-- Packing unit circles in squares: new results
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From: David W. Cantrell on 20 May 2010 13:16 Very recently, Eckard Specht extended Packomania <http://hydra.nat.uni-magdeburg.de/packing/csq/csq.html> by giving 584 "strict regular lattice packings" for 500 < N <= 5000, and commented that "many of them can be improved." Here we discuss a specific way that several of them can be improved trivially, using packings for smaller N already shown at Packomania. ------------------------------------------------------------------- Terminological note: Earlier in this thread, being unfamiliar with the pertinent literature then, I had coined the term "compression" packing to describe what Eckard called a "strict regular lattice" packing and what is called a "grid" packing in _New Approaches to Circle Packing in a Square_. Now preferring the latter term, such packings will henceforth be called "grid packings" in this thread. -------------------------------------------------------------------- As discussed earlier in this thread, some grid packings are not optimal because clumping certain groups of circles into hexagonal lattice substructures can give a more efficient packing. As an example, please take a glance at the packing <http://hydra.nat.uni-magdeburg.de/packing/csq/csq407.html> for N = 407. That presumably optimal packing, with hexagonal lattice clumps, is very closely approximated by the grid packing having 36/21 as the associated fractional underestimate for sqrt(3). Therefore, if there is a larger grid packing having an associated fractional underestimate p/q for sqrt(3) such that p/q can be reduced (by dividing numerator and denominator by k) to 36/21, that larger grid packing is also not optimal and can be improved merely by combining k^2 copies of the packing for N = 407: The grid packing for N' = 1570 has side length 74.55742... and 72/42 as its associated fractional underestimate for sqrt(3). Dividing numerator and denominator of that fraction by k = 2 gives 36/21 [which, of course, is still not in "lowest terms" -- but that's a different matter...] Therefore, combining k^2 = 4 copies of the packing for N = 407, we get an improved packing for N' = 1570, with side length s' = 74.55706...: <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/circs1570sq.gif> (The two dashed lines divide the figure into quadrants. For each quadrant, circles having their centers in it or on its boundary constitute one copy of the packing for N = 407.) [In words: Take the packing for N = 407. Draw an axis through the centers of the circles touching, say, the right side of the square (and discard the semicircles between the axis and the right side). Adjoin to that figure a copy obtained by flipping about the axis; the composite rectangular figure is now symmetric about the axis. Draw an axis through the centers of the circles touching, say, the top side of the rectangle (and discard the semicircles between that axis and the top side). Adjoin to that rectangular figure a copy obtained by flipping about that axis. The resulting figure is the desired packing in a square for N' = 1570.] Another example: The grid packing for N' = 2288 has side length 89.9951... and 87/51 as its associated fractional underestimate for sqrt(3). Dividing numerator and denominator of that fraction by k = 3 gives 29/17. Therefore, combining k^2 = 9 copies of the packing for N = 270, we get an improved packing for N' = 2288, with s' = 89.9937...: <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/circs2288sq.gif> (The dashed lines divide the figure into nine regions. For each region, circles having their centers in it or on its boundary constitute one copy of the packing for N = 270.) The following (incomplete) table gives N' for a grid packing which can be improved by using k^2 copies of a known non-grid packing for N. The symbol # at the end of a row indicates that the non-grid packing for N is irregular (see below). N' k N 513 2 137 581 2 154 # 644 3 80 682 2 180 # 791 2 208 875 4 63 998 2 261 # 1033 2 270 1129 2 295 # 1268 2 330 # 1353 5 63 1415 2 368 # 1570 2 407 1748 3 208 1936 6 63 1985 4 137 2288 3 270 2503 6 80 2622 7 63 3081 4 208 3392 7 80 3413 8 63 3488 3 407 4037 4 270 4307 9 63 4417 8 80 4788 5 208 Knowing the side length s of the packing for N, of course it's trivial to calculate the side length s' of the improved packing for N': s' = k (s - 2) + 2 It seems unnecessary to show other figures of those improved packings since their method of construction has been explained. However, when a packing for N is irregular (that is, it fails to exhibit regularity as seen, for example, in the packings for 407 and 270), caution is needed. For