Repeated Patterns of Dense Packings of Equal Disks in a Square R L Graham, rlg@research.att.com B D Lubachevsky, bdl@research.att.com AT&T Bell Laboratories Murray Hill, New Jersey 07974, USA Submitted: January 17, 1996; Accepted: April 20, 1996 ABSTRACT We examine sequences of dense packings of n congruent non-overlapping disks inside a square which follow specific patterns as n increases along certain values, n = n(1), n(2), n(k), ă Extending and improving previous work of Nurmela and Osterg˚ [NO] where previous patard 2 terns for n = n(k) of the form k , k − 1, k − 3, k(k + 1), and 4k2 + k were observed, we identify new patterns for n = k2 − and n = k2 + k/2 We also find denser packings than those in [NO] for n =21, 28, 34, 40, 43, 44, 45, and 47 In addition, we produce what we conjecture to be optimal packings for n =51, 52, 54, 55, 56, 60, and 61 Finally, for each identified sequence n(1), n(2), n(k), which corresponds to some specific repeated pattern, we identify a threshold index k0 , for which the packing appears to be optimal for k ≤ k0 , but for which the packing is not optimal (or does not exist) for k > k0 Introduction In a previous paper [GL1], the authors observed the unexpected occurrence of repeating “patterns” of dense (and presumably optimal) packings of n equal non-overlapping disks inside an equilateral triangle (see Fig 1.1 for and example) It is natural to investigate this phenomenon for other boundary shapes In particular, this was done by the authors [LG1] for the case of n disks in a circle However, in contrast to the case of the equilateral triangle where the patterns appear to persist for arbitrarily large values of n, for the circle the identified packing patterns cease to be optimal as the number of disks exceeds a certain threshold In this note we describe the situation for the square In a recent paper, Nurmela and ă Osterg [NO] present various conjectured optimal packings of n equal disks in a square for ard up to 50 disks They also point out certain patterns that occur there By using a packing procedure different from theirs, we improve on their best packings for n = 21, 28, 34, 40, 43, 44, 45, and 47 We conjecture that these new packings are optimal, as are the new packings we give for n = 51, 52, 54, 55, 56, 60, and 61 We confirm all the repeated patterns mentioned in [NO], specifically, for n = k2 , k2 − 1, k2 −3, and k(k+1), and we identify two new patterns, namely, for n = k2 −2 and n = k2 + k/2 The latter pattern incorporates the packings of n = 4k2 + k disks as identified in [NO] It was found in [NO] that the obvious “square” pattern of packings of n = k2 disks becomes the electronic journal of combinatorics (1996), #R16 Figure 1.1: The conjectured densest packing of n = 256 disks inside an equilateral triangle, a member of the series n = np(k) = ∆((k + 1)p − 1) + (2p + 1)∆(k) for p = and k = 3, where ∆(m) = m(m + 1)/2 The n − p + = 252 shaded disks can not move (they are “solid”), the p−1 = non-shaded disks are free to move within their local confines (they are “rattlers”) The densest packings of n disks for all checked values of the form n = np (k), p = 1, 2, , k = 1, 2, , have this pattern consisting of one triangle of side (k + 1)p − and 2p + triangles of side k with p − rattlers that are “falling off” the larger triangle non-optimal for n > n0 = 36 This was done by presenting a configuration of k2 = 49 disks with the diameters larger than m = 1/(k − 1) The latter m is disk diameter in the packing that obeys the pattern The standard unit of measure used in most papers on the subject