Maximizing the Packing Density on a Class of Almost...

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Maximizing the Packing Densityon a Class of Almost PeriodicSphere PackingsOliver Knill

Abstract. We consider the variational problem of maximizing the packing densityon some finite dimensional set of almost periodic sphere packings. We show that themaximal density on this manifold is obtained by periodic packings. Since the densityis a continuous, but a nondifferentiable function on this manifold, the variationalproblem is related to number theoretical questions.Every sphere packing in Rd defines a dynamical system with time Rd. If the dynamical system is strictly ergodic, the packing has a well defined density. The packingsconsidered here belong to quasi-periodic dynamical systems, strictly ergodic translations on a compact topological group and are higher dimensional versions of circlesequences in one dimension. In most cases, these packings are quasicrystals becausethe dynamics has dense point spectrum.Attached to each quasi-periodic sphere- packing is a periodic or aperiodic Voronoitiling of Rd by finitely many types of polytopes. Most of the tilings belonging to thed-dimensional set of packings are aperiodic.We construct a one-parameter family of dynamically isospectral quasi-periodic spherepackings which have a uniform lower bound on the density when variing the radiusof the packing. The simultaneous density bound depends on constants in the theoryof simultaneous Diophantine approximation.

0723-0869/96/030227-246$5.50

2 2 8 O l i v e r K n i l l

1. Introduction

Finding sphere packings in Kd with maximal density are important mostly unsolvedproblems with many applications (see [3]). The problem is interesting in geometry,group theory, crystallography or physics. Here, we look at the problem from a dynamical system point of view.It is still controversial, whether the Kepler conjecture claiming that there is nosphere-packing in R3 with density higher than n/y/lS is open or proven ([8, 11]).Also for dimensions bigger than 3, the best densities are still not known evenso thereis a list of candidates (see [2],[3]). According to a general result of Groemer ([6] seealso [7]), there exists a sphere packing which has a maximal density.

Periodic packings and often lattice packings have achieved the highest today knowndensities (see [3]). Following an advise in [16] p. 188, we restrict us here to a subclassof packings, where we know that the density of the packing exists. This family ofquasicrystals, almost periodic sphere-packings in Hd which form a part of the hugespace S of all packings. Quasicrystalline dense sphere packings are of interest inphysics (see for example [14, 1]). The ergodic-theoretical side of the problem comesnatural in the context of aperiodic tilings. Tilings and packings are closely relatedsince to every strictly ergodic sphere-packing is attached a tiling of Rd by Voronoicells consisting of finitely many types of poly topes.The quasicrystals in this article are defined by higher dimensional circle sequencesand have the property that the densities can be computed explicitely from the parameters. This allows for a machine to search through many of these packings andto determine in each case the exact density. The feature of being able to computemacroscopic quantities like densities for aperiodic configurations was also useful whenwe studied cellular automata on circle configurations [10]. The main motivation forthe present work was to investigate dense packings with a new class of packings. Wecould consider many packings with high periodicity which are close to the best knownpacking.

A packing S C 7*d can be described by a configuration n *-+ xn = {0,1}Z . Theclosure Y of all the translates of x in {0,1}Z<< forms a subshift with time Zd. Ingeneral, there exist many shift- invariant measures on Y and the density depends onthe choice of the invariant measure. For the packings, we consider here, the subshifty has a unique shift-invariant measure and it implies that the density is well definedand can explicitly be determined. Moreover, if we make a translation in the i'th coordinate, we get a one-dimensional strictly ergodic subshift for which the dynamicalspectrum (and so the diffraction spectrum) is known to be pure point. The packingsare therefore crystals or quasicrystals (see [13]).

Maximizing the Packing Density on a Class of Almost Periodic Sphere Packings 229

The organization of this article is as follows. In section 2, we relate sphere packingswith dynamical systems and point out, that the uniquely ergodic packings have a welldefined density. In section 3, we consider a finite dimensional class of strictly ergodicsphere packings and compute the density of the packings. We show that the densityfunctional takes its maximum on periodic packings. In section 4, we give classes ofsuch packings and relate the problem to find dense packings to number theoreticalquestions. We also report about packings found by numerical experiments. In section5, we will construct a one-parameter family of isospectral sphere packings whichhave a uniform positive density depending on constants in simultaneous Diophantineapproximation. In section 6, we discuss a related construction for coverings. Relationswith other topics are outlined in section 7 and in section 8, some open problems aresummarized.