example, consider the packing for N = 180 <http://hydra.nat.uni-magdeburg.de/packing/csq/csq180.html> from which we wish to get a packing for N' = 682. If we unthinkingly follow the described procedure, flipping about an axis one unit from the right side and then flipping about an axis one unit from the bottom side, the resulting figure has 681 circles, rather than 682. Furthermore, that figure isn't even a packing because two circles overlap (namely, circle #9 and its image after the second flip). But if our second flip is about an axis one unit from the top side, all is well -- we have N' = 682 and no circles overlap. An unusual case, not included in the table above: It seemed reasonable to try to improve the grid packing for N' = 760 by combining four copies of my irregular packing for N = 200 <http://hydra.nat.uni-magdeburg.de/packing/csq/csq200.html>, flipping about an axis one unit from the left side and then flipping about an axis one unit from the bottom side. But the resulting packing contains a "bonus" circle: We get a packing for N' = 761 which has a side length substantially smaller than that of the grid packing for 760. David W. Cantrell
From: David W. Cantrell on 21 May 2010 14:20 A bound given in NACPS is first corrected and then trivially (but significantly) improved. -------------------------------------------------- A Correction Theorem 10.1 on pp. 138-139 of NACPS [_New Approaches to Circle Packing in a Square_, P.G. Szabo et al. (Springer, 2007)] is intended to give a lower bound for m, the maximal minimum-pairwise-distance between n >= 2 points in a unit square. It states that m >= max(L1(n), L2(n), L3a(n), L3b(n), L4(n), L5(n), L6(n), L7(n), L8(n), L9(n)) where L9(n) = sqrt(2/(sqrt(3) n)), a default value, L8(n) is obtained from grid packings, and L7(n) through L1(n) are based on knowledge about pattern and structure classes, discussed previously in their chapter 10. For example, they state that L2(n) = 1/(ceiling(sqrt(n + 1)) - 3 + sqrt(2 + sqrt(3))) Significantly, that statement is made _without any explicit restriction_ on n. Taking n = 3, we have L2(3) = 1/( -1 + sqrt(2 + sqrt(3))) = 1.073... That happens to be the largest of their L values when n = 3, and thus they would have L2(3) as their lower bound. But in fact, for n = 3, the proven value for m is m = sqrt(6) - sqrt(2) = 1.035... which is _smaller_ than their supposed lower bound! When I noticed this problem, I naturally assumed that I had made a mistake. Looking earlier in chapter 10, I found that L2 is based on their pattern class PAT(k^2 - 1). On p. 118, they say Assertion 5 In the point arrangement problems defined by PAT(k^2 - 1), k >= 3 ... Ah ha! If we must have k >= 3, then their formula for L2 is valid only for n >= 3^2 - 1 = 8. No wonder we had found a contradiction using n = 3. And so then I thought that they must have _intended_ to state appropriate restrictions on n when defining L1 through L7. When such restrictions are applied, their bound for m, when n = 3, becomes their default L9(3) = sqrt(2/(3 sqrt(3))) = 0.620... But when I then looked at their graph (p. 140, fig. 10.12) of the lower bound, I was surprised to see that it had been produced using exactly the definitions which had been stated incorrectly (due to lack of restrictions) on the previous page! So clearly there was an oversight. But it wasn't just a typographical error since they used the incorrect definition of the bound when generating its graph. In the figure at <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/1owerbnd.gif>, the blue curve is a proven lower bound, obtained using necessary restrictions on n, as it presumably should have been shown in NACPS. Erroneous portions of the graph shown in their fig. 10.12 are indicated in red. Points in black show values of m, proven optimal. The dashed curve is an improved lower bound, which we now discuss. ------------------------------------------------------------------ A (Trivially) Improved Lower Bound If we can pack n unit circles in a square of side length s, then obviously we can also pack any smaller number of unit circles in a square of that same size. Equivalently, if n points can be placed in a unit square such that they are separated from one another by at least distance m, then obviously any smaller number of points can also be placed in a unit square such that they are separated from one another by at least that same distance. This shows, of course, that, in the packing problem, s must be a (not necessarily strictly) increasing function of n, and equivalently, in the point arrangement problem, m must be a (not necessarily strictly) decreasing function of n. Quite surprisingly, this trivial fact was overlooked in deriving the lower bound in