhedron, and therefore this set is known as the Ryshkov polyhe-dron [9]. Though the Ryshkov polyhedron has infinitely many vertices, thereare only finitely many orbits under the action of GL d ( Z ). Therefore, by start-ing at some vertex, enumerating the vertices with which it shares an edge, and5epeating the process for any enumerated vertex not of the same orbit as aprevious vertex, one can computationally enumerate all the orbits of vertices ofthe Ryshkov polyhedron. Since the determinant is a quasiconcave function, itslocal minima in the polyhedron can occur only on the vertices, and to find theglobal minimum it is enough to compare its value at all the vertices. To parameterize m -periodic point sets, we can use a similar scheme. An m -periodic point set is the union of m translates of a lattice, Ξ = (cid:83) m − i =0 (Λ + b i ),where Λ = A Z d . Without loss of generality, we can take b = 0. Now, considerthe set M ⊆ Z d + m − given by M = Z d × ( E − E ), where E = { , e , . . . , e m − } is the standard basis of Z m − plus the zero vector. We define the followingfunction J : M → R , which determines Ξ up to rotation: J ( n , . . . , n d , l , . . . , l m − ) = (cid:107) n a + . . . + n d a d + l b + . . . + l m − b m − (cid:107) . (4)This function can be extended uniquely to a quadratic form over R d + m − , rep-resented by a symmetric matrix J ∈ S d + m − , namely J = Q R T R S , (5)where Q ij = a i · a j , R ij = b i · a j , and S ij = b i · b j . The packing radius of Ξis related to the minimum of the quadratic form. Namely, ρ (Ξ) = (min J ) / ,where min J is the minimum of J over M \ { } . Let R ( λ ) = { J : min J ≥ λ } .This set is defined as the intersection of linear inequalities: J ∈ R ( λ ) iff J ( k ) = (cid:104) kk T , J (cid:105) ≥ λ for all k ∈ M \ { } . (6)We want to show that R ( λ ), like the Ryshkov polyhedron, behaves locally likea polyhedron. Lemma 1.
Let Q : R d → R be a positive definite quadratic form, satisfying Q ( n ) ≥ λ for all n ∈ Z d and tr Q ≤ C . Then any vector x ∈ R d satisfying Q ( x ) ≤ also satisfies (cid:107) x (cid:107) ≤ M , where M depends on d , λ , and C . Theorem 1.
Let W = { J ∈ R ( λ ) : | J ij | ≤ L } . There are only finitely many k ∈ M such that for some J ∈ W , J ( k ) ≤ λ .Proof. Let k = ( n , l ). There are only finitely many choices for l . So, we needonly show that for any fixed choice of l , there are finitely many n ∈ Z d suchthat J ( k ) ≤ λ for some J ∈ W . We have J ( k ) = n T Q n + 2 l T R n + l T S l = ( n + q ) T Q ( n + q ) + l T S l − q T Q q , (7)where q = Q − R T l . Therefore, if J ( k ) ≤ λ then ( n + q ) T Q ( n + q ) ≤ λ + q T Q q − l T S l ≤ λ + q T Q q = λ + l T RQ − R T l .Since l T RR T l ≤ dL and tr Q ≤ dL , we have from the lemma that l T RQ − R T l ≤ dL M . Therefore, if J ( k ) ≤ λ then ( n + q ) T Q ( n + q ) ≤ λ + 4 dL M . Again,from the lemma, we have that ( n + q ) T ( n + q ) ≤ λM + 4 dL M . There-fore, there can only be finitely many choices of n for each choice of l such that J ( k ) ≤ λ for some J ∈ W .So, R ( λ ) is locally a polyhedron in the sense that any intersection of R ( λ )with a bounded polyhedron is a bounded polyhedron. We call it the Ryshkov-like polyhedron. The symmetries of R ( λ ) are tightly linked with the symmetries of the set M ⊂ Z d + m − . Namely, if T : R d + m − → R d + m − is a linear map such that T ( M ) = M , then J (cid:55)→ J ◦ T maps R ( λ ) to itself.We decompose T into blocks with a top-left block of size d × d . It is easy tosee that the bottom-left block must be zero, or else there is always some k ∈ M such that T ( k ) = ( n , l ), where l (cid:54)∈ ( E − E ). Therefore we write T = U V W . (8)7s a consequence, we also have that U and W must be invertible as maps Z d → Z d and ( E − E ) → ( E − E ). Therefore, U ∈ GL d ( Z ) and W is apermutation of E , namely W = · · · · · · ... ... ... . . . Π − − · · · · · · · · · ... ... . . . , (9)where Π is a m × m permutation matrix.It is easy to check that whenever U ∈ GL n ( d ), W is a permutation of E ,and V is an arbitrary d × ( m −