2. Sphere packings and dynamical systems

A r-sphere packing 5 is a countable set of points in Rd such that the minimaldistance between two points is > 2r. Let S be the set of all sphere packings. Define a metric on S by d(S,S') — min(l,e), where c is the smallest number such thatTXS = S + x = S' on the ball {\\y\\ < 1/e} for some x G Rd With this metric, whichis adapted from the metric for tilings, (5, d) is a complete metric space. Let Hd acton S by translations Tt(S) = S + t. Given a specific sphere packing S, we can lookat the closure Xs of the orbit Rd(5) = {S +1 \ t G Rd}. If Xs compact in «S, we geta dynamical system (Xs, Rd), where Xs is a compact metric space on which Rd actsby homeomorphisms. Such a system is called minimal or almost periodic, if forevery S G S, the orbit is dense in Xs, it is called uniquely ergodic, if there existsexactly one Rdinvariant measure on Xs, strictly ergodic, if it is uniquely ergodicand minimal.

Remarks.1) A metric on «S, with respect to which the subset of r-packings in S becomes compact is described in [5]. For our purposes, the stronger metric d is good enough.2) In the statistical mechanics literature, a sphere packing is also called a configuration with hard core restriction.3) By normalization of the distance in Rd, we could assume that a sphere packing hasradius r = 1. We keep the additional parameter r since we are interested in packings5 c Z d .

We call a sphere packing S rational, if all points of S belong to a d-dimensionallattice U • Zd C Rd, where U is an invertible d x d-matrix. On a rational spherepacking, there is a natural Zd-action, by identifying Zd with the maximal subgroupof Rd which leaves the lattice invariant. The packing S consists then of a subset


of the lattice and every rational sphere packing defines so a subshift ys C {0,1}Z .This set ys is is invariant under the Zrf-action. A rational sphere packings is calledstrictly ergodic, if the dynamical system (3^5, Zd) is strictly ergodic. If every shiftof the Zd-action is periodic, 5 is called periodic,Remarks.1) Clearly, a periodic packing is rational and also strictly ergodic.2) Periodic packings are dense in S since we can periodically continue a given packingoutside a given box. Periodic packings are also dense in the set of rational packings.3) The name almost periodic which stands as a synonym for minimal has no relationwith the usual almost periodicity of functions or sequences. The expression almostperiodic is however widely used in the topological dynamics and mathematical physicsliterature.

The lower and upper densities of a sphere packing S are defined as

A = h m i n f ' , , A + = h m s u p ,n - > o o | A n | n - f o o | A n |

where An = [—n, n]d. If A = A~ = A+, then A is called the density of S.

Lemma 2.1 For a strictly ergodic sphere packing, the density exists.

Proof. The limits

/^^±( / ) = ± l imsup±|A„ | -1 £ f (Tk(x) )/c(=An

define bounded functionals </>* on C(X) which are invariant under the translationsf ,_>, f(Tk). By Riesz representation theorem, </>*(/) = // dv*y where v± are somemeasures on X which are invariant under the stricly ergodic Zd or Rd dynamicalsystem (X, T, p). It follows from the unique ergodicity that v± — p and hence that thetwo limits coincide. For p : S -» R given by p(S) = l{d(s0,5)<r} with So = {0} C Rd,t h e i n t e g r a l f x p d p i s t h e d e n s i t y o f t h e p a c k i n g . Q

Remarks.1) The above lemma is well known and there are other proofs using more theory. Theresult follows for example also from a multi-dimensional version of Birkhoff 's ergodictheorem (see [4] Chapter VIII). The proof given here uses only lightest tools.2) There exists a dense set of rational packings in S which have no density. Proof.Consider a periodic packing S of radius r having density A. Take an other periodicpacking S" of radius r which has density A' < A. Take a first cubic box Bi centeredat zero and fill it with spheres of radius r according to the first packing. Take a


second larger cubic box B^ and fill B<i — B\ with spheres according to the packing inS'. Make B2 so large that the density of the packing in this finite box B<i is smallerthan (A + A')/2. Take a box S3 and fill S3 — B2 with spheres according to a makeS3 so large that the density in the box S3 is larger than (2A + A')/3. Continueinductively so that the finite volume densities are alternatively below (A-f A')/2 andabove (2A + A')/3.Given e > 0, we can make B\ so large that the distance between the original packingand the modified packing is smaller than e.3) We are forced to define the packing problem on a subclass of packings since thedensity is not a continuous function on the set of all packings for which the densityexists: take such a packing S and define a sequence of packings Sn obtained from Sby deleting all balls in distance less than n from the origin. The packings Sn haveall the same density but Sn converges to the Sqo = 0, which is a packing with zerodensity.

3. Quasi-periodic sphere-packingsWe consider now a specific class of almost periodic sphere packings. These packingscontain also periodic packings but the computations do not rely on the distinctionbetween periodic and aperiodic. In order to find dense packings experimentally, it wassometimes of advantage, not to distinguish between periodic and aperiodic packings.Take a finite union of disjoint half-open intervals J C R1 = R/Z, a rotation vectora G Rd, a radius r > 0 and the standard basis {e,}f=1 in Rd.