NACPS. For n points in the unit square, an improved lower bound for m is therefore max ( max(L1(i), L2(i), L3a(i), L3b(i), L4(i), L5(i), L6(i), L7(i), L8(i), L9(i)) ) i >= n where the L's are defined as in NACPS, except, of course, that necessary restrictions on n are to be applied for L1 through L7. (When a formula given on p. 139 of NACPS for some L function is not applicable, we take it to be 0.) The graph of the improved lower bound is shown as the dashed curve in the figure at the above link. -------------------------------------------------------------------------- The comparison between my previously conjectured lower bound for m and a proven lower bound is still being prepared. David W. Cantrell
From: David W. Cantrell on 22 May 2010 08:06 David W. Cantrell <DWCantrell(a)sigmaxi.net> wrote: > Very recently, Eckard Specht extended Packomania > <http://hydra.nat.uni-magdeburg.de/packing/csq/csq.html> by giving > 584 "strict regular lattice packings" for 500 < N <= 5000, and > commented that "many of them can be improved." Indeed, the great majority of those lattice packings are suboptimal. 1) In my recent post concerning a correction and an improvement to a bound given in NACPS, I made the trivial observation that "If we can pack n unit circles in a square of side length s, then obviously we can also pack any smaller number of unit circles in a square of that same size." With that in mind, consider the first two grid packings shown at Packomania for N > 500: N side length s 507 43.7708541... 513 43.1596604... Obviously, the grid packing for N = 507 is not optimal. We could take the grid packing for N = 513, remove any six unit circles, and thereby get a packing for N = 507 having s = 43.1596604... There are many instances of this phenomenon among the newly added grid packings. In fact, if we remove packings which are thereby known to be suboptimal, based on data at Packomania, only 195 packings remain. The last two of those are N side length s 4992 133.686... 5000 135.254... But if we calculate grid packings a little beyond N = 5000, we find that s = 132.961... for N = 5005. Therefore, we may also eliminate those last two packings, leaving 193 grid packings as possibly being optimal. 2) Using information copied below from my previous post, of those remaining 193, we know that 18 are suboptimal. Thus, 175 (of the original 584) grid packings remain which might be optimal. 3) Of those remaining 175, only 82 satisfy my conjectured upper bound for side length s. Thus, I suspect that, for 500 < N <= 5000, at most 82 grid packings are optimal. 4) Nurmela et al. stated a conjecture which implies that the grid packing for N = 1512 is optimal. In my post on May 2, I broadened their conjecture. If we use power r = -5/3, as I did in that post, to decide which fractional underestimates of sqrt(3) are to be used, it seems reasonable to guess that, in the range 500 < N <= 5000, at least three grid packings are optimal: N = 621, 1512 and 3978. (My next two picks for grid packings in that range which might be optimal would be N = 1235 and 3520.) In conclusion, for 500 < N <= 5000, my guess is that the number of grid packings which are optimal lies somewhere between 3 and 82, and is probably much closer to 3 than to 82. (Please see below the table for an additional comment.) > Here we discuss a > specific way that several of them can be improved trivially, using > packings for smaller N already shown at Packomania. > > ------------------------------------------------------------------- > Terminological note: Earlier in this thread, being unfamiliar with > the pertinent literature then, I had coined the term "compression" > packing to describe what Eckard called a "strict regular lattice" > packing and what is called a "grid" packing in _New Approaches to > Circle Packing in a Square_. Now preferring the latter term, such > packings will henceforth be called "grid packings" in this thread. > -------------------------------------------------------------------- > > As discussed earlier in this thread, some grid packings are not > optimal because clumping certain groups of circles into hexagonal > lattice substructures can give a more efficient packing. As an > example, please take a glance at the packing > <http://hydra.nat.uni-magdeburg.de/packing/csq/csq407.html> for > N = 407. That presumably optimal packing, with hexagonal lattice clumps, > is very closely approximated by the grid packing having 36/21 as the > associated fractional underestimate for sqrt(3). Therefore, if there is a > larger grid packing having an associated fractional underestimate p/q for > sqrt(3) such that p/q can be reduced (by dividing numerator and > denominator by k) to 36/21, that larger grid