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On limits of dense packing of equal spheres in a cube

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On limits of dense packing of equal spheres in a cube Milos Tatarevic Alameda, CA 90, U.S.A. Submitted: Oct 8, 0; Accepted: Jan, 0; Published: Feb, 0 Mathematics Subject Classifications:
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On limits of dense packing of equal spheres in a cube Milos Tatarevic Alameda, CA 90, U.S.A. Submitted: Oct 8, 0; Accepted: Jan, 0; Published: Feb, 0 Mathematics Subject Classifications: C, 0B0 Abstract We eamine packing of n congruent spheres in a cube when n is close but less than the number of spheres in a regular cubic close-packed (ccp) arrangement of p / spheres. For this famil of packings, the previous best-known arrangements were usuall derived from a ccp b omission of a certain number of spheres without changing the initial structure. In this paper, we show that better arrangements eist for all n p /. We introduce an optimiation method to reveal improvements of these packings, and present man new improvements for n 9. Introduction We consider the problem of finding the densest packings of congruent, non-overlapping, spheres in a cube. Equivalentl, we can search for an arrangement of points inside a unit cube so that the minimum distance between an two points is as large as possible. The maimum separation distance of n points in [0, ] we denote b d n. To our knowledge, the optimalit of d n is proved for n =,,,,, 8, 9 [0], n = 0 [] and n = []. Optimalit of d n was conjectured for an infinite famil of packings where p / spheres are arranged in a cubic close-packed (ccp) structure []. We denote b g(p) = p / the number of spheres in these packings, with a maimum separation distance denoted b d p = /(p ). In this paper, we eamine arrangements when n is close, but less than g(p). For this famil of packings d n = d p is often assumed, to mean that the densest known arrangements are derived from ccp b omission of a certain number of spheres without changing the initial structure. Limiting values though were not provided. It was conjectured that d n is constant in the range n [] and 9 n []. A better packing was found for n = []. Similarl, previous search results showed the same trend for 0 n, 0 n 08,..., 8 n 8 [,, ]. In Section we show that most of the listed packings can be improved b proving that d n d p for all n g(p). We also provide a lower bound for these improvements. the electronic journal of combinatorics () (0), #P. In Section, we introduce an optimiation method and improve the lower bound for p. We show that the described procedure can be used as a good packing method when n is slightl smaller than g(p). We run search to determine improvements for other packings when n = g(p) r, for r, r p and p. Eistence of improved packings To simplif our notation, we will assume that the radius of all spheres in the packing is and that our task is to determine the smallest sie of a cube that contains all spheres. We denote the famil of all finite sets of points such that the distance between an two points is at least b F = { S R : s s for all distinct s, s S }. Let S n = {s,..., s n } F and let D c (s i, s j ) be the Chebshev distance between an two points s i, s j S n. The smallest edge length of a cube, with edges parallel to the aes, such that it contains all points S n is equal to D(S n ) = ma {D c (s i, s j ) : i j n}. We notice that the maimum separation distance d n can also be given as d n = ma D(S n ). Theorem. The maimum separation distance of n points contained in a closed region bounded b a unit cube is larger than / (p ) for all n p /. Proof. Let C p be a closed region bounded b a cube with an edge length D p = /d p = (p ), where C p is defined b C p = [0, D p ]. Let G p be a set of g(p) sphere centers in a ccp arrangement, such that G p F, G p C p and 0, 0, 0 G p (see Figure ). For two sets of points A, B F let h(a, B) = min { a b : a A, b B} and let L p = G p \G p (see Figure (a) for an eample). We denote the improved packing of g(p) points b P p = { s,..., s g(p) } F such that P p C p, () h(p p, L p+ ) 0. () Equations () and () directl impl that if these statements are true, then D(P p ) D p. Let T p = τ(l p ) be a set of points created b the translation of points from L p b the function τ such that T p F, T p C p, h(p p, T p ) 0, () the electronic journal of combinatorics () (0), #P. 0 0 (a) G (b) G 0 (c) G Figure : An illustration of arrangements G, G and