is the side of the smallest square that contains the centers of disks We repeat the same procedure for the other patterns identified both in [NO] and in the present paper Namely, for each pattern we state the rule of its formation which allows us to compute the corresponding value of m = m(n) Then we pinpoint the n0 that belongs to the series and such that the packing of n0 disks constructed according to the rule is (presumably) optimal, but for which m(n1 ) for the next value n1 > n0 in the series when the packing is also constructed according to the rule is worse than a certain challenger disk configuration (which may or may not be a solid packing) In this manner, we confirm observation in [NO] that the best packings of n = k2 − disks loose their characteristic pattern for n ≥ n1 = 48 disks We also found that although it was not stated in [NO], the packing of 47 disks presented there (as well as our better packing of 47 disks) challenges the series k2 − Thus, the pattern of the series n = k − becomes non-optimal for n > n0 = 34, and that of the series n = k2 − for n > n0 = 35 We also found challenger disk configurations or packings for other patterns for values of n > 50 which were not identified in [NO] Namely: n0 = 56 for the series k(k + 1) (with the challenger the electronic journal of combinatorics (1996), #R16 n1 = 72), and n0 = 61 for the series k2 − (with the challenger n1 = 78) The situation for the series n = n(k) = k2 + k/2 (n = 5, 10, 18, 27, 39, 52, 68, ) is more complex: the pattern exists for ≤ n(k) ≤ 52 as a solid packing and is probably optimal for these n(k) except the case n = n(3) = 10 which is the subject of several publications ([G], [Sch], [Schl], [Val]) For n = n(8) = 68 the configuration constructed according to the pattern rule has a slight disk overlap, i.e., it does not exist as a disk packing, and the overlap persists for all n = n(k) > 68 In the geometric packing problem, progress in proving lags that of conjecturing Thus, we should warn the reader that almost all our statements are conjectures; they are based on computer experimentations with the so-called “billiards” simulation algorithm [L], [LS] In all series except k2 − 3, the construction rule we found for generating a pattern for a given n is a finite procedure and m(n) can be expressed as the root of a well-defined polynomial The existence of the packing of a given pattern for a fixed n, even if not the pattern’s optimality (when appropriate), should be considered proven However, for the series k2 − we have to resort on an infinite simulation procedure [L], [LS] to construct a pattern for an arbitrary n and to compute m(n) Hence even the pattern’s existence as a solid packing is a conjecture here We point out that some of the proposed methods in the literature attempt to prove a packing optimal or, at the least, prove that a packing with particular parameters exists Usually to fulfill this task, the packing must be actually presented, even if only as a conjecture Thus, we try to present our conjectured packings in a verifiable and reproducible form; we provide 14 decimal digits of accuracy for its parameter m and clearly identify the connectivity pattern (touching disk-disk and disk-wall pairs) Some previous papers provided disk coordinates in the presented packings The interested reader can contact the authors directly for the coordinates (since that would otherwise take up too much space in the paper) Packings The parameter m supplied with each packing is the ratio of the disk diameter to the side length of the smallest square that contains the disk centers Bonds or contact points mark disk-disk or disk-boundary contact In the packing diagrams, bonds are indicated by black dots Most of the packings presented are conjectures This means that a proof is required not only for their optimality, if any, but even for their existence Thus, a bond implies a conjecture that the corresponding distance is zero while the absence of a bond implies a conjecture that the distance is strictly positive We placed or did not place a bond between two disks or a disk and a boundary based on the numerical evidence: the bond was placed when the corresponding distance was less than 10−12 of the disk diameter Such a choice of a threshold is supported by the existence of a well-formed gap between a bond and a no-bond situation: In all cases when the bond is not present between apparently touching surfaces, the computed distance is at least 10−7 of the disk diameter, and, except for the packing of 47 disks in Fig.2.6, it is We were unable to reproduce the best packing of 21 disks for which [MFP] provides the diameter m = 0.27181675 but no other data, e.g no contact diagram Our best packing of 21 disks has a smaller diameter (see Fig 2.1) the electronic journal of combinatorics (1996), #R16 at least 10−5 The existence of this gap also testifies to that in all the packings the double precision resolution we employed for the computations sufficed All solid disks, i.e., those that can not move, are shaded in the packing diagrams; the non-shaded disks are rattlers— they are free to move within their confines Different shadings of disks in some packings and a unique numerical label for each disk on a diagram are provided to facilitate the discussion, These are not part of the packings 5 the electronic journal of combinatorics (1996), #R16 17 18 10 14 16 18 11 22 15 13 21 10 15 21 16 11 17 13 14 20 12 12 19 21 disks 39 bonds 11 m = 0.26795840155072 13 16 20 22 disks m = 0.27181225535931 19 43 bonds 19 18 12 22 20 19 11 23 15 17 16 23 24 14 21 15 10 13 18 22 20 12 14 10 17 21 23 disks 24 disks (2,2) m = 0.25881904510252 13 56 bonds m = 0.25433309503025 13 16 19 18 56 bonds 23 19 22 20 17 11 11 17 12 24 16 21 23 14 12 24 15 10 21 18 20 14 15 10 24 disks (2,3) m = 0.25433309503025 22 24 disks (3,3) 56 bonds m = 0.25433309503025 Figure 2.1: The densest packings found of 21 to 24 disks 56 bonds the electronic journal of combinatorics (1996), #R16 20 18 25 22 12 25 22 13 24 15 12 17 11 23 16 11 17 21 21 24 26 18 20 23 16 19 13 10 15 14 19 14 25 disks 26 disks m = 0.25000000000000 12 60 bonds 19 17 16 18 26 27 14 15 19 10 20 28 55 bonds 12 26 10 27 17 29 disks m = 0.22688290074421 16 25 28 14 20 22 19 13 27 17 24 29 15 57 bonds 18 30 12 11 16 13 29 18 17 19 21 23 14 18 24 m = 0.23053549364267 25 22 15 22 28 disks m = 0.23584952830142 20 13 24 27 disks 12 14 23 10 11 20 23 26 11 21 16 21 56 bonds 28 2 24 25 27 22 m = 0.23873475724122 15 25 13 10 23 11 10 26 21 30 disks 65 bonds m = 0.22450296453109 Figure 2.2: The densest packings found of 25 to 30 disks 65 bonds the electronic journal of combinatorics (1996), #R16 17 24 20 15 18 11 26 10 23 15 27 14 29 22 10 26 24 12 31 13 31 22 32 19 12 21 23 29 25 11 16 30 18 25 20 19 13 28 28 27 30 21 14 16 17 31 disks 32 disks m = 0.21754729161912 55 bonds m = 0.21308235294443 61 bonds 28 22 32 21 34 31 20 12 14 15 23 18 21 11 13 29 10 13 30 28 22 14 15 19 33 18 23 20 12 29 31 32 30 16 33 27 24 17 25 26 25 16 17 26 10 24 19 27 11 33 disks 34 disks (2,4;2,4) m = 0.21132838414326 65 bonds m = 0.20560464675957 80 bonds 21 25 16 26 28 13 23 17 22 14 15 30 33 34 10 18 27 20 12 24 29 19 34 21 31 11 25 32 30 16 23 17 26 20 31 14 15 28 22 13 33 10 18 24 29 27 19 12 11 32 34 disks (2,5;2,4) 34 disks (3,5;2,4) m = 0.20560464675957 80 bonds m = 0.20560464675957 80 bonds Figure 2.3: The densest packings found of 31 to 34 disks the electronic journal of combinatorics (1996), #R16 33 34 14 15 21 30 23 17 24 26 10 18 29 19 20 29 31 11 22 32 27 19 34 disks (2,5;2,5) m= 0.20560464675957 29 16 12 28 17 25 22 23 19 14 27 15 18 26 10 28 27 30 14 33 10 17 35 31 21 13 23 32 34 18 11 25 15 24 20 26 35 disks (2,2) 80 bonds 30 12 16 12 16 22 13 25 28 33 m= 0.20276360086323 80 bonds 29 17 11 35 31 21 13 32 34 22 23 24 20 19 33 11 35 27 25 16 30 12 28 14 15 10 31 34 24 26 13 32 18 21 20 35 disks (2,3) 35 disks (3,3) m= 0.20276360086323 80 bonds m= 0.20276360086323 80 bonds 32 20 23 33 12 13 14 25 13 6 34 11 11 19 22 31 34 36 35 10 21 18 17 35 31 17 27 29 26 15 16 22 36 disks m= 0.20000000000000 32 24 28 30 30 33 28 27 20 10 18 29 36 21 24 12 23 37 14 19 25 15 16 26 37 disks 84 bonds m= 0.19623810145141 Figure 2.4: The densest packings found of 34 to 37 disks 73 bonds the electronic journal of combinatorics (1996), #R16 26 15 35 17 16 36 30 21 32 32 20 23 19 35 26 16 25 28 16 34 18 21 23 37 30 20 40 35 32 19 12 22 25 12 23 38 11 13 29 14 34 29 16 28 26 40 18 13 20 15 33 22 15 17 37 38 31 27 30 24 36 14 39 19 11 m= 0.19436506316151 80 bonds 11 10 36 21 14 36 32 27 31 39 disks 77 bonds 17 31 29 10 30 38 disks m= 0.19534230412691 18 34 24 37 12 24 22 28 20 28 33 34 26 39 29 22 27 12 38 25 35 13 33 23 19 15 11 27 38 37 17 14 10 31 13 18 25 33 41 24 39 21 10 40 disks 41 disks m= 0.18817552201832 85 bonds m= 0.18609951184812 100 bonds 35 34 23 24 21 28 31 14 27 32 10 26 33 20 23 12 16 34 29 22 30 33 26 10 17 11 14 39 42 disks m= 0.18427707211710 42 19 39 24 18 40 19 13 31 29 20 15 25 27 25 15 38 43 30 37 38 22 21 37 42 18 16 35 13 40 12 41 36 17 11 32 36 41 28 43 disks 90 bonds m= 0.18019113545743 Figure 2.5: The densest packings found of 38 to 43 disks 85 bonds 10 the electronic journal of combinatorics (1996), #R16 13 14 38 31 11 41 12 15 43 10 44 16 33 39 25 23 44 28 24 38 29 22 42 34 10 46 27 36 30 44 21 17 32 16 39 19 18 37 20 24 20 31 26 38 18 32 14 41 22 34 14 25 35 31 19 44 16 40 21 45 18 11 10 26 43 46 29 32 24 37 38 12 42 41 11 37 42 36 33 17 24 30 27 15 17 42 25 45 m = 0.17571631417559 94 bonds 35 23 45 disks 12 34 30 41 36 45 82 bonds 33 13 39 11 22 44 disks 28 29 35 31 12 25 26 16 27 23 m = 0.17863924567120 40 43 33 26 14 10 17 35 19 20 28 13 27 29 21 15 37 18 32 21 40 30 22 36 40 34 19 20 15 13 28 23 47 43 39 46 disks 47 disks m = 0.17445936087241 91 bonds m = 0.17126830721141 94 bonds 10 45 48 20 43 27 14 37 34 16 30 21 22 36 12 47 49 31 28 48 47 37 20 16 42 34 10 22 21 41 23 14 33 25 44 17 13 24 40 26 18 15 11 48 disks m = 0.16938210954876 19 19 35 45 32 32 46 43 29 41 25 27 35 44 39 26 38 38 40 17 29 12 33 24 13 18 46 11 42 15 23 39 30 36 31 28 49 disks 101 bonds m = 0.16738607686833 Figure 2.6: The densest packings found of 44 to 49 disks 120 bonds 11 the electronic journal of combinatorics (1996), #R16 25 49 29 11 32 14 13 30 27 38 20 16 23 37 33 31 42 22 44 18 10 12 14 19 52 23 30 36 18 41 20 45 37 10 13 11 50 29 35 33 15 21 11 43 42 25 44 39 10 31 19 33 30 35 23 51 46 27 52 26 37 41 15 13 54 50 24 40 29 16 14 28 18 12 20 48 40 34 17 36 34 45 22 47 21 43 26 33 42 49 38 39 31 25 48 28 24 22 17 12 23 12 49 38 16 44 25 11 m = 0.16561837431260 99 bonds 32 51 27 104 bonds 40 51 disks m = 0.16645462588286 47 15 51 10 26 50 disks 46 18 42 14 34 15 24 45 16 45 27 48 13 48 41 43 32 29 31 20 35 50 43 21 36 19 47 38 36 26 46 40 28 49 44 46 39 28 17 17 30 34 35 24 22 47 41 39 21 37 19 50 53 32 52 disks 54 disks m = 0.16538623796964 105 bonds m = 0.15913951630719 115 bonds 37 16 39 13 10 11 45 44 34 27 28 36 23 17 24 29 20 52 28 30 22 32 32 20 30 18 11 55 disks m = 0.15755574752972 113 bonds 19 13 40 39 25 55 15 42 41 22 47 23 54 12 27 29 26 51 37 50 43 38 21 45 34 31 16 46 38 10 42 53 53 36 25 15 33 52 48 12 33 55 56 35 49 31 35 54 14 21 50 47 48 40 19 49 26 41 51 43 46 17 18 14 24 44 56 disks m = 0.15615650046215 119 bonds Figure 2.7: The densest packings found of 50 to 52, and 54 to 56 disks 12 the electronic journal of combinatorics (1996), #R16 46 18 23 31 41 44 57 24 56 19 15 47 55 59 22 54 13 51 46 10 47 40 32 24 16 48 12 33 25 17 34 26 27 18 40 12 27 24 52 48 53 17 28 23 49 34 38 45 14 60 55 35 16 61 20 22 31 21 36 41 44 56 25 m = 0.14854412669518 62 53 45 54 46 38 39 29 30 31 21 13 61 37 20 19 11 52 28 42 62 52 25 36 15 47 129 bonds 44 35 29 59 50 61 disks 60 51 43 26 58 18 43 33 54 17 42 59 50 42 10 58 49 41 57 40 60 m = 0.14950565404867 56 11 37 60 disks 55 57 39 21 35 52 48 19 58 26 37 13 32 34 28 33 38 53 30 39 25 29 36 51 12 20 43 50 10 30 45 11 16 14 27 49 32 22 14 23 15 71 23 28 61 54 60 21 22 31 58 20 56 65 50 16 42 47 59 72 63 40 67 19 30 34 38 68 27 69 46 48 45 14 44 36 32 53 55 51 41 66 10 15 49 18 12 11 17 70 37 39 24 13 29 35 57 121 bonds 43 26 33 64 62 disks (3,7;2,5) 72 disks m = 0.14569394327531 140 bonds m = 0.13549029317569 169 bonds not the best packing not the best packing 16 15 14 13 12 11 10 24 23 22 21 20 19 18 17 32 31 30 29 28 27 26 25 40 39 38 37 36 35 34 33 48 47 46 45 44 43 42 41 56 55 54 53 52 51 50 49 64 63 62 61 60 59 58 57 51 72 71 70 69 68 57 59 14 61 63 21 65 67 28 69 35 73 75 71 42 77 78 49 76 72 74 13 20 27 34 41 48 68 70 12 19 26 33 40 47 64 66 11 18 25 32 39 46 60 62 67 10 17 24 31 38 45 56 58 66 16 23 30 37 44 52 54 65 72 disks m = 0.13541666666667 152 bonds not the best packing 53 55 15 22 29 36 43 50 78 disks m = 0.12933240481510 155 bonds not the best packing Figure 2.8: The densest packings found of 60 and 61 disks and inferior packings of 62, 72, and 78 disks 13 the electronic journal of combinatorics (1996), #R16 10 4 disks 10 disks m = 0.70710678118655 12 bonds m = 0.41666666666667 21 bonds not the best packing 2 9 12 10 11 10 13 10 disks 13 disks m = 0.42127954398390 21 bonds m = 0.36609600769643 25 bonds 59 14 13 40 24 47 10 74 36 23 18 17 50 62 55 11 12 77 52 38 45 67 22 48 21 61 27 14 64 11 76 12 51 31 18 44 17 49 69 60 39 56 34 43 53 30 63 29 71 73 78 33 32 35 41 70 16 75 65 54 72 66 42 15 57 13 16 15 37 46 10 25 19 58 68 26 20 28 18 disks 78 disks m = 0.30046260628867 38 bonds m = 0.13046077259640 not a rigid packing Figure 2.9: The densest packings of 5, 10, 13, and 18 disks, an inferior packing of 10 disks, and a challenger configuration of 78 disks the electronic journal of combinatorics (1996), #R16 14 Series of packings with similar patterns n = k2 The packings of this series have an obvious square pattern with disk diameter m = 1/(k − 1) The pattern is optimal for n = 1, 4, 9, and 16, and probably 25 and 36 For k1 = this yields value of m = 1/6 which is smaller than the experimental diameter of the challenger packing of 49 disks in Fig 2.6 n = k2 − The pattern can be viewed as a square of (k − 1)2 disks arranged in straight rows and columns into which one row and one column of “shifted” disks are inserted A “shifted” row or column each has k − disks In the diagrams for k = (n = 24, Fig 2.1) and k = (n = 35, Fig 2.4), the “straight” disks are shaded more heavily than the “shifted” ones Several equivalent packings are obtained by inserting the shifted row and column in different places among the straight ones A pair of insertion position (i, j), ≤ i, j ≤ k − 1, identifies the packing, e.g., in packing of 24 disks (2,3) (Fig 2.1), the shifted row is the second and the shifted column is the third (counting from the top left corner) Among the (k − 2)2 packings thus obtained, there are many congruent pairs The pattern is fully developed at n = (one packing) and remains optimal for n = 15 (one packing), n = 24 (three equivalent packings), and n = n0 = 35 (three equivalent packings) For n1 = 48 the pattern looses its optimality, which can be shown as follows The angle at the center of disk 13 in packing (2,2) of 24 disks (see Fig 2.1 ) in the triangle formed by centers of disks 13, 19 and is 15o with √ cos 15o = + Similar angles are the same in all the packings of the series which easily √ implies m = 1/(k − + + 3) For k1 = this yields value of m = 0.168581424 which is smaller than the experimental diameter of the challenger packing of 48 disks in Fig 2.6 n = k2 − This pattern is similar to the pattern of series k2 − 1, only here there are two shifted rows and two shifted columns The (non-optimal) packing of 62 disks depicted in Fig 2.8, shows this pattern for k = There are (k − 3)(k − 4)/2 possible ways to insert a pair of shifted rows (i1 , i2 ) because of the restrictions < i1 < i2 < k and similarly for the columns (j1 , j2 ) Hence, there are ((k − 3)(k − 4)/2)2 possible different index sets (i1 , i2 ; j1 , j2 ) for each of which we can construct an equivalent packing Many of these are congruent The pattern is fully developed at k = (n = 23, one packing) and remains optimal for only one more √ value k = k0 = (n0 = 34, four equivalent packings.) Here m = 1/(k − + 2 + 3) For k = k1 = this yields value m = 0.1705406887 which is smaller than that of the challenger packing of n1 = 47 disks in Fig 2.6 n = k2 − This pattern is represented by conjectured dense packings of 22, 33, 46, and 61 disks (Figs 2.1, 2.3, 2.6, 2.8) A feature of the pattern is a (shaded more heavily on the pictures) densely packed “straight” square of (k − 3)3 disks in the bottom left corner It follows that the pattern is not optimal for k ≥ 10 (n ≥ 97) because the “straight” square itself isn’t However, even for a smaller n = n1 = 78 we were able to get a disk configuration the electronic journal of combinatorics (1996), #R16 15 that challenges the pattern as presented in Fig 2.8, although this configuration is a not fully formed solid disk packing (see Fig 2.9) Looking at the (non-optimal) packing of 78 disks (see Fig 2.8), the pattern can be further described as three alternating columns at the right and three alternating rows at the top with one rattler at the top right corner In each of the three additional rows and the three additional columns, most disks are not touching each other The exceptions are pairs 42-49 in the bottom additional row, 73-77 in the top additional row and 76-72 in the right additional column (Fig 2.8) Also there is almost full contact for pairs of disks between adjacent additional rows and columns, except for the pair 42-75 which are not touching between the first and second additional rows These features are identical in all the packings of the pattern Each such packing can be obtained by “tightening” a certain configuration C described as follows: In C, disks 77 and 76 are not touching each other but all disks in additional rows and columns are touching their neighbors (C is not a solid packing.) We take C as an initial condition for the “billiards” packing algorithm [L], [LS] To “tighten” the configuration, the “billiards” algorithm allows the disks to move chaotically in the square without overlaps while their diameter increases at a common rate until no further growth is possible It is remarkable, that for each n = k2 − 3, k = 5, 6, 7, and 9, this chaotic negotiation always converges (independently of the initial velocities) to the described pattern with parameter m identical to double precision in all runs involving the same n Moreover, when instead of the configuration C, we start with a random initial configuration and zero initial diameters of the disks, the same configuration with the same precision results in the runs that achieve the largest m, but only for k = 5, 6, and 8, excluding value k = Also, for k = (n = 13) which seems to be the smallest n for which the pattern may exist, both methods, the one beginning with a zero-diameter random configuration and the one beginning with the configuration C, lead to the same packing which, however, deviates somewhat from the described pattern, see Fig 2.9 n = k(k + 1) The pattern consists of k + alternating columns with k disks each, see, e.g., the packing of 30 disks in Fig 2.2 Following the path of disks 2, 12, 6, 18, 3, 29 in this packing, we come to the expression k cos α for the horizontal side of the square that contains the disk centers, where the angle α = α(k) is that at the vertex in the triangle 12, 2, The vertical side of the square is k − + sin α, where the latter expression results from the path 12, 2, 5, 30, 24, Equating these two expressions for the side we determine from the resulting √ equation: cos α(k) = (k2 − k + 2k)/(k + 1) and m = (k cos α(k))−1 Looking again at the packing of 30 disks in Fig 2.2, circles and are separated by a √ non-negative distance because α(k) ≤ 30o or cos α(k) ≥ 3/2 for k = The latter inequality holds and the pattern exists as a solid packing of non overlapping disks only for k ≥ 4, i.e., for n = 20, 30, 42, 56, 72, We show that the packing of the pattern is not optimal for n ≥ n1 = 72 as follows We present another regular pattern of packings of k(k + 1) disks; for k = this alternative pattern is depicted in the second row and second column in Fig 2.8 (with one rattler) The alternative the electronic journal of combinatorics (1996), #R16 16 pattern exists for all values of n for which the main pattern exists The diameter m = m(k) ¯ ¯ √ of a disk in the alternative pattern is given by m(k) = 1/((k + 1/2) cos β(k) + ( 3/2) sin β(k)) ¯ where (in the example in Fig 2.8) β(k) for k = is the angle at disk 56 in the triangle 67, 56, 42 As before, we equate the horizontal and vertical sides of the square: k cos β+cos(β+π/3) = (k − 1) sin(β + π/3) + sin β From this equation we easily determine β = β(k) and then we find that m > m for all n ≥ n1 = 72 (but m < m for n = 20, 30, 42, and 56) Note that the ¯ ¯ alternative packing, although it is better than the packing of the pattern for n = 72 disks, is not optimal For example, for 72 disks we found experimentally a disk configuration with an irregular structure (not shown) that is better than both the main or the alternative pattern n = k2 + k/2 The pattern, as exemplified by cases k = (n = 27, Fig 2.2) and k = (n = 39, Fig 2.5), consists of k + alternating columns, odd columns having k disks each and even columns having k − disks each As before, with α denoting the angle, say, at disk 15 in the triangle 7, 15, 10 in the packing of 27 disks in Fig 2.2, we compute the side-length