packing is also not optimal > and can be improved merely by combining k^2 copies of the packing for > N = 407: > > The grid packing for N' = 1570 has side length 74.55742... and 72/42 as > its associated fractional underestimate for sqrt(3). Dividing numerator > and denominator of that fraction by k = 2 gives 36/21 [which, of course, > is still not in "lowest terms" -- but that's a different matter...] > Therefore, combining k^2 = 4 copies of the packing for N = 407, we get an > improved packing for N' = 1570, with side length s' = 74.55706...: > <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/circs1570sq.gif> > (The two dashed lines divide the figure into quadrants. For each > quadrant, circles having their centers in it or on its boundary > constitute one copy of the packing for N = 407.) > > [In words: Take the packing for N = 407. Draw an axis through the > centers of the circles touching, say, the right side of the square > (and discard the semicircles between the axis and the right side). > Adjoin to that figure a copy obtained by flipping about the axis; the > composite rectangular figure is now symmetric about the axis. Draw an > axis through the centers of the circles touching, say, the top side of > the rectangle (and discard the semicircles between that axis and the > top side). Adjoin to that rectangular figure a copy obtained by > flipping about that axis. The resulting figure is the desired packing > in a square for N' = 1570.] > > Another example: The grid packing for N' = 2288 has side length > 89.9951... and 87/51 as its associated fractional underestimate for > sqrt(3). Dividing numerator and denominator of that fraction by k = 3 > gives 29/17. Therefore, combining k^2 = 9 copies of the packing for N = > 270, we get an improved packing for N' = 2288, with s' = 89.9937...: > <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/circs2288sq.gif> > (The dashed lines divide the figure into nine regions. For each > region, circles having their centers in it or on its boundary > constitute one copy of the packing for N = 270.) > > The following (incomplete) table gives N' for a grid packing which can be > improved by using k^2 copies of a known non-grid packing for N. > The symbol # at the end of a row indicates that the non-grid packing > for N is irregular (see below). > > N' k N > 513 2 137 > 581 2 154 # > 644 3 80 > 682 2 180 # > 791 2 208 > 875 4 63 > 998 2 261 # > 1033 2 270 > 1129 2 295 # > 1268 2 330 # > 1353 5 63 > 1415 2 368 # > 1570 2 407 > 1748 3 208 > 1936 6 63 > 1985 4 137 > 2288 3 270 > 2503 6 80 > 2622 7 63 > 3081 4 208 > 3392 7 80 > 3413 8 63 > 3488 3 407 > 4037 4 270 > 4307 9 63 > 4417 8 80 > 4788 5 208 Of the above improved packings, the only ones which might be optimal are N' = 513, 644, 1033, 1415, 1570, 2288 and 3488. And of those, N' = 1415 is the only one which happens to use an irregular packing, N = 368. A slightly improved packing is now given for it, along with the resulting packing for N' = 1415. ------------------------------------------------------------------ N = 368 s = 36.47646514982270891408... 894 contacts <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/circs368sq.gif> The best packing previously known has side length s = 36.47646522... and 893 contacts. ------------------------------------------------------------------ N' = 1415 s' = 70.95293029964541782816... 3448 contacts <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/circs1415sq.gif> The best packing previously known has side length s = 70.9547... and 2832 contacts. ------------------------------------------------------------------ David W. Cantrell > Knowing the side length s of the packing for N, of course it's trivial to > calculate the side length s' of the improved packing for N': > > s' = k (s - 2) + 2 > > It seems unnecessary to show other figures of those improved packings > since their method of construction has been explained. However, when a > packing for N is irregular (that is, it fails to exhibit regularity as > seen, for example, in the packings for 407 and 270), caution is needed. > For example, consider the packing for N = 180 > <http://hydra.nat.uni-magdeburg.de/packing/csq/csq180.html> from which > we wish to get a packing for N' = 682. If we unthinkingly follow the > described procedure, flipping about an axis one unit from the right side > and then flipping about an axis one unit from the bottom side, the > resulting figure has 681 circles, rather than 682. Furthermore, that > figure isn't even a packing because two circles overlap (namely, circle > #9 and its image after the second flip). But if our second flip is about > an axis one unit from the top side, all is well -- we have N' = 682 and > no circles overlap.
