On the Density of Packings of Spheres in Spherical 3-Space

Transcription

1 Sitzungsber. Abt. II (2005) 214: On the Density of Packings of Sheres in Sherical 3-Sace By August Florian (Vorgelegt in der Sitzung der math.-nat. Klasse am 13. Oktober 2005 durch das w. M. August Florian) Abstract In an n-dimensional Euclidean, sherical or hyerbolic sace consider a acking of at least four sheres with given radius r. The well-known simlicial density bound d n ðrþ gives an uer bound to the density of such a acking. In sherical 3-sace there are exactly four ackings for which the density d 3 ðrþ is attained. These ackings are formed by the insheres of the regular tilings of tye fa; 3; 3g, for a ¼ 2; 3; 4 and 5. In this aer it is roved that d 3 is a strictly decreasing function of r. This imlies the existence of the uer bound to the density of any acking of at least four congruent sheres in S 3. Mathematics Subject Classification (2000): 52C17, 05B40. Key words: Sherical sace, acking of sheres, density. 1. Introduction We shall use S n, E n and H n to denote the n-dimensional sherical, Euclidean and hyerbolic sace, resectively. The sace S n can be viewed as the surface of the ðn þ 1Þ-dimensional unit shere in E nþ1. A set of oen sheres is said to form a acking in S n, E n or H n if each oint of the sace belongs to at most one shere of the set. The most frequently emloyed method of measuring the efficiency of a For my brother HELMUT FLORIAN.

2 90 A. Florian acking is to determine its density [6]. As usual, the density of a system of subsets of S n is defined as the ratio of the total n-dimensional volume of the sets to the volume of S n. The density of a system of bodies in E n is defined, roughly seaking, as the ratio of the total volume of the members of the system to the volume of the whole sace. This is made recise by an aroriate limit rocess. A general definition of the density of an arbitrary arrangement of subsets of H n with resect to the whole sace is not available. The difficulties arising in this connection are discussed in some detail in [6]. Thus only statements of a local character can be exected concerning density in H n. In S n, E n or H n ðn 2Þ consider a regular simlex of edge length 2r (in S n : 2r<arccosð 1=nÞ), and the system of n þ 1 sheres of radius r, with their centres at the vertices of the simlex. The sheres of the system do not overla. Let d n ðrþ denote the ratio of the volume of the art of the simlex covered by the sheres to the volume of the whole simlex. In a aer of 1959, L. FEJES TOTH [10] stated the conjecture that in S n, E n or H n, each acking of at least n þ 1 sheres of radius r has a density d d n ðrþ: ð1þ In the case of the hyerbolic sace, (1) has to be relaced by an aroriate local version. As is well known, inequality (1) is true for n ¼ 2; see [8] or [11]. In two earlier aers L. FEJES TOTH [9] and COXETER [5], indeendently of each other, had ut forward conjecture (1) in the case n ¼ 3. They advanced some arguments to suort it and ointed out some interesting consequences, esecially for ackings in non-euclidean saces. About the time when FEJES TOTH s conjecture was ublished, ROGERS [15, 16] succeeded in roving (1) for ackings in E n ðn 2Þ. Since in E n the bound d n ðrþ does not deend on r, we shall simly write d n for d n ðrþ. A acking of congruent sheres is said to be saturated if any additional shere of the same size overlas some member of the acking. Let us consider a saturated acking of n-dimensional congruent sheres. To each shere S of the acking we assign the set PðSÞ of all oints of sace, whose distance from the centre of S is not greater than their distance from the centre of any other shere of the system. The set PðSÞ is a closed, bounded and convex olytoe containing S. The olytoes PðSÞ, corresonding to the different sheres of the system, fit together, without overlaing and without gas, to fill u the whole sace; they are called the Voronoi olytoes of the acking.

3 On the Density of Packings of Sheres in Sherical 3-Sace 91 The required inequality (1) is roved for E n by showing that, for each shere S of the acking, the ratio of the volume of S to the volume of the assigned Voronoi olytoe PðSÞ does not exceed d n. Note that BARANOVSKIĬ [1] indeendently roved the simlicial bound (1) for E n. In the secial case when n ¼ 2, we have d 2 ¼ = ffi 1 The incircles of the regular tiling f6; 3g form a acking with maximum density = ffi 12 [11]. When n ¼ 3, inequality (1) shows that the density of each acking of equal sheres in E 3 satisfies d d 3 ¼ ffi 18 arccos 1 3 ¼ 0: : ð2þ 3 The dihedral angle of a regular tetrahedron, i.e. arccos 1=3, lies between 2=6 and 2=5. Therefore, the density d 3 cannot be attained by utting together congruent regular tetrahedra to fill u the whole sace. ROGERS uer bound on the density of acking in E 3 has been successively reduced. MUDER s result [13] d 0: ðe 3 Þ ð3þ is at resent the best comletely ublished uer bound to the density of ackings of equal sheres in E 3. The review articles [6], [7], [2], [12] contain more information on acking of sheres (also in higher dimensions) and related toics. Let us now consider ackings of equal sheres in sherical and hyerbolic saces. Let Mðn; rþ denote the maximum number of sheres with radius r<=2 which can be laced on S n without overlaing. RANKIN [14] determined the exact values of Mðn; rþ for all n 2 and r =4. In 1963 BÖRÖCZKY [3] roved FEJES TOTH s conjecture (1) for n ¼ 3. More recisely: In S 3, E 3 or H 3 consider a saturated acking of at least 4 sheres with radius r (on S 3 : r<=4). Then the density of each shere of the acking with resect to its Voronoi olyhedron is not greater than d 3 ðrþ. This formulation does not refer to a concet of global density and thus also alies to hyerbolic sace. The statement clearly imlies that the density of acking with resect to the whole sace S 3 or E 3 is also bounded by d 3 ðrþ. Several years later, BÖRÖCZKY [4] roved FEJES TOTH s conjecture (1) for ackings of equal sheres in n-dimensional saces (on S n : r<=4).

4 92 A. Florian Let us return to ackings in 3-dimensional saces, first in H 3. In the common aer with BÖRÖCZKY [3], FLORIAN roved that d 3 ðrþ is a strictly increasing function for 0<r<1. Thus the (local) density of any acking of equal sheres in H 3 satisfies d< lim d 3 ðrþ: r!1 The elegant reresentation of the uer bound lim r!1 d 3ðrÞ ¼ 1 þ þ þ þþ 1 ¼ 0: is due to COXETER [5]. The bound is attained by the acking of horosheres inscribed in the cells of the 3-dimensional regular tiling f6; 3; 3g. Finally let us turn to ackings of equal sheres in S 3 and BÖRÖCZKY s tetrahedral density bound d 3 ðrþ. It is convenient to consider, in addition to the side-length 2r, the dihedral angle of a regular tetrahedron. In S 3, a non-degenerate regular tetrahedron with dihedral angle 2 exists if and only if arccos 1 3 <2<: The lower bound is the dihedral angle of a regular tetrahedron in E 3. When 2 ¼, the tetrahedron Tð2Þ degenerates into a half-sace of S 3. The edge-length 2r is related to 2 by the equation cos 2r cos 2 ¼ ð5þ 1 þ 2 cos 2r showing that is a strictly increasing function of r. Inequality (4) is equivalent to 0<2r< arccos 1 3 ¼ 2 arctan 2 : ð6þ In this aer we rove the following theorem. Theorem. In S 3, the density bound d 3 ðrþ is a strictly decreasing function, for 0<r arctan Corollary. The density of any acking of at least 4 equal sheres in S 3 is less than lim d 3ðrÞ ¼ ffi 18 arccos 1 r!0 3 ¼ 0: : ð7þ 3 ð4þ

5 On the Density of Packings of Sheres in Sherical 3-Sace 93 On S 3 there are exactly 4 regular tilings, the cells of which are tetrahedra. They corresond to the dihedral angle 2=k, for k ¼ 2; 3; 4; 5: f3; 3; 2g; f3; 3; 3g; f3; 3; 4g; f3; 3; 5g: The vertices of the tilings are centres of 4, 5, 8 and 120 equal sheres touching each other in each case which form a densest acking of sheres with the resective radius. They are the insheres of the cells of the dual tilings f2; 3; 3g, f3; 3; 3g, f4; 3; 3g, f5; 3; 3g. The following list contains the radii and corresonding densities of ackings. f2; 3; 3g r 2 ¼ 1 2 arccos 1 3 ; dðr2 Þ¼0: ; f3; 3; 3g r 3 ¼ 1 2 arccos 1 4 ; dðr3 Þ¼0: ; f4; 3; 3g r 4 ¼ =4; dðr 4 Þ¼0: ; f5; 3; 3g r 5 ¼ =10; dðr 5 Þ¼0: : There is no other acking of sheres with radius r and density d 3 ðrþ, for 0<r arctan This follows from BÖRÖCZKY s aer [4] for 0<r<=4, and RANKIN [14] for r ¼ =4. The cases when r>=4 can be settled in a direct way, since only 5 or 4 sheres are involved. Combining the Theorem with RANKIN s result [14] for n ¼ 3 we see that BÖRÖCZKY s density bound d d 3 ðrþ holds for 0<r arctan ffi It is remarkable that dðr 5 Þ is greater than the density of any acking of equal sheres in E 3, see (3). This is in contrast to the corresonding situation in dimension 2, see [11]. The insheres of f5; 3; 3g form a densest acking of at least 4 and at most 120 equal sheres. Observe that dðr 5 Þ is rather close to lim r!0 d 3 ðrþ ¼0: It may be that the insheres of f5; 3; 3g even form a densest acking of at least 4 equal sheres in S 3 (see [10]). At resent, however, we are far from being able to rove this conjecture. Observe that the density of the densest acking of sheres with radius r is not decreasing in any subinterval of 0<r< arctan Thus, the behaviour of this function is quite different from that of d 3 ðrþ. 2. Proof of the Theorem Let us recall the definition of the tetrahedral density bound d 3 ðrþ in the case of the 3-dimensional sherical sace. Consider a system of 4 sheres of radius r ð0<r arctan 2 Þ touching each other. Let T be the tetrahedron with its vertices at the centres of the sheres. Then d 3 ðrþ is the ratio of the volume of the