1) integer matrix, then T is a symmetry of M .Let us call this group of symmetries Γ, and use its members to act on elementsof M by k (cid:55)→ T k or elements of R ( λ ) by J (cid:55)→ J ◦ T . We conjecture that for any λ > R ( λ ) has finitely many faces (in particular vertices) up to the action ofΓ, but we do not have a proof, except in the cases where we have enumeratedthe vertices (see Sec. 4), and found, by the fact of the algorithm halting, thatthere were finitely many vertices. Equation (5) described how to obtain a quadratic form J ∈ R ( λ ) from an m -periodic packing Ξ of packing radius ρ (Ξ) ≥ λ / . However, the reverseoperation is not always possible. Clearly, rank J = d is a necessary condition.In fact, it is also sufficient, since J ∈ R ( λ ) for λ > Q is posi-tive definite, and therefore a , . . . , a d can be recovered through, e.g., Choleskydecomposition, and b , . . . , b m − by solving R ij = b i · a j .Let R ( λ ) = { J ∈ R ( λ ) : rank J = d } . For m = 2, we can replace thecondition rank J = d by det J = 0 or λ min ( J ) = 0, where λ min denotes thesmallest eigenvalue. In general, we can write it as the vanishing of the Schurcomplement S − RQ − R T = 0. 8 .6. The density objective We wish to find the maximum of δ (Ξ) among m -periodic sets Ξ of packingradius at least ρ . This is equivalent to finding the minimum of f ( J ) = det Q among J ∈ R (4 ρ ), where Q is the top-left block of J . While f ( J ), like theobjective in Voronoi’s algorithm, is quasiconcave, the nonlinearity of the rankconstraints does not allow for a straightforward characterization of the localminima.A vivid illustration of a new type of local minima that can arise is the9-dimensional fluid diamond packing, D +9 ( t ) = D ∪ ( D + t ). The lattice D = { n ∈ Z n : n + n + . . . + n ∈ Z } has a packing radius ρ = 1 / √ , , . . . , ) is at a distance3 / > ρ from the closest lattice point, and so when t is in the neighborhood ofthis deep hole, D +9 ( t ) also has packing radius ρ = 1 / √
2. As a 2-periodic pointset, this is clearly a local maximum of the density since a nearby 2-periodicpoint set would have a lattice of symmetry nearby the D lattice and thereforea smaller packing radius or a larger determinant. Thus we have a 9-dimensionalfamily of locally optimal 2-periodic point sets, which achieves the highest knownpacking density for spheres in 9 dimensions. (In fact, the set of values of t suchthat the packing radius is 1 / √ D , and the twolattices can flow past each other an unbounded distance). A similar situation ina lower dimension can be constructed using the extreme A lattice and its deephole.This vivid example should discourage us from attempting to transport themethods of the Voronoi algorithm, where a critical result was that all localoptima lie on vertices and are therefore isolated. However, we will show that, atleast in the case m = 2, the fluid diamond example illustrates the only possibleproblem we may encounter. Namely, for m = 2, a local optimum of f ( J ) in R ( λ ) either lies on an edge of R ( λ ) or is part of a fluid family, where twocopies of a constant extreme lattice flow relative to each other.To start, we define a sufficient condition for local optimality by linearizing9ll of the constraints, which we call algebraic extremeness , and prove that it isindeed a sufficient condition. Let J ∈ R ( λ ). The set { J (cid:48) : rank J (cid:48) = d } is a[ ( d + m )( d + m − − m ( m − J , and its tangent space is given by T J = J + ( S [ N ( J )]) ⊥ = { J (cid:48) : (cid:104) J (cid:48) − J, uu T (cid:105) = 0 for all u ∈ N ( J ) } ,where N ( J ) is the null space of J , and S [ N ( J )] is the space of quadratic formsover N ( J ). The linear inequality constraints, encoded in the polyhedron R ( λ ),give rise in the neighborhood of J to the cone C J = { J (cid:48) : J (cid:48) ( k ) ≥ λ for all k ∈ M such that J ( k ) = λ } .Finally, the gradient of the objective is proportional to G J = Q −
00 0 . Definition 1.
A configuration J ∈ R ( λ ) is called algebraically extreme if (cid:104) J (cid:48) − J, G J (cid:105) > J (cid:48) ∈ C J ∩ T J except J (cid:48) = J . Theorem 2.
Let J ∈ R ( λ ) be algebraically extreme. Then J is an isolated minimum of f ( J (cid:48) ) among J (cid:48) ∈ R ( λ ) . J lies on a m ( m + 1) -dimensional face of R ( λ ) .Proof. The first claim follows as the usual sufficient optimality criterion for asmooth optimization problem with smooth equality and inequality constraints[14].To prove the second claim, note that if J lies on a r -dimensional face of R ( λ )(considering the polyhedron itself as a full-dimensional face) and r > m ( m +1),then C J contains an affine space passing through J of dimension r . Since T J is an affine space of codimension m ( m + 1) passing through J , then C J ∩ T J must include a positive-dimensional affine subspace, and it is not possible for (cid:104) J (cid:48) − J, G J (cid:105) to be positive everywhere on this subspace, even with J excepted.Therefore, J is not algebraically extreme.10 efinition 2. A configuration J ∈ R ( λ ) is called a fluid packing if it sits ona continuous, nonconstant curve J (cid:48) : [0 , → R ( λ ), such that J (cid:48) (0) = J , theupper-left block Q (cid:48) ( t ) is constant, and J (cid:48) ( t ) is a local minimum of f over R ( λ )for all 0 ≤ t ≤ Theorem 3.