From: David W. Cantrell on 31 May 2010 02:18 A family of packings, closely related to grid packings having p/q = 5/3, is presented. This family improves on the grid packings by using rhombic hexagonal-lattice groups of unit circles. ------------------------------------------------------------------------- For this family, the following simple program prints the number n of unit circles, the precise side length s of the square and a decimal approximation of s. (Eckard Specht has recently extended Packomania with some select packings up to n = 10000, and so members of the new family were generated that far.) Please see comments and links added after various lines of the output. Do[ n = 30*k^2 + 68*k + 39; s = 2 + (15/17)*((3 + 5*Sqrt[3])*k + Sqrt[(30*Sqrt[3] - 52)*k^2 + 136]); Print[{n, FullSimplify[s], N[s, 12]}], {k, -1, 17}] {1, 2, 2.00000000000} Obviously! {39, 2 + 30*Sqrt[2/17], 12.2899151086} This is just the known grid packing. {137, (2*(62 + 75*Sqrt[3]))/17, 22.5769188903} This packing is already known: <http://hydra.nat.uni-magdeburg.de/packing/csq/csq137.html> {295, (2*(62 + 75*Sqrt[3] + 15*Sqrt[6*(-3 + 5*Sqrt[3])]))/17, 32.8610101090} This is the only member of the family which is known to be suboptimal. {513, (169 + 225*Sqrt[3] + 15*Sqrt[-332 + 270*Sqrt[3]])/17, 43.1421850507} This is new. See <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/circs513sq.gif> {791, (2*(107 + 150*Sqrt[3] + 15*Sqrt[6*(-29 + 20*Sqrt[3])]))/17, 53.4204375077} This packing was recently found by Eckard: <http://hydra.nat.uni-magdeburg.de/packing/csq/csq791.html> All of the following packings are new. But since they all look much like either n = 513 or n = 1527, those are the only new figures presented here. {1129, (259 + 375*Sqrt[3] + 15*Sqrt[-1164 + 750*Sqrt[3]])/17, 63.6957587524} {1527, (2*(152 + 225*Sqrt[3] + 15*Sqrt[-434 + 270*Sqrt[3]]))/17, 73.9681374993} <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/circs1527sq.gif> {1985, (349 + 525*Sqrt[3] + 15*Sqrt[6*(-402 + 245*Sqrt[3])])/17, 84.2375598567} {2503, (2*(197 + 300*Sqrt[3] + 15*Sqrt[-798 + 480*Sqrt[3]]))/17, 94.5040092652} {3081, (439 + 675*Sqrt[3] + 15*Sqrt[-4076 + 2430*Sqrt[3]])/17, 104.767466425} {3719, (2*(242 + 375*Sqrt[3] + 15*Sqrt[-1266 + 750*Sqrt[3]]))/17, 115.027909210} {4417, (529 + 825*Sqrt[3] + 15*Sqrt[-6156 + 3630*Sqrt[3]])/17, 125.285312565} {5175, (2*(287 + 450*Sqrt[3] + 15*Sqrt[2*(-919 + 540*Sqrt[3])]))/17, 135.539648392} {5993, (619 + 975*Sqrt[3] + 15*Sqrt[-8652 + 5070*Sqrt[3]])/17, 145.790885416} {6871, (2*(332 + 525*Sqrt[3] + 15*Sqrt[6*(-419 + 245*Sqrt[3])]))/17, 156.038989036} {7809, (709 + 1125*Sqrt[3] + 15*Sqrt[-11564 + 6750*Sqrt[3]])/17, 166.283921153} {8807, (2*(377 + 600*Sqrt[3] + 15*Sqrt[6*(-549 + 320*Sqrt[3])]))/17, 176.525639977} {9865, 47 + 75*Sqrt[3] + 15*Sqrt[-876/17 + 30*Sqrt[3]], 186.764099814} -------------------------------------------------------------------------- David W. Cantrell
From: David W. Cantrell on 31 May 2010 11:13
*************************************************************************** *** This thread is dedicated to Martin Gardner, whose wonderful columns *** *** in Scientific American first piqued my interest in packing problems.