6 94 A. Florian art of T covered by the sheres of the system to the volume of T. The dihedral angle 2 of T is connected with r by the equation cos 2r cos 2 ¼ 1 þ 2 cos 2r : ð8þ Then 4ð6 Þ d 3 ðrþ ¼ ð2r sin 2rÞ: 4VðTð2ÞÞ ð9þ The volume of the tetrahedron Tð2Þ follows from Schläfli s differential form by integration ð cos 2u VðTð2ÞÞ ¼ 6 2rðuÞ du; cos 2rðuÞ ¼ cos 2u ; ð10þ where 2 0 ¼ arccosð1=3þ is the dihedral angle in E 3. Formula (9) takes the form 6 d 3 ðrþ ¼ ð2r sin 2rÞ; ð11þ 6I where I ¼ 2rðuÞ du: 0 Referring to (8), (10), (11) and (12) we shall rove that d3 0 ðrþ<0 for 0<r< arctan For brevity, we write tan r ¼ t; so that 0<t< ffi 2 : From (8) and (12) we obtain d dr ¼ ð tð1 þ t 2 Þ 2 t 2 ð3 t 2 Þ ; di d ¼ 2r dr dr : ð12þ ð13þ ð14þ ð15þ Here and in the following we omit straightforward calculations. Using (15) we get from (11) 6I 2 d3 0 24t2 ðrþ ¼ 1 þ t 2 gðrþ f 1ðrÞ; ð16þ

7 where and On the Density of Packings of Sheres in Sherical 3-Sace 95 gðrþ ¼I r ð1 þ t 2 Þ 2 ð2r sin 2rÞ 2 t 2 t 2 ð3 t 2 Þ ð17þ f 1 ðrþ ¼ ð1 þ t2 Þ 2 ð2r sin 2rÞI 4t 2 t 2 ð3 t 2 ÞgðrÞ þ 6 : Since 2r>sin 2r, t>r and t 2 <2, we see that ð18þ g 0 ðrþ¼ ð1þt2 Þ 2 ð2r sin2rþ þ12t 4 Þ tð3 t 2 Þð2 t 2 ÞŠ<0: 2t 2 ð2 t 2 Þ 3=2 ð3 t 2 Þ 2½rð6 30t2 Because, moreover, lim r!0 gðrþ ¼0wehave gðrþ<0 ð19þ for 0<r< arctan Introducing the abbreviation ffi ffi 4t 2 t 2 ð3 t 2 ÞgðrÞ¼4t 2 t 2 ð3 t 2 ÞI 2rð1 þ t 2 Þ 2 ð2r sin 2rÞ ¼ NðrÞ ð20þ we obtain from (18) N 2 I f 1 0 ðrþ ¼ð1þt2 Þg 1 ðrþf 2 ðrþ; ð21þ where 1 g 1 ðrþ ¼ ½3ð1 þ t 2 Þð 1 þ 5t 2 2t 4 Þð2r sin 2rÞ 2 t 2 þ 4t 3 ð2 t 2 Þð3 t 2 ÞŠ ð22þ and f 2 ðrþ ¼8I þ 2ð1 þ t 2 Þ 2 ð1 þ t 2 Þð2r sin 2rÞ 8t 2 r ð2r sin 2rÞ: g 1 ðrþ ð23þ First we show that g 1 ðrþ>0 ð24þ for 0<r< arctan ffi This is clear 1 þ 5t 2 2t 4 0. Let now 1 þ 5t 2 2t 4 <0, i.e., t 2 <ð5 ffi 17 Þ=4< 1. Since r<t< 2,we 4

8 96 A. Florian have 2r<2 ffi 2 and ð2rþ 2 <8. Thus and From 2r sin 2r ¼ ð2rþ3 3! ð2rþ5 5! 3ð2r sin 2rÞ<4r 3 <4t 3 : þ< 4 3 r3 j 1 þ 5t 2 2t 4 j¼1 5t 2 þ 2t 4 <1 þ 2t 4 < 9 8 and 1 þ t 2 < 5 4 we conclude that jð1 þ t 2 Þð 1 þ 5t 2 2t 4 Þj< ¼ : On the other hand we have ð2 t 2 Þð3 t 2 Þ> ¼ > ; as required. From (22) it follows that g 1 ðrþ lim r!0 2r sin 2r ¼ 15 ; 2 and from (23) lim f 2 ðrþ ¼0: ð25þ r!0 Differentiation of f 2 ðrþ yields in a first ste g 2 1 f 2 0ðrÞ 2ð1 þ t 2 Þ 2 ð2r sin 2rÞ ¼ g 8rt 1 ð2 t 2 Þð3 t 2 Þ ð 3 þ 15t2 6t 4 Þ þ g 1 4t½ð1 þ t 2 Þð2r sin 2rÞ 8t 2 rš þ g 1 ½2tð1 þ t 2 Þð2r sin 2rÞ 16rtð1 þ t 2 ÞŠ ½ð1þt 2 Þð2r sin 2rÞ 8t 2 ršg 0 1 ðrþ; ð26þ where g 0 1ðrÞ can be calculated from (22), g 0 1 ðrþ ¼ tð1 þ t2 Þ ð1 þ t 2 Þð 3 þ 15t 2 6t 4 Þð2r sin 2rÞ ð2 t 2 Þ 3=2 þ 4t 3 ð2 t 2 Þð3 t 2 Þ 1 þ 2tð1 þ t 2 Þð12 þ 18t 2 18t 4 Þð2r sin 2rÞ 2 t 2 þ 4t 2 ð15 þ 8t 2 24t 4 þ 7t 6 Þ : ð27þ

9 On the Density of Packings of Sheres in Sherical 3-Sace 97 Collecting the terms containing the same ower of 2r sin 2r, from (26) and (27) we obtain g 2 1 f 2 0ðrÞ 2ð1 þ t 2 Þ 2 ð2r sin 2rÞ ¼ A ð2r sin 2rÞ2 þ Bð2r sin 2rÞ ð28þ with A ¼ tð1 þ t2 Þ 2 ð2 t 2 Þ ð 81 þ 3=2 138t2 63t 4 þ 6t 6 Þ; B ¼ 4t2 ð1 þ t 2 Þ ð 15 þ 25t 2 8t 4 Þ 2 t 2 8rtð1 þ t 2 Þ þ ð2 t 2 Þ 3=2 ð3 t 2 Þ ð45 57t2 6t 4 þ 57t 6 33t 8 þ 6t 10 Þ: ð29þ If we relace sin 2r by 2t=ð1 þ t 2 Þ, it follows from (28) and (29) that g 2 1 f 0 2 ðrþ 2ð1 þ t 2 Þ 2 ð2r sin 2rÞ 2 2tð1 þ t 2 Þ ¼ ð2 t 2 Þ 3=2 ð3 t 2 Þ ftð63 45t2 49t 4 þ 49t 6 10t 8 Þ þ rð 63 þ 24t 2 þ 144t 4 18t 6 57t 8 þ 18t 10 Þg 2tð1 þ t 2 Þ 2 ¼ ð2 t 2 Þ 3=2 ð3 t 2 Þ ½tP 1ðtÞþrP 2 ðtþš: ð30þ Here we have used the abbreviations P 1 ðtþ ¼ 10ðt 2 3Þðt 2 1:5Þðt 2 1:4Þ; P 2 ðtþ ¼18t 8 75t 6 þ 57t 4 þ 87t 2 63: ð31þ First let us assume that 0<t 1: ð32þ Obviously, P 1 ðtþ>0, while P 2 ðtþ changes from negative to ositive values. We shall now show that tp 1 ðtþþrp 2 ðtþ>0: ð33þ Let P 2 ðtþ<0. Since r ¼ arctan t<t t3 3 þ t5 5 ;

10 98 A. Florian we have tp 1 ðtþþrp 2 ðtþ> tp 1 ðtþþ t t3 3 þ t5 P 2 ðtþ 5 ¼ t 5 ð3:6t 8 21t 6 þ 54:4t 4 86:6t 2 þ 74:4Þ: It can easily be roved that the last term is ositive for 0<t 1. Second let us assume that 1 t ffi 2 : ð34þ Observe that, by (31), P 1 ðtþ is ositive, excet for the interval 1:4 <t< 1:5, where P1 ðtþ<0. Because P 2 ðtþ>0 in the whole interval and r ¼ arctan t>t t3 3 ; we have tp 1 ðtþþrp 2 ðtþ> tp 1 ðtþþ t t3 P 2 ðtþ 3 ¼ t 5 ð 6t 6 þ 43t 4 104t 2 þ 87Þ: It can easily be shown that the last term is ositive, so that (33) also holds for 1 t The last two results combined with (30) imly that f2 0 ðrþ>0 ð35þ for 0<t< From (35) and (25) it follows that f 2 ðrþ>0: ð36þ From (36), (24) and (21) we conclude that f1 0 ðrþ>0; ð37þ for 0<r< arctan In order to calculate lim r!0 f 1 ðrþ we refer to (18). Making use of (15) we obtain the exansion of IðrÞ into a ower series IðrÞ ¼ r3 þ : ð38þ Hence ð2r sin 2rÞIðrÞ ¼ r6 þ : ð39þ

11 On the Density of Packings of Sheres in Sherical 3-Sace 99 On the other hand, we get from (20) and (38) 4t 2 t 2 ð3 t 2 ÞgðrÞ ¼ 12 5 r6 þ: ð40þ Then, by (18), (39) and (40) in conjunction with (5), lim f 1 ðrþ ¼ 10 r!0 81 þ arccos ¼ 0: >0: ð41þ The combination of (37) and (41) shows that f 1 ðrþ>0 ð42þ for 0<r< arctan The desired inequality (13) is a consequence of (16), (19) and (42). Acknowledgement I would like to thank Prof. J. LINHART for his careful reading of the aer and for correcting an error. References [1] BARANOVSKIĬ, E. P. (1964) On acking of n-dimensional Euclidean saces by equal sheres (Russian). Izv. Vyss. Ucebn. Zaved. Matematika 39: [2] BEZDEK, K. (to aear) Shere ackings revisited. Euro. J. Comb. [3] BÖRÖCZKY, K., FLORIAN, A. (1964) Uber die dichteste Kugelackung im hyerbolischen Raum. Acta Math. Acad. Sci. Hungar. 15: [4] BÖRÖCZKY, K. (1978) Packing of sheres in saces of constant curvature. Acta Math. Acad. Sci. Hungar. 32: [5] COXETER, H. S. M. (1954) Arrangements of equal sheres in non-euclidean saces. Acta Math. Acad. Sci. Hungar. 5: [6] FEJES TOTH, G., KUPERBERG, W. (1993) Packing and covering with convex sets. In: Handbook of Convex Geometry (GRUBER, P. M., WILLS, J. M., eds.), North-Holland, Amsterdam [7] FEJES TOTH, G. (2004) Packing and covering. In: Handbook of Discrete and Comutational Geometry (GOODMAN, J. E., et al., eds.), 2nd enlarged Ed., CRC Press, Boca Raton [8] FEJES TOTH, L. (1953) Kreisausfüllungen der hyerbolischen Ebene. Acta Math. Acad. Sci. Hungar. 4: [9] FEJES TOTH, L. (1953) On close-ackings of sheres in saces of constant curvature. Publ. Math. Debrecen 3: [10] FEJES TOTH, L. (1959) Kugelunterdeckungen und Kugelüberdeckungen in Räumen konstanter Krümmung. Arch. Math. 10: [11] FEJES TOTH, L. (1964) Regular Figures. Pergamon Press, Oxford [12] FLORIAN, A. (to aear) On the density of ackings of sheres in saces of constant curvature. Rend. Circ. Mat. Palermo

12 100 A. Florian: On the Density of Packings of Sheres in Sherical 3-Sace [13] MUDER, D. J. (1993) A new bound on the local density of shere ackings. Discrete Comut. Geom. 10: [14] RANKIN, R. A. ( ) The closest acking of sherical cas in n dimensions. Proc. Glasgow Math. Assoc. 2: [15] ROGERS, C. A. (1958) The acking of equal sheres. Proc. London Math. Soc. 8: [16] ROGERS, C. A. (1964) Packing and Covering. Cambridge University Press, Cambridge Author s address: Prof. Dr. August Florian, Deartment of Mathematics, University of Salzburg, Hellbrunner Straße 34, 5020 Salzburg, Austria. august.florian@sbg.ac.at.