of the square obtained in two different ways: k cos α = 2(k − 1) sin α From this equation we easily determine cos α(k) and then m = 1/(k cos α(k)) The pattern exists when, as in the example of 27 disks (Fig 2.2), disks 12 and 13 are separated by a non-negative distance This occurs when sin α(k) ≥ 1/2 Thus, the pattern exists only for k = 2, 3, 4, 5, 6, and (n = 5, 10, 18, 27, 39, and 52) and those packings (shown in Figs 2.9, 2.2, 2.5, and 2.8) are, indeed, optimal, except the case of n = 10: proved [GMPW] for n = 5, 10, 18 and conjectured for n = 27, 39, and 52 For k ≥ the “ideal” pattern yields overlaps in pairs of disks The overlap increases with k For k = (n = 68) the overlap is less than 1% of the disk diameter so it is naturally to expect that the optimal packing will be a small deviation from the main pattern Experiments, indeed show that best configurations are of this sort Unfortunately, none of them is a solid packing, because it becomes very difficult to find exactly in which pairs the disks are touching, and not merely just very close to each other References [CFG] H T Croft, K J Falconer and R K Guy, Unsolved Problems in Geometry, Springer Verlag, Berlin, 1991, 107–111 [GMPW] C de Groot, M Monagan, R Peikert, and D Wurtz, Packing circles in a square: a review and new results, in System Modeling and Optimization (Proc 15th IFIP Conf Zurich 1991), 45–54 [FG] J H Folkman and R L Graham, A packing inequality for compact convex subsets of the plane, Canad Math Bull 12 (1969), 745–752 the electronic journal of combinatorics (1996), #R16 17 [GL1] R L Graham and B D Lubachevsky, Dense packings of equal disks in an equilateral triangle: from 22 to 34 and beyond, The Electronic Journ of Combinatorics (1995), #A1 [GLNO] ă R L Graham, B D Lubachevsky, K J Nurmela, and P R J Osterg˚ Packing ard, congruent circles in a circle by stochastic optimization methods, (In preparation.) [G] M Goldberg, The packing of equal circles in a square, Math Mag 43 (1970), 24–30 [L] B D Lubachevsky, How to simulate billiards and similar systems, J Computational Physics 94 (1991), 255–283 [LG1] B D Lubachevsky and R L Graham, Dense packings of 3k(k+1)+1 equal disks in a circle for k = 1,2,3,4, and (Submitted for publication.) [LS] B D Lubachevsky and F H Stillinger, Geometric properties of random disk packings, J Statistical Physics 60 (1990), 561–583 [MFP] C D Maranas, C A Floudas, P M Pardalos, New results in the packing of equal circles in a square, Discrete Mathematics 142 (1995), 287293 [NO] ă K J Nurmela and P R J Osterg˚ Packing up to 50 equal circles in a square, ard, Discrete & Computational Geometry, submitted [O] N Oler, A finite packing problem, Canad Math Bull (1961), 153–155 [Sch] J Schaer, On the packing of ten equal circles in a square, Math Mag 44 (1971), 139–140 [Schl] K Schlăter, Kreispackung in Quadraten, Elem Math 34 (1979), 1214 u [Val] G Valette, A better packing of ten circles in a square, Discrete Math 76 (1989), 57–59  ... not touching between the first and second additional rows These features are identical in all the packings of the pattern Each such packing can be obtained by “tightening” a certain configuration... non-shaded disks are rattlers— they are free to move within their confines Different shadings of disks in some packings and a unique numerical label for each disk on a diagram are provided to facilitate... described as follows: In C, disks 77 and 76 are not touching each other but all disks in additional rows and columns are touching their neighbors (C is not a solid packing.) We take C as an initial