*** *************************************************************************** David W. Cantrell <DWCantrell(a)sigmaxi.net> wrote: > A family of packings, closely related to grid packings having p/q = 5/3, > is presented. This family improves on the grid packings by using rhombic > hexagonal-lattice groups of unit circles. Another part of the same family is presented below. Conjecture: This family contains infinitely many optimal packings. (Indeed, it might contain only finitely many suboptimal ones.) > ------------------------------------------------------------------------- > > For this family, the following simple program prints the number n of unit > circles, the precise side length s of the square and a decimal > approximation of s. (Eckard Specht has recently extended Packomania with > some select packings up to n = 10000, and so members of the new family > were generated that far.) Please see comments and links added after > various lines of the output. > > Do[ > n = 30*k^2 + 68*k + 39; > s = 2 + (15/17)*((3 + 5*Sqrt[3])*k + Sqrt[(30*Sqrt[3] - 52)*k^2 + 136]); > Print[{n, FullSimplify[s], N[s, 12]}], > {k, -1, 17}] > > {1, 2, 2.00000000000} > Obviously! > > {39, 2 + 30*Sqrt[2/17], 12.2899151086} > This is just the known grid packing. > > {137, (2*(62 + 75*Sqrt[3]))/17, 22.5769188903} > This packing is already known: > <http://hydra.nat.uni-magdeburg.de/packing/csq/csq137.html> > > {295, (2*(62 + 75*Sqrt[3] + 15*Sqrt[6*(-3 + 5*Sqrt[3])]))/17, 32.8610101090} > This is the only member of the family which is known to be suboptimal. > > {513, (169 + 225*Sqrt[3] + 15*Sqrt[-332 + 270*Sqrt[3]])/17, 43.1421850507} > This is new. See > <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/circs513sq.gif> > > {791, (2*(107 + 150*Sqrt[3] + 15*Sqrt[6*(-29 + 20*Sqrt[3])]))/17, 53.4204375077} > This packing was recently found by Eckard: > <http://hydra.nat.uni-magdeburg.de/packing/csq/csq791.html> > > All of the following packings are new. But since they all look much like > either n = 513 or n = 1527, those are the only new figures presented > here. > > {1129, (259 + 375*Sqrt[3] + 15*Sqrt[-1164 + 750*Sqrt[3]])/17, 63.6957587524} > > {1527, (2*(152 + 225*Sqrt[3] + 15*Sqrt[-434 + 270*Sqrt[3]]))/17, 73.9681374993} > <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/circs1527sq.gif> > > {1985, (349 + 525*Sqrt[3] + 15*Sqrt[6*(-402 + 245*Sqrt[3])])/17, 84.2375598567} > > {2503, (2*(197 + 300*Sqrt[3] + 15*Sqrt[-798 + 480*Sqrt[3]]))/17, 94.5040092652} > > {3081, (439 + 675*Sqrt[3] + 15*Sqrt[-4076 + 2430*Sqrt[3]])/17, 104.767466425} > > {3719, (2*(242 + 375*Sqrt[3] + 15*Sqrt[-1266 + 750*Sqrt[3]]))/17, 115.027909210} > > {4417, (529 + 825*Sqrt[3] + 15*Sqrt[-6156 + 3630*Sqrt[3]])/17, 125.285312565} > > {5175, (2*(287 + 450*Sqrt[3] + 15*Sqrt[2*(-919 + 540*Sqrt[3])]))/17, 135.539648392} > > {5993, (619 + 975*Sqrt[3] + 15*Sqrt[-8652 + 5070*Sqrt[3]])/17, 145.790885416} > > {6871, (2*(332 + 525*Sqrt[3] + 15*Sqrt[6*(-419 + 245*Sqrt[3])]))/17, 156.038989036} > > {7809, (709 + 1125*Sqrt[3] + 15*Sqrt[-11564 + 6750*Sqrt[3]])/17, 166.283921153} > > {8807, (2*(377 + 600*Sqrt[3] + 15*Sqrt[6*(-549 + 320*Sqrt[3])]))/17, 176.525639977} > > {9865, 47 + 75*Sqrt[3] + 15*Sqrt[-876/17 + 30*Sqrt[3]], 186.764099814} > > ------------------------------------------------------------------------- Note that the differences between the previously presented part of the family and the new part are small: The number of circles packed is now n = 2*(3*k + 2)*(5*k + 3) = 30*k^2 + 38*k + 12, rather than 30*k^2 + 68*k + 39. And in the formula for side length s, the only change is that the constant term under the radical is now 34, rather than 136. Again: Some comments and links have been added after various lines of the output. Since members of this family look so much alike, only two new figures (n = 644 and 952) are presented. Do[ n = 2*(3*k + 2)*(5*k + 3); s = 2 + (15/17)*((5*Sqrt[3] + 3)*k + Sqrt[(30*Sqrt[3] - 52)*k^2 + 34]); Print[{n, FullSimplify[s], N[s, 12]}], {k, 0, 17}] {12, 2 + 15 Sqrt[2/17], 7.14495755428} This is just the known grid packing. {80, 2 + 15/17 (3 + 5 Sqrt[3] + