Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space

Discrete & Computational Geometry, Feb 2012

The contact graph of an arbitrary finite packing of unit balls in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings. In particular, we prove that if C(n) denotes the largest number of touching pairs in a packing of n>1 congruent balls in Euclidean 3-space, then \(0.695<\frac{6n-C(n)}{n^{\frac{2}{3}}}< \sqrt[3]{486}=7.862\dots\) for all \(n=\frac{k(2k^{2}+1)}{3}\) with k≥2.

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Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space

Discrete Comput Geom Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space Károly Bezdek 0 1 2 0 K. Bezdek Institute of Mathematics, Eötvös University , Budapest , Hungary 1 K. Bezdek Department of Mathematics, University of Pannonia , Veszprém , Hungary 2 K. Bezdek ( ) Department of Mathematics and Statistics, University of Calgary , Calgary , Canada The contact graph of an arbitrary finite packing of unit balls in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings. In particular, we prove that if C(n) denotes the largest number of touching pairs in a packing of n > 1 congruent balls in Euclidean 3-space, then 0.695 < 6n−nC32 (n) < √3486 = 7.862 . . . for all n = k(2k32+1) with k ≥ 2. Congruent sphere packing; Contact number; Density; (truncated) Voronoi cell; Union of balls; Isoperimetric inequality; Spherical cap packing 1 Introduction Let Ed denote the d -dimensional Euclidean space. Then the contact graph of an arbi trary finite packing of unit balls (i.e., of an arbitrary finite family of non-overlapping balls having unit radii) in Ed is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if and only if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have in Ed . In 1974 Harborth [6] proved the following optimal result in E2: the maximum number c(n) of touching pairs in a packing of n congruent circular disks in E2 is precisely 3n − √12n − 3 implying that lim n→+∞ 3n − c(n) √n = Some years later the author [2] has proved the following estimates in higher dimensions. The number of touching pairs in an arbitrary packing of n > 1 unit balls in Ed , d ≥ 3 is less than 21 τd n − 21d δd− d−d1 n d−d1 , where τd stands for the kissing number of a unit ball in Ed (i.e., it denotes the maximum number of non-overlapping unit balls of Ed that can touch a given unit ball in Ed ) and δd denotes the largest possible density for (infinite) packings of unit balls in Ed . Now, recall that on the one hand, according to the well-known theorem of Kabatiansky and Levenshtein [7] τd ≤ 20.401d(1+o( 1 )) and δd ≤ 2−0.599d(1+o( 1 )) as d → +∞ on the other hand, τ3 = 12 (for the first complete proof see [12]) moreover, π . Thus, by combinaccording to the recent breakthrough result of Hales [5] δ3 = √18 ing the above results together we find that the number of touching pairs in an arbitrary packing of n > 1 unit balls in Ed is less than 1 20.401d(1+o( 1 ))n − 21 2−0.401(d−1)(1−o( 1 ))n d−d1 2 as d → +∞ and in particular, it is less than for d = 3. The main purpose of this note is to improve further the latter result. In order, to state our theorem in a proper form we need to introduce a bit of additional terminology. If P is a packing of n unit balls in E3, then let C(P) stand for the number of touching pairs in P , that is, let C(P) denote the number of edges of the contact graph of P and call it the contact number of P . Moreover, let C(n) be the largest C(P) for packings P of n unit balls in E3. Finally, let us imagine that we generate packings of n unit balls in E3 in such a special way that each and every center of the n unit balls chosen, is a lattice point of the face-centered cubic lattice Λfcc with shortest non-zero lattice vector of length 2. Then let Cfcc(n) denote the largest possible contact number of all packings of n unit balls obtained in this way. Before stating our main theorem we make the following comments. First, recall that according to [5] the lattice unit sphere packing generated by Λfcc gives the largest possible density for unit ball packings in E3, namely √π18 with each ball touched by 12 others such that their centers form the vertices of a cuboctahedron. Second, it is easy to see that Cfcc( 2 ) = C( 2 ) = 1, Cfcc( 3 ) = C( 3 ) = 3, Cfcc( 4 ) = C( 4 ) = 6. Third, it is natural to conjecture that Cfcc( 9 ) = C( 9 ) = 21. If this were true, then based on the trivial inequalities C(n + 1) ≥ C(n) + 3, Cfcc(n + 1) ≥ Cfcc(n) + 3 valid for all n ≥ 2, it would follow that Cfcc( 5 ) = C( 5 ) = 9, Cfcc( 6 ) = C( 6 ) = 12, Cfcc( 7 ) = C( 7 ) = 15, and Cfcc( 8 ) = C( 8 ) = 18. Furthermore, we note that C( 10 ) ≥ 25, C( 11 ) ≥ 29, and C( 12 ) ≥ 33. In order to see that, one should take the union U of two regular octahedra of edge length 2 in E3 such that they share a regular triangle face T in common and lie on opposite sides of it. If we take the unit balls centered at the nine vertices of U, then there are exactly 21 touching pairs among them. Also, we note that along each side of T the dihedral angle of U is concave and in fact, it can be completed to 2π by adding twice the dihedral angle of a regular tetrahedron in E3. This means that along each side of T two triangular faces of U meet such that for their four vertices there exists precisely one point in E3 lying outside U and at distance 2 from each of the four vertices. Finally, if we take the twelve vertices of a cuboctahedron of edge length 2 in E3 along with its center of symmetry, then the thirteen unit balls centered about them have 36 contacts implying that C(13) ≥ 36. Whether in any of the inequalities C( 10 ) ≥ 25, C( 11 ) ≥ 29, C( 12 ) ≥ 33, and C(13) ≥ 36 we have equality is a challenging open question. In the rest of this note we give a proof of the following theorem. Theorem 1.1 (i) C(n) < 6n − 0.695n 32 for all n ≥ 2. (ii) Cfcc(n) < 6n − 3 √3π18π n 3 = 6n − 3.665 . . . n 3 for all n ≥ 2. 2 2 (iii) 6n − √3486n 32 < Cfcc(n) ≤ C(n) for all n = k(2k32+1) with k ≥ 2. As an immediate result we get Corollary 1.2 0.695 < 6n − C(n) 2 n 3 √ < 3 486 = 7.862 . . . for all n = k(2k32+1) with k ≥ 2. The following was noted in [2]. Due to the Minkowski difference body method (see for example, Chap. 6 in [11]) the family PK := {t1 + K, t2 + K, . . . , tn + K} of n translates of the convex body K in Ed is a packing if and only if the family PKo := {t1 + Ko, t2 + Ko, . . . , tn + Ko} of n translates of the symmetric difference body Ko := 21 (K + (−K)) of K is a packing in Ed . Moreover, the number of touching pairs in the packing PK is equal to the number of touching pairs in the packing PKo . Thus, for this reason and for the reason that if K is a convex body of constant width in Ed , then Ko is a ball of Ed , Theorem 1.1 extends in a straightforward way to translative packings of convex bodies of constant width in E3. For the sake of completeness we mention that the nature of contact numbers changes dramatically for non-congruent sphere packings in E3. For more details on that we refer the interested reader to the elegant paper [8] of Kuperberg and Schramm. Last but not least, it would be interesting to improve further the estimates of Theorem 1.1. In the last section of this paper we mention a particular packing conjecture that could lead to a significant improvement on the estimate (i) in Theorem 1.1. 2 Proof of Theorem 1.1 2.1 Proof of (i) Let B denote the (closed) unit ball centered at the origin o of E3 and let P := {c1 + B, c2 + B, . . . , cn + B} denote the packing of n unit balls with centers c1, c2, . . . , cn in E3 having the largest number C(n) of touching pairs among all packings of n unit balls in E3. (P might not be uniquely determined up to congruence in which case P stands for any of those extremal packings.) Now, let rˆ := 1.81383. The following statement shows the main property of rˆ that is needed for our proof of Theorem 1.1. Lemma 2.1 Let B1, B2, . . . , B13 be 13 different members of a packing of unit balls in E3. Assume that each ball of the family B2, B3, . . . , B13 touches B1. Let Bˆi be the closed ball concentric with Bi having radius rˆ, 1 ≤ i ≤ 13. Then the boundary bd( Bˆ1) of Bˆ1 is covered by the balls Bˆ 2, Bˆ3, . . . , Bˆ 13, that is, bd(Bˆ 1) ⊂ 13 j=2 Bˆj . Proof Let oi be the center of the unit ball Bi , 1 ≤ i ≤ 13 and assume that B1 is tangent to the unit balls B2, B3, . . . , B13 at the points tj ∈ bd(Bj ) ∩ bd(B1), 2 ≤ j ≤ 13. Let α denote the measure of the angles opposite to the equal sides of the isosceles triangle o1pq with dist(o1, p) = 2 and dist(p, q) = dist(o1, q) = rˆ, where dist(·, ·) denotes the Euclidean distance between the corresponding two points. Clearly, cos α = 1rˆ with α < π3 . Proposition 2.2 Let T be the convex hull of the points t2, t3, . . . , t13. Then the radius of the circumscribed circle of each face of the convex polyhedron T is less than sin α. Proof Let F be an arbitrary face of T with vertices tj , j ∈ IF ⊂ {2, 3, . . . , 13} and let cF denote the center of the circumscribed circle of F . Clearly, the triangle o1cF tj is a right triangle with a right angle at cF and with an acute angle of measure βF at o1 for all j ∈ IF . We have to show that βF < α. We prove this by contradiction. Namely, assume that α ≤ βF . Then either π3 < βF or α ≤ βF ≤ 3 π . First, let us take a closer look of the case π3 < βF . Reflect the point o1 about the plane of F and label the point obtained by o1. Clearly, the triangle o1o1oj is a right triangle with a right angle at o1 and with an acute angle of measure βF at o1 for all j ∈ IF . Then reflect the point o1 about o1 and label the point obtained by o1 furthermore, let B1 denote the unit ball centered at o1 . As π3 < βF therefore dist(o1, o1) < 2, and so one can simply translate B1 along the line o1o1 away from o1 to a new position say, B1 such that it is tangent to B1. However, this would mean that B1 is tangent to 13 non-overlapping unit balls namely, to B1 , B2, B3, . . . , B13, clearly contradicting to the well-known fact [12] that this number cannot be larger than 12. Thus, we are left with the case π . By repeating the definitions of o1, o1 , and B1 , the inequality βF ≤ when α ≤ βF ≤ 3 π3 implies in a straightforward way that the 14 unit balls B1, B1, B2, B3, . . . , B13 form a packing in E3. Moreover, the inequality α ≤ βF yields that dist(o1, o1) ≤ 4 cos α = 4 crˆo=ntr2a.d2i0c5ts27to83th3e36fo9l1lo. w..in<g2r.e2c0e5n2t 7re9s2u1l7t7o0f5B. öFriöncazllkyy, nanodticSeztahbaót [th3e].latter inequality Lemma 2.3 Let B1, B2, . . . , B14 be 14 different members of a packing of unit balls in E3. Assume that each ball of the family B2, B3, . . . , B13 touches B1. Then the distance between the centers of B1 and B14 is at least 2.205279217705. This completes the proof of Proposition 2.2. Now, we are ready to prove Lemma 2.1. First, we note that by projecting the faces F of T from the center point o1 onto the sphere bd(Bˆ 1) we get a tiling of bd(Bˆ 1) into spherically convex polygons Fˆ . Thus, it is sufficient to show that if F is an arbitrary face of T with vertices tj , j ∈ IF ⊂ {2, 3, . . . , 13}, then its central projection Fˆ ⊂ bd(Bˆ 1) is covered by the closed balls Bˆj , j ∈ IF ⊂ {2, 3, . . . , 13}. Second, in order to achieve this it is sufficient to prove that the projection cˆF of the center cF of the circumscribed circle of F from the center point o1 onto the sphere bd(Bˆ 1) is covered by each of the closed balls Bˆ j , j ∈ IF ⊂ {2, 3, . . . , 13}. Indeed, if in the triangle o1oj cˆF the measure of the angle at o1 is denoted by βF , then Proposition 2.2 implies in a straightforward way that βF < α. Hence, based on dist(o1, oj ) = 2 and dist(o1, cˆF ) = rˆ, a simple comparison of the triangle o1oj cˆF with the triangle o1pq yields that dist(oj , cˆF ) < rˆ holds for all j ∈ IF ⊂ {2, 3, . . . , 13}, finishing the proof of Lemma 2.1. Next, let us take the union in=1(ci + rˆB) of the closed balls c1 + rˆB, c2 + rˆB, . . . , cn + rˆB of radii rˆ centered at the points c1, c2, . . . , cn in E3. Lemma 2.4 nvol3(B) vol3( in=1(ci + rˆB)) < 0.7785, where vol3(·) refers to the 3-dimensional volume of the corresponding set. Proof First, partition in=1(ci + rˆB) into truncated Voronoi cells as follows. Let Pi denote the Voronoi cell of the packing P assigned to ci + B, 1 ≤ i ≤ n, that is, let Pi stand for the set of points of E3 that are not farther away from ci than from any other cj with j = i, 1 ≤ j ≤ n. Then, recall the well-known fact (see for example, [11]) that the Voronoi cells Pi , 1 ≤ i ≤ n just introduced form a tiling of E3. Based on this it is easy to see that the truncated Voronoi cells Pi ∩ (ci + rˆB), 1 ≤ i ≤ n generate a tiling of the non-convex container in=1(ci + rˆB) for the packing P . Second, as 3 2 = 1.2247 . . . < rˆ = 1.81383 therefore the following recent result (Corollary 3 in [1]) of the author applied to the truncated Voronoi cells Pi ∩ (ci + rˆB), 1 ≤ i ≤ n implies the inequality of Lemma 2.4 in a straightforward way. Lemma 2.5 Let F be an arbitrary (finite or infinite) family of non-overlapping unit balls in E3 with the unit ball B centered at the origin o of E3 belonging to F . Let P stand for the Voronoi cell of the packing F assigned to B. Moreover, let r := 23 = 1.2247 . . . and let rB denote the (closed) ball of radius r centered at the origin o of E3. Then vol3(B) vol3(B) vol3(P) ≤ vol3(P ∩ rB) 20√6 arctan( √22 ) − 2(2√6 − 1)π ≤ 5√2 + 3π − 15 arctan( √22 ) This finishes the proof of Lemma 2.4. The well-known isoperimetric inequality [10] applied to in=1(ci + rˆB) yields Lemma 2.6 n n , i=1 i=1 where svol2(·) refers to the 2-dimensional surface volume of the corresponding set. Thus, Lemmas 2.4 and 2.6 generate the following inequality. Corollary 2.7 . n i=1 disjoint open spherical caps of Sˆi ; moreover, Now, assume that ci + B ∈ P is tangent to cj + B ∈ P for all j ∈ Ti , where Ti ⊂ {1, 2, . . . , n} stands for the family of indices 1 ≤ j ≤ n for which dist(ci , cj ) = 2. Then let Sˆi := bd(ci + rˆB) and let cˆij be the intersection of the line segment ci cj with Sˆi for all j ∈ Ti . Moreover, let CSˆi (cˆij , π6 ) (resp., CSˆi (cˆij , α)) denote the open spherical cap of Sˆi centered at cˆij ∈ Sˆi having angular radius π6 (resp., α with 0 < 1 ). Clearly, the family {CSˆi (cˆij , π6 ), j ∈ Ti } consists of pairwise α < π2 and cos α = rˆ j∈Ti svol2(CSˆi (cˆij , π6 )) svol2( j∈Ti CSˆi (cˆij , α)) = svol2( j∈Ti C(uij , α)) where uij := 21 (cj − ci ) ∈ S2 := bd(B) and C(uij , π6 ) ⊂ S2 (resp., C(uij , α) ⊂ S2) denotes the open spherical cap of S2 centered at uij having angular radius π6 (resp., α). Now, Molnár’s density bound (Satz I in [9]) implies that In order to estimate svol2(bd( in=1(ci + rˆB))) from above let us assume that m members of P have 12 touching neighbors in P and k members of P have at most nine touching neighbors in P . Thus, n − m − k members of P have either 10 or 11 touching neighbors in P . (Here we have used the well-known fact that τ3 = 12, that is, no member of P can have more than 12 touching neighbors.) Without loss of generality we may assume that 4 ≤ k ≤ n − m. First, we note that svol2(C(uij , π6 )) = 2π(1 − cos π6 ) = 2π(1 − √23 ) and √ svol2(CSˆi (cˆij , π6 )) = 2π( 1 − 23 )rˆ2. Second, recall Lemma 2.1 according to which if a member of P say, ci + B has exactly 12 touching neighbors in P , then Sˆi ⊂ j∈Ti (cj + rˆB), i.e., svol2(bd( in=1(ci + rˆB))) has zero contribution coming from Sˆi . These facts together with (1) and ( 2 ) imply the following estimate. Finally, as the number C(n) of touching pairs in P is obviously at most therefore ( 3 ) implies that finishing the proof of (i) in Theorem 1.1. k ( 3 ) Although the idea of the proof of (ii) is similar to that of (i) they differ in the com binatorial counting part (see ( 9 )) as well as in the density estimate for packings of spherical caps of angular radii π6 (see ( 8 )). Moreover, the proof of (ii) is based on the new parameter value r¯ := √2 (replacing rˆ = 1.81383). The details are as follows. First, recall that if Λfcc denotes the face-centered cubic lattice with shortest nonzero lattice vector of length 2 in E3 and we place unit balls centered at each lattice point of Λfcc, then we get the fcc lattice packing of unit balls, labeled by Pfcc, in which each unit ball is touched by 12 others such that their centers form the vertices of a cuboctahedron. (Recall that a cuboctahedron is a convex polyhedron with 8 triangular faces and 6 square faces having 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron, i.e. an Archimedean solid, being vertex-transitive and edge-transitive.) Second, it is well known (see [4] for more details) that the Voronoi cell of each unit ball in Pfcc is a rhombic dodecahedron (the dual of a cuboctahedron) of volume √32 (and of circumradius √2). Thus, the density 4π of Pfcc is √332 = √π18 . Now, let B denote the unit ball centered at the origin o ∈ Λfcc of E3 and let P := {c1 + B, c2 + B, . . . , cn + B} denote the packing of n unit balls with centers {c1, c2, . . . , cn} ⊂ Λfcc in E3 having the largest number Cfcc(n) of touching pairs among all packings of n unit balls being a sub-packing of Pfcc. (P might not be uniquely determined up to congruence in which case P stands for any of those extremal packings.) As the Voronoi cell of each unit ball in Pfcc is contained in a ball of radius r¯ = √2 therefore, based on the corresponding decomposition of n i=1(ci + r¯B) into truncated Voronoi cells, we get nvol3(B) vol3( in=1(ci + r¯B)) As a next step we apply the isoperimetric inequality [10]: n i=1 Thus, ( 4 ) and ( 5 ) in a straightforward way yield n i=1 n i=1 (ci + r¯B) . ( 4 ) ( 5 ) ( 6 ) 2.2 Proof of (ii) (ci + r¯B) Now, assume that ci + B ∈ P is tangent to cj + B ∈ P for all j ∈ Ti , where Ti ⊂ {1, 2, . . . , n} stands for the family of indices 1 ≤ j ≤ n for which dist(ci , cj ) = 2. Then let S¯i := bd(ci + r¯B) and let c¯ij be the intersection of the line segment ci cj with S¯i for all j ∈ Ti . Moreover, let CS¯i (c¯ij , π6 ) (resp., CS¯i (c¯ij , π4 )) denote the open spherical cap of S¯i centered at c¯ij ∈ S¯i having angular radius π6 (resp., π4 ). Clearly, the family {CS¯i (c¯ij , π6 ), j ∈ Ti } consists of pairwise disjoint open spherical caps of S¯i ; moreover, j∈Ti svol2(CS¯i (c¯ij , π6 )) , ( 7 ) svol2( j∈Ti C(uij , π4 )) ≤ 6 1 − with equality when 12 spherical caps of angular radius π6 are packed on S2. Finally, as svol2(C(uij , π6 )) = 2π(1 − cos π6 ) and svol2(CS¯i (c¯ij , π6 )) = 2π × √ ( 1 − 23 )r¯2 therefore (7) and ( 8 ) yield that n i=1 From ( 10 ) the inequality Cfcc(n) < 6n − 3 √3π18π n 23 = 6n − 3.665 . . . n 3 follows in 2 a straightforward way for all n ≥ 2. This completes the proof of (ii) in Theorem 1.1. 2.3 Proof of (iii) It is rather easy to show that for any positive integer k ≥ 2 there are n(k) := 2k33+k = k(2k2+1) lattice points of the face-centered cubic lattice Λfcc such that their convex 3 hull is a regular octahedron K ⊂ E3 of edge length 2(k − 1) having exactly k lattice points along each of its edges. Now, draw a unit ball around each lattice point of Λfcc ∩ K and label the packing of the n(k) unit balls obtained in this way by Pfcc(k). It is easy to check that if the center of a unit ball of Pfcc(k) is a relative interior point of an edge (resp., of a face) of K, then the unit ball in question has seven (resp., nine) touching neighbors in Pfcc(k). Last but not least, any unit ball of Pfcc(k) whose center is an interior point of K has 12 touching neighbors in Pfcc(k). Next we note that the number of lattice points of Λfcc lying in the relative interior of the edges (resp., faces) of K is 12(k − 2) = 12k − 24 (resp., 8( 12 (k − 3)2 + 21 (k − 3)) = 4(k − 3)2 + 4(k − 3)). Furthermore the number of lattice points of Λfcc in the interior of K is equal to 23 (k − 2)3 + 31 (k − 2). Thus, the contact number C(Pfcc(k)) of the packing Pfcc(k) is equal to ( 11 ) √ 2 Finally, as 2k3 < n(k) therefore 6k2 < 3 486n 3 (k), and so ( 11 ) implies (iii) of 3 Theorem 1.1 in a straightforward way. 3 On Improving (i) in Theorem 1.1 Let δ(K) denote the largest density of packings of translates of the convex body K in Ed , d ≥ 3. The following result has been proved by the author in [2]. Lemma 3.1 Let Ko be a convex body in Ed , d ≥ 2 symmetric about the origin o of Ed and let {c1 + Ko, c2 + Ko, . . . , cn + Ko} be an arbitrary packing of n ≥ 1 translates of Ko in Ed . Then nvold (Ko) vold ( in=1(ci + 2Ko)) < δ(Ko). Let B denote the unit ball centered at the origin o of E3 and let P := {c1 + B, c2 + B, . . . , cn + B} denote the packing of n unit balls with centers c1, c2, . . . , cn having the largest number C(n) of touching pairs among all packings of n unit balls in E3. (P might not be uniquely determined up to congruence in which case P stands for any of those extremal packings.) The well-known result of Hales [5] according to which δ3 = √π18 and Lemma 3.1 imply in a straightforward way nvol3(B) vol3( in=1(ci + 2B)) < δ(B) = √ π 18 . The isoperimetric inequality [10] yields Lemma 3.2 Lemma 3.3 Corollary 3.4 Thus, Lemmas 3.2 and 3.3 generate the following inequality. . √ 2 4 3 18π n 3 < svol2 bd . n i=1 n i=1 Now, assume that ci + B ∈ P is tangent to cj + B ∈ P for all j ∈ Ti , where Ti ⊂ {1, 2, . . . , n} stands for the family of indices 1 ≤ j ≤ n for which ci − cj = 2. Then let Si := bd(ci + 2B) and let CSi (cj , π6 ) denote the open spherical cap of Si centered at cj ∈ Si having angular radius π6 . Clearly, the family {CSi (cj , π6 ), j ∈ Ti } consists of pairwise disjoint open spherical caps of Si ; moreover, j ∈Ti svol2(CSi (cj , π6 )) svol2( j ∈Ti CSi (cj , π3 )) = svol2( j ∈Ti svol2(C(uij , π6 )) j ∈Ti C(uij , π3 )) , where uij := 21 (cj − ci ) ∈ S2 and C(uij , π6 ) ⊂ S2 (resp., C(uij , π3 ) ⊂ S2) denotes the open spherical cap of S2 centered at uij having angular radius π6 (resp., π3 ). Now, we are ready to state the main conjecture of this section. open spherical caps of angular radii π6 in S2. Then Conjecture 3.5 Let {C(um, π6 ), 1 ≤ m ≤ M } be a family of M pairwise disjoint svol2( 1≤m≤M svol2(C(um, π6 )) 1≤m≤M C(um, π3 )) ≤ 6 1 − with equality when M = 12 spherical caps of angular radii π6 are packed on S2. Clearly, M ≤ τ3 = 12. Moreover, if true, then Conjecture 3.5 can be used to improve the upper bound for C(n) in (i) of Theorem 1.1 as follows. First, Conjecture 3.5 implies in a straightforward way that Second, the above inequality combined with Corollary 3.4 yields from which the inequality follows. Clearly, this would be a significant improvement on (i) in Theorem 1.1. Acknowledgements The author wishes to thank an anonymous referee for a number of helpful comments and suggestions. Partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. 1. Bezdek , K. : On a stronger form of Rogers' lemma and the minimum surface area of Voronoi cells in unit ball packings . J. Reine Angew . Math. 518 , 131 - 143 ( 2000 ) 2. 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Károly Bezdek. Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space, Discrete & Computational Geometry, 2012, 298-309, DOI: 10.1007/s00454-012-9405-9