Compact Packings of the Space with Two Spheres
Thomas Fernique
A packing of Rn (n ≥ 2) by k spheres is a set of interior-disjoint closed spheres ofkdifferent sizes. Its densityδ is defined by
δ:= lim sup
r→∞
proportion of Brcovered by the spheres
volume ofBr ,
whereBr denotes the closed ball of radius r centered on the origin. The max- imal density has been extensively studied fork = 1 ([CS99]), but it is known in only few dimensions, namelyn= 2 ([Thu10, FT43]),n= 3 ([Hal05]),n= 8 ([Via17]) andn= 24 ([CKM+17]). Some cases have also been settled fork= 2 andn= 2 ([Hep00, Hep03, Ken04, Ken06]). In all these cases the density turned out to be maximized for so-calledcompact packings, except for (k, n) = (1,3), where no such packing does exist. This issue is of great interest in chemistry to model new material structures (see,e.g., [HO11] and references therein).
Compact packings have been defined in dimension two in [FT64]: they are disc packings whosecontact graph,i.e., the graph which connects the centers of adjacent discs, is a triangulation. It can be naturally extended as follow: a com- pact packing ofRnis a packing by spheres whose contact graph is a homogeneous simplicial complex of dimension n, i.e., a set of n-dimensional simplices such that the intersection of two simplices is either empty or a lower dimensional face.
The compacity hypothesis yields strong constraints on the radius of spheres, and rather few sizes of spheres may allow a compact packing. Here, we search for all the compact packings in the case (k, n) = (2,3),i.e., two spheres in the Euclidean space. We prove that there is only one such packing. Its structure is actually well known: this is the one of the salt crystal (Fig. 1).
Theorem 1 The only compact packing in R3 with two sizes of spheres is the one with spheres of radius1centered on theface-centered cubicorcheckerboard lattice
D3:={(x, y, z)∈Z3 |x+y+z= 2k√
2, k∈Z} and spheres of radius r = √
2−1 centered on the same lattice translated by (√
2,0,0),i.e., each hole between six large spheres contains a small sphere.
This packing has density (53−√
2)π'0.79 (to compare with the maximal density π
3√
2 '0.74 with only one size of sphere). Whether this compact pack- ing maximizes the density amongall the packings with the same spheres (not necessarily compacts) remains an open question.
Figure 1: Crystal structure of sodium chloride.
1 Necklaces
Any pair of adjacent spheresB andH in a compact packing yields a sequence B1, . . . , Bk of spheres, whereBk is adjacent toBk+1,B andH (Fig. 2). We call such a sequence anecklace, seingB as the body,H as the head and theBk’s as the beads. A necklace is coded by the word over{1, r}whose k-th letter gives the radius of thek-th bead. Among the possible codings, we usually choose the lexicographically minimal one.
B H
B
kB
k+1Figure 2: The bodyB, the headH and two beads of a necklace.
A necklace is said to belargeifB andH have radius 1, small if they have radiusr, andskewotherwise. Any compact packing by spheres of radius 1 and rcontains two different spheres in contact, thus at least one skew necklace.
2 Skew necklaces and possible values of r
Any skew necklace contains at most 5 beads. Indeed, the spheres of a necklace can be rolled onHuntil their center are all in the same plane as the center ofH. Since, in the plane, at most 6 unit discs can be adjacent to a unit discs, there is at most 6 beads. Actually there is at most 5 beads because when we rolled the beads onH we enlarge the necklace. There is thus finitely many potential skew necklaces; they can be listed (Tab. 1). We shall see that each skew necklace characterize the radiusr.
11111 1111r 111rr 11rrr 1rrrr rrrrr 1111 111r 11r1r 1r1rr rrrr
111 11r 11rr 1rrr
1r1r rrr 1rr
Table 1: Potential skew necklaces (same number of r in each column).
Letδs,t denote the dihedral angle between the triangle which connects the centers ofB,H and a beadBk of radiuss, and the triangle which connects the centers ofB, H and a beadBk+1 of radius t. Letxyzd denote the angle iny of a contact graph between three spheres of radiusx,y andz. Spherical geometry allows to express the cosine of δs,t as a function of the angles of faces of the tetrahedron:
cosδs,t= cos(s1t)c −cos(dr1s) cos(r1t)c sin(dr1s) sin(r1t)c .
Sines can be expressed with cosines via sin2a+ cos2a= 1, and cosines can be expressed withrvia the cosine theorem:
cos(d111) = 1
2, cos(d11r) = 1
1 +r, cos(dr1r) = 1− 2r2 (1 +r)2. A computation then yields:
cosδ1,1= r2+ 2r−1
2r(2 +r) , cosδ1,r=
r r
(2 +r)(2r+ 1), cosδr,r= 1 + 2r−r2 2(2r+ 1) . The sum of the dihedral angles given by all the consecutive beads in a neck- lace must be equal to 2π: this yields an equation inr. For example, the skew necklace 111rr yields
2δ1,1+ 2δ1,r+δr,r = 2π.
Take the cosine of both sides, fully expand the left-hand side and substract cos(2π) = 1 to both sides. This yields a polynomial equation in cosines and sines of theδs,t’s. We can then substitute the cosines by the above computed expressions inr, and use sin2a+ cos2a= 1 to express the sines in function of
r. However, we prefer to use auxiliary variables for square roots in order to get algebraic equations. Namely, letX0, . . . , X3 be the auxiliary variables defined by the algebraic equations:
0 = (2 +r)(2r+ 1)X02−r, 0 = X12−3r2−6r+ 1, 0 = (2 +r)(1 + 2r)X22−2, 0 = X32+r2−6r−3.
This allows to write:
cosδ1,r = X0, sinδ1,1 = 2r(2+r)1+r X1, sinδ1,r = (1 +r)X2, sinδr,r = −4r+21+rX3.
Thus, for each skew necklace we get a system of 5 algebraic equations (one for the sum of dihedral angles and 4 for the auxiliary variables) that we can solve exactly via Gr¨obner basis. This yields a total of 16 possible values of r.
For each one, we check by interval arithmetic that not only the cosine of the equation on dihedral angles is satisfied, but the equation itself: this amounts to check that the sum of the angles is equal to 2πand no 2kπ for somek6= 1.
This reduces to 10 the number of possible values ofr(Tab. 2).
11111 X4+ 4X3+X2−6X+ 1 0.902 1111r 4X4+ 8X3−4X2−6X+ 1 0.849 111rr X4+ 4X3+ 3X2−6X+ 1 0.720 11r1r 4X3−20X2+ 9X+ 2 0.690 11rrr X4−2X3−5X2+ 1 0.420
1111 X2+ 2X−1 0.414
111r 2X2+ 3X−1 0.280
111 2X2+ 4X−1 0.224
1r1rr 2X3+ 9X2−20X+ 4 0.223
11rr X2−6X+ 1 0.171
Table 2: Skew necklace (left), minimal polynomial of the value ofr it charac- terizes (middle) and approximated value ofr(right).
3 Large and small necklaces
Letδs,tbe the analogous ofδs,twhen both B andH have radius 1. A compu- tation yields (as for cosδs,t):
cosδ1,1= 1
3, cosδ1,r= 1
p3r(2 +r), cosδr,r =2−r 2 +r.
The large necklaces correspond to the non-zero triples (i, j, k) of non-negative integers satisfying
iδ1,1+jδ1,r+kδr,r= 2π.
Given a value r, we first compute a lower approximation of the δs,t’s which yields an upper bound on the possible values fori,j andk. We then use arith- metic interval to find triples (i, j, k) which could be solutions. Last, we check these triples exactly by taking the cosine of the equation, expanding it and use the exact values of the cosδs,t’s (and the sines via sin2a+ cos2a= 1).
Computing this shows that onlyr=√
2−1 allows large necklaces, namely 11r11r (this is the one appearing in the packing described in Theorem 1).
We do the same withδs,t being the analogous of δs,t when both B and H have radiusr, that is, for small necklaces. We compute
cosδ1,1= 1 + 2r−2r2
1 + 2r , cosδ1,r= r
p3(1 + 2r), cosδr,r =1 3. The we search for non-zero triples (i, j, k) of non-negative integers satisfying
iδ1,1+jδ1,r+kδr,r= 2π.
An exhaustive search shows that, for each of the 10 possible values ofr, there is no solution to this equation, that is, there is no small necklace.
4 Proof of the theorem
In any compact packings by spheres of radius 1 andr, there is a skew necklace.
There are only 10 possible skew necklaces. Since any of them contains two adjacent beads of radius 1 (see Tab. 2), there must be two adjacent spheres of radius 1 in the packing, thus a large necklace. The only r which allows a large necklace is r = √
2−1. This is thus the only value which allows a compact packing. It allows only two necklaces: the skew one 1111 and the large one 11r11r. The only compact packing satisfying this is the one described in Theorem 1.
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