Define R(r) = {k G Zd | 0 < || £?=1 *»e«ll2 < rh where II' II is the Euclidean /2-normin Rd. If (J + R{r)a) 0 J = 0, we get for any 9 G T1 a sphere-packing

S={keZd I Y^0 + kiai€J}1=1

If we put a sphere of radius r/2 at each point of H, we get a r-packing because themaximal distance between two different points in H is by construction < r. Thispacking is not periodic if one of the numbers a, is irrational. We call such a packingquasi- periodic.

Lemma 3.1 A quasi-periodic sphere packing is stricly ergodic.

Proof. Consider the dynamical system (T1 ,Ti(6) = 9 + a,-, dx). If all the a, arerational, then every orbit of this Zd-action is finite. Every 9 G T1 defines a configuration xn = lj(0 + n • a) and the closure is a finite set in {0,1}Z on which the shiftsact as permutations. Clearly every orbit is dense and the equal distribution is the


only invariant measure. If at least one number a, is irrational, then every orbit of Tis infinite and is dense in T1. The shift action on the closure Xj of the configurationxn = Ij(9 + n • a) is strictly ergodic since the irrational rotation a, alone has onlyone invariant measure p on Xj and is minimal (see [9] Proposition A.I.). □

Remark.The density of a quasi-periodic sphere-packing can be approximated explicitely byperiodic sphere-packings because every vector a is a limit of rational vectors and thedensity depends continuously on a. It is not excluded that in some dimension, thehighest possible packing is aperiodic, but we will see that if J consists of a singleinterval, the maximal density is obtained by a periodic packing.

The center density S of a sphere-packing is defined as A/V(d), where A is the densityand V(d) is the volume of the unit sphere in Kd.

Proposition 3.2 Assume (J + R(r)a) fl J = 0. The center density of the quasi-periodic or periodic packing is (r/2)d\J\, the density is V(d)(r/2)d\J\.

Proof. The orbit of a configuration xn = lj(9 + net) is a subset of {0,1}Z whichis shift-invariant. By the above lemma, its closure X is a strictly ergodic set. Thedensity of the configuration x G {0, l}2* is therefore \J\ and is independent of 0 G J.T h e r / 2 - p a c k i n g S = { n | x n = 1 } h a s d e n s i t y V ( d ) { r / 2 ) d \ J \ . □

Remark. Note that evenso the map a h* 5(a) is not continuous, the density iscontinuously depending on a.We can replace the standard basis by any other basis /, = f/e, with U G GL(d, R).

Corollary 3.3 Given any basis /, = Ut{ with U G GL(d, R). Assume (J + R(r)a)ClJ = 0 with R(r) = {k G Zd | ||£?=i Art/,||2 < r}. The density of this packing isV(d)(r/2)d\J\/det(U).

Proof. Every sphere of the packing is in bijective correspondence with a parallelepipedspanned by the basis /i,..., /<*, attached to the center of each ball. This parallelepipedhas volume det(U) so that the measure of the sphere divided by det(x7) gives thed e n s i t y i n t h e c a s e w h e n \ J \ = 1 . D

Given a, r, we find J consisting of one interval such that (J + R(r)a) fl J — 0. Startwith 0O = 0 and form the finite set

R ( r ) a = { 9 u . . . , 9 N } c T l .


The points 9\ — c,9n = —c nearest to 90 are symmetric to 0. Form J = [0,c). Wecheck that R(r)JDJ = 0. The center density of the packing with this interval is thenc(r/2)«.

Remarks.1) We can add a second interval J' as follows. Assume the interval (0,-, 0,+i) of length/, is the largest interval disjoint from R(r) • a. There are three cases. If/, > 2c, takeJ' = [a, 6) = [0,-+i - 2c, 0t+i - c). If c < U < 2c, define J' = [a, b) = [0t, 0,+i - c). Inthe other case, take J' = 0. We show now that, [ii(r)a + (J U J')] n (J u «H = 0- For9 G J, we have 0 + /i(r)a g J and 0 -I- #(r)a g J', since J' is disjoint from J + ii(r)a.For 0 G J7, we have also that 0 + R(r)a is disjoint from J'. It is also disjoint from J\,since R(r)a = -#(r)a. The center density of the packing is then (c + 6 - a)(r/2)d.2) We could proceed as follows to get larger and larger sets J. Given Jn, take anypoint 0 outside Jn U R{r)Jn. If fl(r) + 0 mod 1 is disjoint from Jn, we can find amaximal interval /# which has the property (R + Ie) H 1$ = 0. Form Jn+i = Jn U 1$.Repeating this construction leads to a finite or countable union J of half-open intervalswhich has the property that (R + J) fl J = 0. In other words, we are adding spheresuntil the covering is saturated, in the sense that there is no longer room with positivedensity for adding additional spheres. Usually, in our experiments if the first intervalis large enough, then J is already saturated and no second interval is needed.3) For getting high densities, we have to choose r such that the number of latticepoints inside the ball of radius r is just below a point of discontinuity. Necessary isr = y/n, where n G N and that there is a lattice point on the boundary of the ballwith radius r so that a further increase of r increases the number of lattice points.By a theorem of Lagrange in number theory, this is always true if d > 4 and r = y/n.In other words, by extending the variational problem and variing r also, a packingwhich maximizes the density has the property that r2 is an integer.4) We expected to get high densities for radii r, for which many lattice points are onthe boundary of a sphere of radius r because of a possible large kissing number. Theexperiments confirm this in some cases like r = 5, d = 2, 3, but it was not the rule.

Proposition 3.4 There exists a quasi-periodic sphere-packing in Rd with density

V ( d ) r - N ( d , r , a )

whereN(d,r,a) = min In a (mod 1)1

0<||n||2<r

and where V(d) is the volume of the d-sphere. More generally, given a general basisft — Uti in Rd. There exists a quasi-periodic sphere packing with density

V[d)2d' det(£0

Oliver Knill

where Nu(d, r, a) = minO x I l ^ n . U e . l U x r

n ■ a (mod 1)|.

Proof. Do the above construction with one interval J of length N(d,r,a) rsp.N u ( d , r , a ) . I

Fig. 1. The piecewise linear function (01,02) >-▶ N(2,r, a) — min o<||n||2<r I" • « (mod 1)|in the case r = 5. As larger the value, as brighter is the point (01, o2) G T2. On theblack resonance lines, the value is zero.

The next proposition deals with the variational problem to maximize the nondiffer-entiable function N(d,r,a) over the d dimensional torus of parameters a, when thedimension d and the radius r are fixed. The proposition says that this variationalproblem has only rational "critical points". It means that on some finite dimensionalmanifolds of almost periodic packings, the maximal density is achieved by periodicpackings.

Proposition 3.5 The functiona G Td >->• N(d, r, a) = min \n ■ a (mod 1)1

0<| |n | |2<r

takes its global maximum on a rational point a G Qd/Zd fl Td. Also all local maximaa are rational.

Proof. Since TV is a continuous function on the compact manifold Td, the maximumis attained. Let j3 maximize o >-▶ N(a) = N(d,r,a). There exist n,n' G Z such


that n'(3 = n/3 (mod 1), otherwise a *-> N(a) would be strictly monotone and f3 werenot maximal. There exists therefore m = (mi,..., m^), k G N such that m • /? = k.Consider now the (d — l)-dimensional surface

f/(1) = {a£Td\ma = k}

which contains the maximum. U^ is topologically a (d - 1) dimensional torus.Choose coordinates on U^ so that a is a (d - l)-dimensional vector. Call f3^the maximum /3 in these coordinates. Again, a »->• N(d,r,a) is a linear function on[/(^Ma | 3m, Ar, ma = k} so that the maximum /?(1) on (7i satisfies m*1) -0^ = k^for some integers m^,..., m^, fc(1). Define the (d - 2)-dimensional surface

x7(2) = {aGt/(1)|^(1)-^ = *(1)}

containing the maximum with local coordinates /3^2KInductively, after d — 1 such steps, we know that the maximum is on a one dimensionline U^d~^ which is the intersection of (d- 1) rational surfaces projected from Rd toTd. Claim: the point on this line, which has maximal distance from 0 has rationalcoordinates in Td. Proof, working in the covering of Td, this claim can be rephrasedas follows: given a line segment in Rd which connects two lattice points P, Q G Zd.Any point on this line segment, which maximizes the distance from Zd has rationalcoordinates. This statement is true because the linear projection matrix from Rd tothis line has rational entries (if the basis vector on the line is the vector from P to Q)and therefore every lattice point in Zd is mapped onto a rational point.The same proof works also for local maxima instead of global maxima. □

Remark.The same argument shows that for any finite set R C Zd not containing the origin,the function

a G Td *-> N(d, r, a) — min \n • a (mod 1)|

takes its global maximum on a rational point Qd/ZdC\Td. We can consider thereforeany packing problem, where the spheres are replaced by some compact set K C Rd.The analogue problem is to find for a given r, the densest packing of Rd with copiesn + rK, n G Zd of r • K. The same construction gives a a?-dimensional manifold ofalmost periodic packings for which the density exists. The maximal density on thismanifold is obtained by periodic packings.

4. Examples of quasi-periodic packingsLattice packings. Assume the packing is a lattice packing defined by the generatormatrix U. We can take r = 1 so that R(l) = 0 and J = T1 leading to N(d, 1, a) — 1


independent of a. The density of the lattice packing is V(d)/(2ddet(U)). For example,for d = 2 and with the generator matrix

the density is1 N { 2 , l , a ) _ j -

which is known to be the highest density in two dimensions.

Packings with radius r = y/2 give the lattices Dd- Take r = \J2 and the standardbasis e,. We have

f i (V2) = { (± l , . . . ,0 ,0) , (0,± l , . . . ,0) , . . . , (0 ,0, . . . ,± l ) } .

With a = (1,1,..., l)/2 and J = [0,1/2], we get TV(3, \/2, a) = 1/2 and the densityis V(d)/2dl2+l corresponding to a center density 2~W2+1). These packings are lattice packings called Dd- In dimensions d = 3,4,5, these packings are the packingswith the highest known density like for d = 3, the density is V(3)>/2 /23N(2,1, a) —

|tt • 1/(V§) and in four dimensions 1/(4)^/2^(2, l,a) = tt2/2 • 1/8.

Packings with radius r = \/3- With r = >/3, and a = (1,2,3,..., d)/2d, we getAT(d, >/3, a) = l/(2d) and a center density (\/3/2)d/(2d). This density is larger thanthe density of Dd for d > 12. These packings were the densest we found numericallyfor d = 2, 3,4, 5 in the class of quasi-periodic packings with r — \/3.

Packings with radius r — 2. We have the problem to find integers p and ai,.. .a<*such that in the group Zp, no sum Ylniai gives zero if ]Ci n2 < r- 1° °^ner words,all sums atl ± a,2 ± a^3 are different from zero modulo p. We can build solutions bydefining recursively a sequence an with the linear difference equation a\ = 1,<Z2 = 2,an = an_i -h an_2 + 1 and define the vector a = (ai,<i2,. •., a^)/(2arf). In the cased = 2, a = (1,2)/4 or d = 3, a = (1,2,4)/8, or rf = 4, a = (1,2,4,7)/14, these were thedensest 2- packings we found in our class. For d = 5, the 2-packing a = (1, 3, 5, 7,9)/18was denser than the 2-packing determined by a = (1,2,4, 7,12)/24.

This construction of families of packings with increasing dimension can be generalizedfor any r. The problem is to find the smallest p — p(d,r), such that there exist dnumbers ai,...,a^ G Zp such that for any n G Zd with ||n||2 < r, the equationn - a = 0 in Zp has no solution. The center density of the corresponding packing isthen (r/2)d/p(d,r). We rephrase the result in the following proposition.


Proposition 4.1 Fix the dimension d and the radius r. If we can find d numbersa\,...,ad in the group Zp = Z/(pZ) such that for any n G Zd with 0 < ra2 + .. .H-n^ <r2, the equation

n - a = 0 mod p

has no solution, then there exists a rational r-packing S C Zd with density V(d)(r/2)d/p

Proof We do the construction with the vector a = (cti,... , a<*)/p and the intervalJ = [0,1/p). Given a point 0 G J and n with ||n||i < r, then 0 + n • a is not in J. □

Remark.For prime p, a packing in the proposition defines a code in the vector space Zd. Incoding theory, one considers however rather the packing problem with the Hammingmetric instead of the Eucledean metric.

We found some good packings in the special case, when r is an integer and a\ =l,a2 = r,a3 = r2y...1ad = rd~l.

Corollary 4.2 Let r be an integer and let p be the smallest integer such that

n\ + n2r + n3r2 + ... + rdrd~l = 0 mod p

has no solution for 0 < £^=1 nf < r. Then there exists a sphere packing with densityV(d)(r/2)d/p.

Remarks.1) Because of the periodicity of the packing, we have for even r that p - r/2 is amultiple of r and that for odd r, p- (r+ l)/2 is a multiple of r.Example: for r = 89, p = 502984 = (r + l)/2 + 5651 • r, this gives a packing withdensity 0.73386212.2) The construction can be modified by taking for example a = (1, r + 1, r(r 4- l))/p'with suitable //, which gives for some r denser packings than a = (1, r, r2)/p.


Danal tyO . M r

;tuhu\\\\\\̂ ^^^^

0.73

0.7

. #•

0.«8

0.<«

• •

' <c 2 0 4 0 6 0 t O

Fig. 2. Packing densities for special packings in the case d = 2, d = 3. For each r,we plot the density obtained by taking a = (l,r)/p, where p = p(r) is the integerdescribed in corollary 4.2.

The densities seem to accumulate at the maximal known density 7r/\/l2 ~ 0.9069 intwo dimensions and to 7r/>/l8 ~ 0.74048 in three dimensions.We were also looking for dense packings using a multiscale Monte Carlo method. Werestricted ourselves to the standard basis in Rd and to packings defined by one interval and went only up to dimension 5. One method was to shoot randomly vectors afor fixed r and to adapt the search step near good parameters. We did not assumethat the best density is obtained for a rational a. Indeed, we conjectured and provedonly after having done some experiments that the maxima are rational. For largervalues of r and especially for higher dimensions, the searches need a lot of computingtime. The reason is that for fixed r and d, the function a *-> N(d, r, a) is a piecewiselinear function having gradients of the order rd. We would need therefore to checkroughly at (rd)d points in order to find the global maximum since so many local maxima are expected. This is an very time consuming task also for quite small r and gPs.The huge number of local minima is also the reason why the common methods forminimizing multidimensional functions like the downhill simplex method would notwork well here. The function a •->• N(d, r, a) is just too wiggly.


Some good packings were obtained by taking a good solution (d, r, (c*i,..., ad)) andchoosing a^+i so that the packing (<i+1, r, (ai,..., c*d+i)) has maximal density. Thisis motivated (evenso there is no direct relation) by the laminated lattice construction(see [3]), which also constructs d-dimensional packings by building up layers of (d— 1)-dimensional packings which are known to be dense.

We never found a denser packing while using two intervals. The search is also algo-rithmically more expensive, since we have to order the set R(r) • a (mod 1).

Table 1. Some examples of packings in three dimension. The packing with radius y/2is the Kepler packing. The packing with radius 120 gives a slightly denser packing asthe packing reported in [17] using Penrose tilings which have densities accumulatingby 0.7341.

r o t density

V2 ( l , l , l ) / 2 0.7404803 (2 ,5 ,6 ) /20 0.706858\ / l 3 (8,17,23)/35 0.7012065 (19,5,37)/91 0.7192297 (21,150,194)/252 0.71267612 (60, 630,783)/1270 0.71242431 (1,31,961)/21623 0.72138638 (1,38,1444)/39729 0.72317265 (1,65,4225)/196463 0.73191089 (l,89,7921)/502984 0.733862120 (1,120,14400)/1231260 0.734830

Table 2. Some examples of packings in four dimension. The first packing is thepacking which is believed to be the densest.

' a density

V2 ( l , l , l , l ) / 2 0.616850v/3 ( l , 2 , 3 , 4 ) / 8 0.3469782 (2,7,10,13)/14 0.3524863 (48,275,395,419)/435 0.4594474 (81,105,139,169)/185 0.4267945 (20, 76,148,167)/364 0.52955719 (1,19, 19M93)/72020 0.558099

240 Oliver Knill

Table 3. Some examples of packings in five dimension. The first packing is the packingwhich is believed to be the densest.

r a density

V2 ( l , l , l , l , l ) / 2 0.465260Vs (1,2,3,4,5) /10 0.2564202 (3,9,11,13,17)/18 0.292433v^ (1,2,4,8,15)/31 0.2966283 (2165,1835,1154,1575,499)/2178 0.2385705 (80, 304,592,668,687)/1456 0.353051

5. Isospectral packings with a simultaneous densitybound

If we allow to vary r but fix the dynamical system of the packing, the constructionof packings with simultaneous lower density bound has a relation with simultaneousDiophantine approximation.

A vector a = (ai,.. -,(*d) is called badly approximable by rational numbers if thereexists a constant C such that for all n G Zd and p G Z

| n - a - p | > C - | | fi | | r d f

where ||n||i = X^Li Kl- Such numbers exist (see Theorem 6F in [12]). Denote withC(d) the maximum of all possible constants C.

Proposition 5.1 There exists a one-parameter family Sr of dynamically isospectralquasi-periodic sphere-packings in Hd with spheres of radius r and Sr C Zd such thatthe center density is bounded below by C(d)2~dd~dl2, independent ofr.

Proof. We show that there exists a G Td, such that N(d, ra) > C(d) • r"dd~d/2 forall r. By Holder's inequality, ||n||i < ||n||2||l||2 = IMkv^. We get therefore for such

\n-a-p \>C{d) .d-d l2 \ \n \ \ idfor all n G Zd and p G Z. This means that

|no| mod 1 > C(d) • <Td/2r-d


for all ||ra||2 < r.The subshift X C {0,1}Z obtained by closing the set of translates of xn = lj(0 +na) in the product topology is strictly ergodic and the shift Ti on X is measuretheoretically isomorphic to a factor of 0 •-» 0 + a,. If the set J is an interval of lengthless than 1/2, the shift Ti is measure theoretically isomorphic to 0 *-+ 9 + a,- andhas the point spectrum {e27n/ca« | k G Z}. The spectrum of the packing is therefore{ e 2 n i k a > \ k € Z , l < i < d } . □

Remark.It is trivial to get in any dimension packings with a center-density higher thanC(d)2~dd~d/2 because C(d) < 1. The uniform bound for the density in r givesnot dense packings and the nontriviality of Proposition 5.1 lies in the fact that thepackings are dynamically isospectral in the sense that they have the same dynamicalpoint spectrum and that we can do the packing simultaneous for any r > 0. Theestimate N(d, ra) > C(d) • r~dd~d/2 is crude for small r.

6. CoveringsThe algorithm to compute packings can be modified to get coverings. Assume, wehave an interval J C T1 such that R(r)J = T1. In this case, the spheres with centersin

d

5(0) = {kezd\ J2e + kiai€J)i = l

and radius r are covering all Zd and the spheres of radius r + y/d/2 are covering thewhole space Rd. The density of the covering is

V(d)\J\(r+Vd/2)d.

Proposition 6.1 There exist sphere-coverings ofRd with densities

V(d)(r+y/d/2)d ■ M{d,r,a) ,

whereM(dyr,a) = max min \(n — m)a (mod 1)| > 0 .

||n||2<r \\m\\2<r,m^n

Proof Given parameters d, r and a, we can find M(d, r, a) such that (n—m)a mod 1 >M(d,r,a) for all ||n||2, ||m||2 < r satisfying na ^ ma (mod 1). We can choose forthe interval J the largest gap in the finite set R(r) • a C T1, since then, R(r)J is ac o v e r i n g o f T 1 . T h e l e n g t h o f J i s M ( d , r , a ) . □


Remark.This proposition implies that in order to get a good covering, the parameters r anda have to be chosen in such a way that R(r) • a C T1 is as homogeneous as possible.The computations for coverings are more involved as the computations for packingssince the determination of the set R(r) • a mod 1 includes a time- consuming sorting.

7. Related topics

Crystallographic considerations. The packing question for spheres has been formulated by Hilbert for general compact subsets K of Rd. An example is to replace theEucledean norm by the l? norm ||n||p = (£?=1 \n^)1^ and to pack Rd with with Pspheres and clearly for p — 1 or p = oo, the maximal packing density is 1 and differentp will probably lead to different lattices with optimal density. Parameters p, wherethe optimal packing type changes should be considered as a bifurcation parameter.For general K, the construction of quasiperiodic packings is the same as alreadydescribed. We have only to replace ||n||2 < r by n G int(A") to obtain the corresponding densities. General packing problems with different type of subsets K areinteresting and relevant in crystallography or chemistry, where K takes the shape of amolecule. The same proof as shown above shows that the maximal packing density ond-dimensional manifolds of quasi-periodic packings is attained by periodic packings ifwe take only copies of K which are obtained from K by translation by n G Zd. Removing the last restriction might lead in general to aperiodic packings with maximaldensity: an indication is that there exist convex sets in R3, the Schmitt-Conway-Danzer tiles, for which the densest packing is aperiodic with density 1. We mentionalso a result in [15] which says that for a generic set of convex sets in the plane, thedensests packing is not lattice like.

Quasicrystals. Crystals in nature are often periodic. A situation closest to theperiodic case are quasi-periodic crystals which are a family of so called quasicrystals.Our result that in a class of quasi-periodic crystals, the maximal density is attainedon periodic crystals can have physical relevance since high densities are favored bystable crystalline structures.

Defects. If the packing is almost periodic, there are large domains, where the packing looks periodic, but there must be defects, if the lattice is not periodic. For c > 0,consider the e-collar Ke of the packing S which consists of all points in Hd which havedistance < c from any ball. There exists a largest e > 0, such that the complement ofK€ still percolates. The defect, the complement of Kt with critical e is not necessarilyalmost periodic since the topology of Ke does not depend continuously on e.


Minimizing energy. Moving on the manifold of almost periodic rational r-packingsparametrized by a G Td corresponds to (non continuous) deformations of the crystal.The density, when considered as a negative "energy" of the crystal is a piecewise linearcontinuous function on the parameters. Variational methods do not apply in orderto find the maximal density which correspond to minimizing the energy because theenergy is not differentiate at the maximum. However, we have seen that the minimalenergy on this manifold is obtained by periodic crystals. Since atomic dense packingsare believed to be fundamental in condenced matter, our toy model can in principleexplain an aspect of the stability of crystals. Moreover, the periodicity of some ofthe crystals can be so large that it would be hard to distinguish them experimentallyfrom a true quasicrystal.