Let m = 2 , and let J ∈ R ( λ ) . If J is a local minimum of f ( J (cid:48) ) among J (cid:48) ∈ R ( λ ) then one of the following is true: J is algebraically extreme. J is a fluid packing.Proof. Without loss of generality, let λ = 4. Let { k , . . . , k s } = { k : J ( k ) =4 } = J − (4). The quadratic form J is a local optimum of the problemMinimize f ( J (cid:48) ),subj. to g i ( J (cid:48) ) (cid:44) − (cid:104) k i k Ti , J (cid:48) (cid:105) ≤ i = 1 , . . . , s , h ( J (cid:48) ) (cid:44) λ min ( J (cid:48) ) = 0. (10)As a consequence, it satisfies the Karush-Kuhn-Tucker condition. Namely,there exist u k ≥ v such that ∇ f + (cid:80) si =1 u i ∇ g i + v ∇ h = 0. We con-sider the derivative with respect to J (cid:48) d +1 ,d +1 , which we denote ∂ d +1 ,d +1 . Since ∂ d +1 ,d +1 f = 0, ∂ d +1 ,d +1 g i ≤
0, and ∂ d +1 ,d +1 h >
0, we have that v ≥
0. TheLagrangian corresponding to these KKT coefficients is L = f + (cid:80) u i g i + vh .Let C be the marginal cone C = { ˜ J : (cid:104)∇ g i , ˜ J (cid:105) = 0 for all i ∈ I + , (cid:104)∇ g i , ˜ J (cid:105) ≥ i ∈ I , (cid:104)∇ h, ˜ J (cid:105) = 0 } , (11)where I + = { i : u i > } and I = { i : u i = 0 } . Suppose that J is notalgebraically extreme, then the marginal cone is nontrivial. Consider the Hessianof the Lagrangian H = hess L = hess f + v hess h , where, e.g., (cid:104) ˜ J, (hess f ) ˜ J (cid:105) =( d /dt ) f ( J + t ˜ J ). A necessary condition for J to be a local optimum of (10)is that (cid:104) ˜ J, H ˜ J (cid:105) is nonnegative for all ˜ J ∈ C [14, p. 216]. However, bothhess f and hess h are negative semidefinite (for f , this is the quasiconcavity11f the determinant function; for h , this is a well-known result of second-ordereigenvalue perturbation theory). It follows that C must lie in the null-space ofhess f , that is, the Q component of any vector in the marginal cone is zero.Consider a nonzero element ˜ J ∈ C , with blocks ˜ Q = 0, ˜ R , and ˜ S , andlet J (cid:48) ( t ) be a one-parameter family such that Q (cid:48) ( t ) = Q , R (cid:48) ( t ) = R + t ˜ R ,and S (cid:48) ( t ) = S + t ˜ S + t ˜ R T Q − ˜ R . By the Schur complement condition, wesee that rank J (cid:48) ( t ) = d for all t . Also, g i ( J (cid:48) ( t )) = 4 − k Ti J (cid:48) ( t ) k i = (cid:104)∇ g i , ˜ J (cid:105) − t l T ˜ R T Q − ˜ R l is nonpositive for t > Q − . There-fore, for some (cid:15) , J (cid:48) ( t ) ∈ R (4) for all 0 ≤ t < (cid:15) . Since f ( J (cid:48) ( t )) = f ( J ) and J is a local minimum of f over R (4), so must J (cid:48) ( t ) for all sufficiently small t .Therefore J is a fluid packing.Theorems 2 and 3 suggest a straightforward generalization of Voronoi’s al-gorithm to 2-periodic sets, which we elaborate in the next section.
3. The generalized Voronoi algorithm
Our algorithm seeks to enumerate all the locally optimal 2-periodic sets in d dimensions. As we proved in Theorem 3, those are either fluid packings oralgebraically extreme packings. The problem of enumerating the fluid packingsis rather straightforward for m = 2, since the two component lattices mustthemselves be extreme. Therefore, for each extreme d -dimensional lattice, eachhole (circumcenter of Delone cell) that is deep enough to accommodate anothertranslate of the lattice without reducing the packing radius can give rise to afluid packing, and these different fluid packings may or may not be connectedto each other via their flow. This informal description is the extent to whichwe will discuss this part of the problem in this paper, and we will devote theremainder to the enumeration of the algebraically extreme lattices. In fact,in the dimensions where we have implemented our algorithm and present theresults of the full enumeration, there are no fluid packings.12he enumeration of the algebraically extreme 2-periodic sets of packing ra-dius 1 consists of two steps that can be conceptually thought of as occurring oneafter the other, but in our implementation are actually interleaved. The firststep is to enumerate all the vertices and edges of R (4) up to equivalence underthe action of Γ. The second step is to take each edge, represented in the form { J + tJ (cid:48) : 0 ≤ t ≤ } or { J + tJ (cid:48) : t ≥ } , solve for all t such J + tJ (cid:48) ∈ R (4),and check whether J + tJ (cid:48) is algebraically extreme. We elaborate on these stepsbelow. Throughout the algorithm, we need to solve a problem analogous to the prob-lem known as the shortest vector problem (SVP) in the context of lattices [15].In our context, given a quadratic form J : R d + m − → R with positive definiteupper-left d × d block Q , we wish to find its minimum over nonzero vectors in M = Z d × ( E − E ). We also, in some cases, want to enumerated the vectorsattaining this minimum, or more generally enumerate all vectors that attain avalue below some threshold.For k = ( n , l ) ∈ M with fixed l , J ( k ) is an inhomogeneous quadratic formof n (see Eq. (7)). Finding the minimum of an inhomogeneous quadratic formover the integer vectors is a problem known as the closest vector problem (CVP)for lattices and is closely related to the SVP [15]. Since there are only finitelymany possible values of l , we have thus reduced the SVP problem for m -periodicforms to a finite number of instances of the CVP problem for the underlyinglattice.As in the case of lattices, it might be expected that reduction of the quadraticform (using elements of Γ) would significantly decrease the average time tocompute the SVP. To reduce the form, we first reduce the underlying lattice(the Q block) by applying an operation T of the form (8), where V = 0 and W = 1. We then perform size reduction of the translation vectors, by applyingthe appropriate operation T , where U = 1 and W = 1, so that Q − R T hasentries in [ − , ]. 13 .3. Enumeration of vertices The algorithm to enumerate the vertices of R (4) is similar to the one ac-complishing the analogous task in Voronoi’s algorithm. We start with a knownvertex of R (4), denoted J . We compute the extreme rays of its cone C J [16].For each such ray, { J + tJ (cid:48) : t ≥ } there are two possibilities: either it liesentirely in R (4) or there is some t > J + tJ (cid:48) is another vertexof R (4), which we say is contiguous to J . The first possibility does not existin the original algorithm for lattices, but does occur in our case, and we mustcheck for this possibility. We discuss this problem in Section 3.4.For each contiguous vertex, we check if it is equivalent to J (more on thisstep in Section 3.5), and if not, we add it to a queue of vertices to be processedand to our partial enumeration of vertices. At each subsequent step of thealgorithm, we remove a vertex from the queue, compute its contiguous vertices,check them for equivalence against all vertices in our partial enumeration, andadd the ones that are not equivalent to previously enumerated vertices to thequeue and to the partial enumeration. The enumeration is complete when thequeue is empty. Since vertices of R (4) are necessarily rational, we can performthese calculations using exact arithmetic.One way to obtain a starting vertex to initialize the algorithm is as follows:consider A d (or D d , or any extreme, nonfluid d -dimensional lattice). It can berepresented as 2-periodic set by taking a sublattice of index 2 instead of theprimitive lattice. This 2-periodic set is necessarily algebraically extreme [13],and so it lies on an edge of R (4), which must terminate at a vertex in at leastone direction. Given a vertex J of R (4) and an extreme ray of C J , of the form { J + tJ (cid:48) : t ≥ } , we want to determine if the ray lies entirely in R (4). This is the caseif and only if J (cid:48) ( k ) ≥ k ∈ M . So, we have a problem very similarto the SVP above, except that we may not assume that Q (cid:48) is positive definite.If Q (cid:48) is not even positive semidefinite, then there is some k = ( n ,
0) such that14 (cid:48) ( k ) = Q (cid:48) ( n ) <
0, and the ray is not unbounded. So the only remaining caseis when Q (cid:48) is positive semidefinite and has nontrivial null space, N ( Q (cid:48) ).We break this remaining case into two cases. First, consider the case thatthere exist l ∈ ( E − E ) and u ∈ N ( Q (cid:48) ) with c = u T ( R (cid:48) ) T l >
0. Let v ( t ) =( − tc u , l ), let [ v ]( t ) ∈ M be the closest integer vector to v ( t ), and let ( e ,
0) = v ( t ) − [ v ]( t ) be the remainder. Then J (cid:48) ([ v ]( t )) = t c u T Q (cid:48) u − tc u T Q (cid:48) e T + e Q (cid:48) e − tc + l T S l . (12)All terms except − tc are either zero or bounded as t → ∞ and so J (cid:48) ([ v ]( t )) < t and the ray is not unbounded.The final case is that u T ( R (cid:48) ) T l = 0 for all l ∈ ( E − E ) and u ∈ N ( Q (cid:48) ). Inthis case, the problem reduces to the first case, where Q (cid:48) is positive definite,albeit in a smaller dimension: Since Q (cid:48) is a rational matrix, there a unimodulartransformation U ∈ GL d ( Z ) such that Q (cid:48) ◦ U has span { e r +1 , . . . , e d } as its nullspace and is positive definite on span { e , . . . , e r } . We can find this transfor-mation by computing the Hermite normal form or the Smith normal form of Q (cid:48) scaled to an integer matrix. Let n = n (cid:48) + n (cid:48)(cid:48) with n (cid:48)(cid:48) ∈ N ( Q (cid:48) ◦ U ) and n (cid:48) ∈ N ( Q (cid:48) ◦ U ) ⊥ . Then J (cid:48) ( U ( n ) , l ) = ( n (cid:48) ) T U T Q (cid:48) U n +2( n (cid:48) ) T U T ( R (cid:48) ) T l +2( n (cid:48)(cid:48) ) T U T ( R (cid:48) ) T l + l T S (cid:48) l . (13)Since, by assumption of this case, ( n (cid:48)(cid:48) ) T U T ( R (cid:48) ) T l = 0, the value of J (cid:48) ( U ( n ) , l )depends only on n (cid:48) and l , and we may find its minimum using our SVP method. Given two quadratic forms J and J , we wish to determine if J = J ◦ T = T T J T for some T ∈ Γ. Let us denote this relation as J ∼ J . This problemmay apply to vertices of R (4), as part of the algorithm for enumerating vertices,but we may also apply it to any pair of quadratic forms in R (4) that are notnecessarily vertices. Let us first prove some useful results. Definition 3.