G. for all p . If such a set T p eists, then we can state that h(t p, L p+ ) 0 () P p = P p T p, for all p . We can easil show that P eists b constructing it eplicitl. If we want to give P in such a wa to maimie the capabilit to translate points from L and thus produce a the electronic journal of combinatorics () (0), #P. 0 0 (a) L (b) L, 0 0 (c) L, (d) L, Figure : An illustration of arrangements L, L,, L, and L,. better packing P, we have to find the positive root of the polnomial a + a + 8a 8 = 0, a 0 = a = then P is given as P = { 0, 0, a, b, b, 0 }, where b = a /. While P p and L p are separated (), we can tr to translate all points from L p and the electronic journal of combinatorics () (0), #P. keep (), and () true. We split L p into three subsets, and perform the translations on each of them while maintaining the given conditions. Let L p = L p, L p, L p, such that { L p, = l, l, l L p : l i, l j (p ) } for some distinct i, j {,, }, L p, = { l, l, l L p \ L p, : l, l, l 0}, L p, = L p \ L p, \ L p,. For an illustration of sets L, L,, L, and L,, see Figure. Now we can give T p as T p = T p, T p, T p,, T p,i = { l u i, τ i (p), l u i, τ i (p), l u i, τ i (p) : l, l, l L p,i }, { if l j = (p ) { if l j 0 u,j =, u,j =, u,j = 0 otherwise 0 otherwise, where τ (p), τ (p) and τ (p) are small numbers such that τ (p) τ (p) τ (p) and conditions () and () hold. If we additionall state that h(p p, T p, ) = 0, h(p p T p,, T p, ) = 0 and h(p p T p, T p,, T p, ) = 0, we can give eplicit solutions for τ, τ and τ as follows τ () = a, ( τ (p) τ (p) = + τ (p) + ) τ (p) +, for all p, ( τ τ (p) = (p) + τ (p) + ) τ (p) +, for all p, τ (p) = + τ (p ) τ (p ) + τ (p ) +, for all p . We notice that D(P p ) = D p τ (p), hence the maimum separation distance can be given as d n (p ), for all n g(p) τ (p) and while τ (p) 0, then d n /(p ). We denote the lower bound of improvements b I p = d g(p) d p. B performing the calculations, we get particular values such as I 8. 0, I . 0 9,.... This is a rough approimation while condition () needs to remain true for all p and does not allow us to further improve the packing for particular p. In the net section, we show that I p is usuall above this approimation. We did not find a wa to improve the packings when n = g(p), and we conjecture that in this case d n = d p. the electronic journal of combinatorics () (0), #P. Optimiation Approach Most of the eisting packing methods focus on searching for a completel new arrangement of spheres, usuall performing a search from a randoml given initial position of spheres. Such approach assumes that the packing of higher densit can be reached after a certain number of iterations and multiple runs of the search procedure using different initial parameters [,, ]. The large number of iterations often limits the search procedure to the use of double or quadruple floating-point precision, to maintain computation speed. This precision is insufficient to detect improvements in man packings. Man of these approaches are adapted and modified from widel known procedures for packing congruent circles in a square or a circle [,, 8, 9, ]. The method we suggest is based on a hpothesis that an improved packing can be reached just b the omission of two or more spheres from the ccp and b performing a translation of spheres using the available space made after we remove the spheres. The initial positions of the sphere centers we denote b S p,r G p, as a set of g(p) r points such that at least one sphere with a center s i S p,r can be continuousl translated inside a cube container without overlapping with other spheres. More precisel, g(p) r i= {q C p : q / S p,r, (S p,r \ {s i }) {q} F}. We can see that the construction of S p,r is possible for r, and we eperimentall determine solutions based on improvements reached for certain arrangements. To simplif the search procedure, instead of tring to figure out the best performing arrangements S p,r for each pair (p, r), we find removal patterns R r = G p \ S p,r and use them to search for improvements for an p. Table shows patterns in a simplified notation where R r = { s : s R r that onl R is the most likel an optimal pattern for all p . r R r { 0,,,,, 0 } { 0, 0, 0,, 0,,, 0, 0 } { 0,,,,, 0,,,,,, } { 0,,,, 0,,,, 0,, 0, 0,,, } { 0, 0, 0, 0,,,, 0,,,, 0,, 0, 0,,, } Table : Eperimentall determined patterns R r }. We notice Using the initial arrangement S p,r we tr to perform the translation of each sphere using the limited set of translation vectors denoted b T. This algorithm can be described as follows: the electronic journal of combinatorics () (0), #P. For a given initial set S n S p,r repeat until D(S n ) D p : For each s i S n do: Randoml choose t T, Let v = {s i + kt : k [, ]}, Let v i = {q v : (S n \ {s i }) {q} F, D(S n {q}) = D(S n )}, Find endpoints a and b of the largest line segment ab v i such that s i ab, New position of s i is given as s i (a + b)/. After we tr to move all points from S n, we sa that we completed one iteration. Because of the ver limited space to which we translate the spheres, and in order to minimie the number of required iterations, we usuall set T = {0, } \ { 0, 0, 0 }. We also tested performances when T takes different values such as {, 0, }, more or less reduced sets, but the improvements gained were alwas slightl worse. In practice, D(S p,r ) is slightl larger than D p while coordinates are given with finite precision. The described procedure stops when the first improvement is detected, but if we continue the search, we can improve the packing even more. It is also important to set the precision above the epected value of D p D(S p,r ), otherwise the improvement cannot be registered. Choosing the higher precision enables us to reach an improvement with less iterations, but onl up to a certain level. We usuall set the precision. times higher than the epected improvement. If the precision is too high, the search can be ver slow, thus often we have to guess the range of possible improvements using a lower precision at first, and increase it if an improvement cannot be reached. This approach is different from the procedures used for sphere packing in the past, as in [,,, ], while we focus onl on tin changes/improvements in the high densit structure. We cannot consider this approach as a good general packing method for r p. Its main weakness is that, because of the small available space where spheres can be moved, random perturbations are hard to implement, or at least we did not find an good method to do it. Still, this method allows us to find improved packings in less than one second for some well eamined cases even when high precision is not required, as for eample n = 9, 9 or 0. The improvements attained are shown in the Tables and. Table lists values obtained for I p with p. Because of the slow computation times for p, we ran a search with approimatel 000 iterations when improvement gains started to slow down. Table lists other improved packings for r, r p and p using R r patterns described in Table. The results are listed as the best known values performed after a large number of iterations and multiple runs of the search procedure. the electronic journal of combinatorics () (0), #P. n p I p Table : Improved values of I p n p r d n d p n p r d n d p Table : Improvements reached using patterns R r the electronic journal of combinatorics () (0), #P. 8 References [] D. Boll, J. Donovan, R. L. Graham, B. D. Lubachevsk. Improving Dense Packings of Equal Disks in a Square. Electron. J. Combin. : #R, 000. [] Th. Gensane. Dense Packings of Equal Spheres in a Cube. Electron. J. Combin. : #R, 00 [] M. Goldberg. On the densest packing of equal spheres in a cube. Mathematics Magaine : 99 08, 9 [] A. Grosso, A. R. M. J. U. Jamali, M. Locatelli, F. Schoen. Solving the problem of packing equal and unequal circles in a circular container. J. Glob. Optim. : 8, 00 [] W. Huang, L. Yu. Serial Smmetrical Relocation Algorithm for the Equal Sphere Packing Problem. arxiv:0.9, 0 [] A. Joos. On the packing of fourteen congruent Spheres in a cube. Geom. Dedicata 0: 9 80, 009 [] M. Locatelli, M. Maischberger, F. Schoen. Differential evolution methods based on local searches. Comput. Oper. Res. : 9 80, 0 [8] M. C. Markot, T. Csendes. A New Verified Optimiation Technique for the Packing Circles in a Unit Square Problems. SIAM J. Optim. : 9 9, 00 [9] M. C. Markot, T. Csendes. A Reliable Area Reduction Technique for Solving Circle Packing Problems. Computing :, 00 [0] J. Schaer. On the densest packing of spheres in a cube. Canad. Math. Bull. 9: 0,, 80, 9 [] J. Schaer. On the densest packing of ten congruent spheres in a cube. In: Intuitive Geometr (Seged, 99), Colloq. Math. Soc. Janos Bolai : 0, 99 [] E. Specht. Packings of equal spheres in fied-sied containers with maimum packing densit. 0 [] P. G. Sabo, M. C. Markot, T. Csendes, E. Specht, L. G. Casado, I. Garcia. New Approaches to Circle Packing in a Square. Springer, 00 the electronic journal of combinatorics () (0), #P. 9
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