Aperiodic Voronoi tilings. For strictly ergodic packings, there exists a constant Csuch that if xn = 1 then there exists xm — \ with some m satisfying \\m — n\\2 < C.In other words, there are only finitely many sphere-configurations which can occur ina ball of given radius. The Voronoi tiling of Hd is built up by finitely many Voronoisimplices. This Voronoi decomposition is an aperiodic tiling of Rn if a rationallyindependent.

Diffraction spectrum. The diffraction spectrum of the quasicrystals consideredhere can be computed explicitely: it is a point measure with weights given by aFourier transform of lj (see [10]).

Random sphere packings. We should compare the quasi-periodic packings in R3with random sphere packings, which are interesting from the practical point of view.In three dimensions, computer simulations and experiments with large containers ofsteel balls give a density of 0.6366 (see [18]). Higher densities are no more believed tobe random, since the high density is then obtained by crystalisation. Almost periodicsphere packings in three dimensions with higher densities are obtained for most radiir.Generalization of the construction. The construction could be generalized inthe following way. Take any compact abelian group G (generalizing G = T1) and dcommuting group translations 7\,.. .Td- Consider a ring J of measurable sets (generalizing the ring of half open intervals), where each element in J different from haspositive Haar measure and such that the boundary of every J £ J has zero Haarmeasure.

Take J G 3 with maximal measure such that [J + R(Jyr)a]D J = 0 and do the sameconstruction as before. We get a strictly ergodic subshift defining the centers of asphere packing with spheres of radius r/2 and the packings define quasicrystals withpure point dynamical spectrum and so pure point diffraction spectrum. We think


however that no other group is as convenient as G = T1. Taking the p-adic groupwould be a natural choice from the dynamical systems point of view.

A replacement of the irrational rotation with d commuting homeomorphisms of thecircle does not lead to more general packings: if one of the rotation numbers ofthese homeomorphisms is irrational, then this Zdaction is topologically conjugated tothe action given by the irrational rotations. The subtle and still unsolved problem,whether d commuting circle diffeomorphisms are smoothly conjugated to irrationalrotations is not involved.

8. QuestionsA number theoretical question is: what is the maximal density of a quasi-periodicsphere packing in Rd with spheres of radius r centered on a subset of Zd.

Related (but not equivalent) to this question is the problem to find

max min \n • a (mod 1)|a€Td 0<||n||2<r

in dependence of r. We know that the maximum is always taken for rational a.This leads to the number theoretical problem to find for fixed r > 0, the smallestcyclic group Zp = Z/(pZ) such that there exist a\,...,ad G Zp such that for anyn G Zd with 0 < n\ + ... -f n2d < r2, the equation

V^ n, • a, = 0 mod p»=i

has no solution in Zp.

We also do not know if we can in all dimensions get close to the highest densities bychoosing a special sequence of a's. In dimensions up to 6, a class of dense packings canbe obtained by a = (1, r, r2,..., rd~l)/p with odd r and p = p(r, d) = (r + l)/2 + qris the smallest integer, such that the density is not vanishing. The densities wereconverging to the highest known densities for r —> oo.

Acknowledgements: I would like to thank A. Hof for discussion and literature hints.


References

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[3] J.H. Conway and N.J.A. Sloane. Sphere packings, Lattices and Groups, volume290 of A series of Comprehensive Studies in Mathematics. Springer Verlag, NewYork, 2.nd edition edition, 1993.

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[11] W-Y. Hsiang. A rejoinder to Hales's article. The Mathematical Intelligencer,17:35- 42, 1995.

[12] W.M. Schmid. Approximation to algebraic numbers. Monographic no. 19 deL'Enseignement mathematique. Serie des conferences de l'Union matherriatiqueinternationale, no. 2. Geneve, Universite, L'Enseignement mathematique, 1972.

[13] M. Senechal. Quasicrystals and geometry. Cambridge University Press, 1995.

[14] A.P. Smith. The sphere packing problem in quasicrystals. Journal of Non-Crystalline Solids, 153:258-263, 1993.

[15] G. Fejes Toth. Densest packings of typical convex sets are not lattice like. Discreteand Computational Geometry, 14:1-8, 1995.


[16] L. Fejes Toth. Lagerungen in der Ebene, auf der Kugel und im Raum. Berlin,New York, Springer, Second edition, 1972.

[17] J.M. Wills. Dense sphere packings in cylindrical Penrose tilings. Phys. Rev. B,42:4610-4612, 1990.

[18] A.Z. Zinchenko. Algorithm for random close packings of spheres with periodicboundary conditions. J. Comp. Physics, 114:298-307, 1994.

Received: 05.09.95

Division of Physics, Mathematics and AstronomyCaltech

Pasadena, CA, 91125USA

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