A set M ∈ M is perfect if J ( k ) = J (cid:48) ( k ) for all k ∈ M implies J = J (cid:48) . 15 set M is perfect if and only if the set { kk T : k ∈ M } spans the space ofsymmetric ( d + m − × ( d + m −
1) matrices. Denote by J − ( A ) ⊂ M the setof vectors in M attaining values in the set A . If A = { } and J is a vertex of R (4), then J − ( A ) is perfect. A direct consequence of the definition of a perfectset is the following lemma: Lemma 2.
Let A be a set of real values, and let M = J − ( A ) and M = J − ( A ) be the set of vectors in M that achieve these values. If M and M areperfect, then the following are equivalent: J = J ◦ T for some T ∈ Γ . T ( M ) = M , and J | M = ( J ◦ T ) | M for some T ∈ Γ .In particular a T that satisfies one condition also satisfies the other.Proof. First, suppose J = J ◦ T , and let k ∈ M . Since J ( k ) = J ( T k ) ∈ A ,we have that T k ∈ M . Similarly, if k ∈ M , then T − k ∈ M . So T ( M ) = M ,and clearly J | M = ( J ◦ T ) M follows a fortiori from the unrestricted equality.The other direction follows immediately from the definition of a perfect set.Therefore, a simple algorithm to check for equivalence is as follows: first,construct a set A such that M = J − ( A ) is perfect. When J is a vertex of R (4), the set A = { } suffices. Otherwise, we find the smallest a > A = [4 , a ] suffices. Next, compute M = J − ( A ). If M is not perfect, | M | (cid:54) = | M | , or J does not take values in A over M with the same frequencyas J does over M , then J (cid:54)∼ J . We give labels 1 , . . . , s to the elementsof M = { k , . . . , k s } and M = { k (cid:48) , . . . , k (cid:48) s } , such that k , . . . , k d + m − arelinearly independent (there must be such a linearly independent subset for M to be perfect). We now try to construct an injective map σ : { , . . . , d + m − } →{ , . . . , s } such that( k (cid:48) σ ( i ) ) T J k (cid:48) σ ( j ) = k Ti T T J T k j for all 1 ≤ i, j ≤ d + m −
1. (14)We can do this by a backtracking search, constructing σ on 1 , . . . , n < d + m − n , and backtracking when no possible assignment of σ ( n + 1)16atisfies (14) for 1 ≤ i, j ≤ n + 1. For each such complete map produced by thebacktracking search, there is a unique linear map T (cid:48) such that T (cid:48) k i = k (cid:48) σ ( i ) for1 ≤ i ≤ d + m −
1. If T (cid:48) ∈ Γ and T (cid:48) ( M ) = M , we are done and J ∼ J .Otherwise, we continue with the backtracking search. If the backtracking searchconcludes without finding any equivalence, then J (cid:54)∼ J . Given an edge of R (4) of the form J + tJ (cid:48) , where 0 ≤ t ≤ t ≥
0, wewish to identify the points of this edge that lie in R (4). As we are limitingourselves to the case m = 2, we simply need to solve det( J + tJ (cid:48) ) = 0 for t ,which is a univariate polynomial equation. This equation can be solved for t asa generalized eigenvalue problem J x = − tJ (cid:48) x . It is possible that the entire edgelies in R (4), as happens for one edge in our enumeration for d = 5. In thatcase, either Q is constant over the edge, and any locally optimal packing on theedge is necessarily a fluid packing, or Q is not constant, and therefore neitheris f , so only points that minimize f over the edge — necessarily endpointsby quasiconcavity — can be locally optimal packings. In any case, we have afinite number of points for which we need to determine if they are algebraicallyextreme.Given J ∈ R (4), we can certify that it is algebraically extreme by lookingat the dual problem. Namely, J is algebraically extreme if and only if the cone { (cid:80) k ∈ J − (4) η k kk T + xx T : η k > , x ∈ N ( J ) } is full-dimensional and includes G J . Otherwise, we can certify the opposite by the direct problem of finding J (cid:48) that gives a counterexample for the definition: J (cid:48) (cid:54) = J , (cid:104) J (cid:48) − J, G J (cid:105) ≤
0, and J (cid:48) ∈ C J ∩ T J . Since t comes from the root of a univariate polynomial, it is not,in general, rational. In our calculation, we use floating point representation forthe candidate algebraic extreme sets.
4. Numerical results
We use an implementation of the algorithm described in the previous sectionto fully enumerate the vertices of the Ryshkov-like polyhedron R (4) and the17ertices of R (4) 4 (2) 10 (6) 34 (25)algebraically extreme 2-periodic sets 3 (1) 7 (3) 29 (20)highest density 1 / (2 √
2) 1 / / (8 √ Table 1: Summary of enumeration results. For d = 3 , , algebraically extreme m -periodic sets for m = 2 and d = 3 , ,
5. Our attemptedenumeration for d = 6 did not terminate after over a month of running. Themain bottleneck appears to be the enumeration of extreme rays of C J for vertices J with large number of minimal vectors J − (4) (see Sec. 4.4). This is similar tothe main bottleneck in the enumeration of perfect lattices in d = 8, where onlyby exploiting the symmetries of these high-kissing-number lattices, was the fullenumeration made tractable [17]. We are hopeful that a similar approach couldbe used for m = 2 to make full enumeration in higher dimensions than d = 5tractable, but we do not attempt to implement it in this work. The enumeration in d = 3 gave 4 inequivalent vertices and 3 inequivalentalgebraically extreme 2-periodic sets. All the algebraically extreme 2-periodicsets in 3 dimensions of packing radius 1 have the same density δ = 1 / (2 √ ≈ . J , = 2 −
10 2 0 10 0 4 2 − J , = 2 − − (15)18 igure 1: The three algebraically extreme 2-periodic packings in 3 dimensions. We use redand blue spheres to represent the two orbits under translation by lattice vectors. Both theleft and middle packings are representations of the fcc lattice as a 2-periodic set, but theyare not equivalent when the two orbits are distinguished. The right packing is the hexagonalclose-packed 2-periodic arrangement and is the only algebraically extreme 2-periodic packingin 3 dimensions that is not also a lattice. J , = 23 (16)The forms J , and J , are two inequivalent representations of the fcc latticeas a 2-periodic set. J , is a stacking of square layers, whereas J , is a stackingof triangular layers (see Figure 1). The forms are not equivalent under theaction of Γ because the corresponding two sublattices of the fcc lattices are notequivalent under the symmetries of the fcc lattice. The form J , represents thehexagonal-close-packing 2-periodic arrangement, which is not a lattice. Notethat a form J is the representation of a lattice as a 2-periodic set if and only if Q − R T ∈ ( Z ) d .All the algebraically extreme 2-periodic sets have the same kissing num-ber, 12, but as quadratic forms they have different number of minimal vectors: | J − , (4) | = 20, | J − , (4) | = | J − , (4) | = 18. This difference occurs because con-tacts between spheres in the same orbit contribute the same minimal vector asthe analogous contact in a different orbit. Denote by | · | ∗ a counting measure19 igure 2: The vertices of the Ryshkov-like polyhedron for d = 3 and m = 2 can be interpretedas binary packings of nonadditive hard spheres. that gives weight m to vectors of the form k = ( n , κ = | J − (min J ) | ∗ /m .The vertices of the Ryshkov-like polyhedron are also interesting to consider.They are J , v = 2 −
10 0 2 11 − J , v = 2 (17) J , v = 2 − − J , v = 2 − − − − . (18)They have full rank, so they do not correspond to 2-periodic sets in 3 dimensions.However, because they are positive definite, they correspond to 4-dimensionalsets that are periodic (with two orbits) under a 3-dimensional lattice. Whenthey are projected to the space spanned by the lattice, they can be interpretedas binary packings of non-additive spheres, where the two sphere species haveequal self-radius, but smaller radius when interacting with each other. The firstis a simple cubic lattice of one species with its body-center holes filled withspheres of the other species. The second is a hexagonal lattice with one of thetwo inequivalent triangular prism-shaped holes in each unit cell filled. The third20 igure 3: The Voronoi graph of 2-periodic sets in d = 3, a generalization of the Voronoigraph for lattices. The blue nodes are the vertices of R (4), the Ryshkov-like polyhedron; theorange nodes are points of R (4) lying on unbounded edges (rays) of R (4). Two vertices areconnected in the graph if they are contiguous, and edge points are connected to the vertices onwhich their edge is incident. The labels V n and
P n corresponds to J ,nv and J ,n respectively.In d = 3 all the points of R (4) lying on polyhedron edges are on unbounded edges and theyare all algebraically extreme. and fourth are the fcc lattice with its octahedral or (one of its) tetrahedral holesfilled, giving, respectively, a simple cubic lattice with alternating species (theNaCl crystal structure) and the diamond crystal structure (see Figure 2).Finally, we point out as an example of an unbounded edge, the edge con-necting J , and J , v . In 4 dimensions, we obtain 7 inequivalent algebraically extreme 2-periodicsets. Six of those lie in the relative interior of edges of the Ryshkov-like polyhe-dron, and one is a vertex. Two algebraically extreme sets achieve the maximumdensity, δ = 1 / . D as a21-periodic set. They are represented by the forms, J , = 2 −
10 0 2 0 −
10 0 0 2 − − − − J , = 2 − − − − − − .(19)The form J , is a vertex of the Ryshkov-like polyhedron. Both have the kiss-ing number of D , κ = 24, as the average kissing number. The next-highestdensity achieved is δ = 2 / (cid:112)
144 + 64 √ ≈ . J , = 2 − − − − − − − τ − τ − − τ , (20)where τ = (1 + √ / κ = 22.Finally, there are four algebraically extreme 2-periodic sets that achieve thesame density as the A lattice, δ = 1 / (4 √ ≈ . A lattice, but the other two are not also lattices. Allfour have the same kissing number as the lattice, κ = 20. Including J , , thereare 10 inequivalent vertices of the Ryshkov-like polyhedron. In 5 dimensions, we obtain 29 inequivalent algebraically extreme 2-periodicsets. Five inequivalent 2-periodic sets achieve the maximum density δ = 1 / (8 √ ≈ . D as a 2-periodic set,and two are nonlattices that achieve the same density. The ones that represent22 igure 4: The Voronoi graph in d = 4. The blue nodes are polyhedron vertices that are notin R (4). The green and orange nodes are points of R (4) lying on vertices and unboundededges of the polyhedron, respectively. In d = 4, as in d = 3, all the points of R (4) in therelative interior of polyhedron edges are on unbounded edges and are all algebraically extreme.We omit the labels for neatness. D are given by the forms, J , = 2 − − − − −
10 0 0 2 0 −
10 0 0 0 2 −
11 0 − − − (21) J , = 2 − − − − − − − − (22) J , = 2 − − − − − − − − − − − − , (23)where J , is also a vertex of the Ryshkov-like polyhedron (the only vertex thatis also in R ). The two that are not lattices but achieve the same density are J , = − − − − − − − − − − − − − − − − (24)24 , = 25
10 5 5 − − − − − − −
50 0 0 0 16 −
80 5 5 − − . (25)All have the same kissing number κ = 40.The next highest density, δ = ( (cid:112) − √ / ≈ . J , = 25 − − − − − − − − − − − − − − √ √ − − √ (26) J , = 25
10 5 − − − − − − − − − − − − √ √
610 10 − − √ (27) J , = 12 − − − − − − − √ − − √
60 4 4 − − − √ . (28)The first two of these have a kissing number of κ = 35, and the third has κ = 34.The third largest density is that achieved by one of the three extreme lattices25n 5 dimensions, δ = 1 / (9 √ ≈ . A , has the lowest densityof all extreme lattices, δ = 1 / (8 √ ≈ . R (4) onan edge of R (4) that has lower density, but fails to be algebraically extreme.Including J , , the Ryshkov-like polyhedron has 34 vertices. Two of thevertices are not positive semidefinite, and this is the lowest dimension wheresuch vertices occur. When interpreted as nonadditive binary sphere packings,these packings would have a nonself-radius larger than the self-radius. In thissense they are similar to the fluid packings (the distance between orbits is largerthan the distance within each orbit), but the underlying lattices are not extreme.Another interesting phenomenon that first occurs in 5 dimensions is an edge of R (4) that is completely contained in R (4). Such an edge is the one connecting J , to J , ◦ T , where T = − − − − , (29)as well as all the equivalent edges. Along the edge J , + t ( J , ◦ T − J , ), theobjective determinant is f ( t ) = 512(1 + t − t ), so any internal point cannot bea locally optimal 2-periodic set. In d = 6 we were only able to perform partial enumeration starting from avertex incident on the edge on which a representation of A lies. The partial26 δ δ L/N κ / (8 √
2) L 402 0.0884 1 / (8 √
2) L 403 0.0884 1 / (8 √
2) L 404 0.0884 1 / (8 √
2) N 405 0.0884 1 / (8 √
2) N 406 0.0795 (cid:112) − √ / (cid:112) − √ / (cid:112) − √ / / (9 √
2) L 3010 0.0786 1 / (9 √
2) L 3011 0.0786 1 / (9 √
2) L 3012 0.0786 1 / (9 √
2) N 3013 0.0786 1 / (9 √
2) N 3014 0.0771 (cid:112)
419 + 1011 √ / −√ √ / −√ √ / / (2 (cid:112)
126 + 4 √
19 cos α −
266 cos 2 α ) N 3018 0.0750 3 /
40 N 3019 0.0748 1 / (36 (cid:112) −
58 + 26 √
5) N 2620 0.0748 (3 √ / (( (cid:112) (3 − α − α ))(4(5 + 4 cos α ))) N 2821 0.0738 59049 / (4 √ x ) N 2522 0.0737 5 / (48 √
2) N 3023 0.0722 1 / (8 √
3) L 3024 0.0722 1 / (8 √
3) L 3025 0.0722 1 / (8 √
3) L 3026 0.0722 1 / (8 √
3) N 3027 0.0722 1 / (8 √
3) N 3028 0.0722 1 / (8 √
3) N 3029 0.0722 1 / (8 √
3) N 30
Table 2: Some invariants of the 29 algebraically extreme 2-periodic sets in five dimensions,including density and kissing number. The
L/N column indicates whether this is the repre-sentation of a lattice as a 2-periodic set ( L ) or a truly nonlattice arrangement ( N ). Here α = (tan − (3 √ − π ), α = (tan − (9 √ / − π ), α = (tan − (486 √ / − π ),and x = 8189832078 + 22432320 √
661 cos α − α − α +6990736 √
661 cos 5 α . igure 5: The Voronoi graph in d = 5. The notation is the same as in Figs. 3 and 4. This isthe lowest dimension where a point of R (4) lying in the relative interior of an edge connectingtwo vertices appears (red nodes). This is also the lowest dimension where an entire edge ofthe polyhedron is in R (4), but since only endpoints of such edges can be locally optimal, wedo not include such edge points in the graph. We omit the node labels for neatness. R (4) and 692 points of R (4) on edgesof R (4) (including 8 vertices). There are also 4 edges connecting 3 vertices lyingcompletely in R (4). The obstacles that stalled the full enumeration are threevertices with particularly complex cones. One is a representation of E as a2-periodic set (with | J − (4) | = 124) and two vertices with | J − (4) | = 112 and | J − (4) | = 126 minimal vectors: J ,E = 2 − − − − − − − − − − − − , (30) J , = 2 − − − − − − − − − − − − − − − − , (31) J , = 2 − − − − − − − − − − − − − − − − − − . (32)The number of faces of the cone is half the number of minimal vectors. Forcomparison, the E lattice (as a 1-periodic set) has 126 minimal vectors, andthe enumeration of the extreme rays of its cone is already barely tractable by29rute force. Since these forms are highly symmetric (have large automorphismgroups), the number of symmetry orbits of the extreme rays is much smallerthan the total number of extreme rays. By using a method that exploits thesymmetry of these forms, the calculation would hopefully become tractable.This strategy was successful, for example, in making the enumeration of perfectlattice in d = 8 tractable [17].
5. Conclusion
Using our generalization of the Voronoi algorithm to 2-periodic sets, wewere able to produce a complete enumeration of the locally optimal 2-periodicsphere packings in dimensions d = 3 , , and 5. In particular, we show that itis impossible to obtain a higher density using 2-periodic arrangements in thesedimensions than is possible with lattices. However, in d = 3 and d = 5 (but notin d = 4), there are nonlattice 2-periodic arrangement that match the optimallattice packing density.Our work leaves a number of important open questions, whose solution willenable application of this method to higher values of m and d :1. We were not able to prove a priori that the Ryshkov-like polyhedron musthave a finite number of faces (in particular vertices) up to the action ofΓ. For m = 2 and d = 3 , ,
5, this follows directly from the fact thatour enumeration halts. However, it would be good to know that theenumeration is always guaranteed to halt for any m and d .2. Theorem 3 is proven only for m = 2, but we conjecture that it holds alsofor m >
2. If this conjecture holds, our algorithm can be immediatelyextended to m >
2. However, it would now involve looking for pointsof R (4) that lie on m ( m + 1)-dimensional faces of R (4). This wouldsignificantly increase the complexity of two steps the algorithm. First,instead of enumerating extreme rays of C J for vertices J , we need toenumerate the m ( m + 1)-dimensional faces of C J . Second, instead of30olving for S − RQ − R T = 0 over a line, we need to solve over a m ( m +1)-dimensional space.3. To make the enumeration for m = 2 and d > J , we need only enumerate the orbits of the extreme rays of C J under the automorphism group of J , Aut( J ) ⊂ Γ. Such methods wereused to make the enumeration of perfect lattices tractable in d = 8 [17].4. In dimensions where a full enumeration might not be tractable, heuristicoptimization methods could be useful for discovering new packing arrange-ments as well as providing empirical backing to conjectures about certainarrangements being optimal. Again, in the case of lattices ( m = 1), suchmethods have been remarkably successful, reproducing the densest knownlattices in up to d = 20 dimensions. Stochastic enumeration, traversingthe 1-skeleton of the Ryshkov polyhedron by picking a random contiguousvertex at each step [11, 18], can be applied to the Ryshkov-like polyhe-dron treated here. Sequential linear programming methods [19, 20] andsimulated annealing methods [21] can also be used to sample periodic ar-rangements. Our results and those of Sch¨urmann [12, 13] can be used tocertify locally optimal packings found. Acknowledgments
A. A. was supported by Project Code (IBS-R024-D1). Y.K. was supported by an Omidyar Fellowship at the Santa Fe Institute.
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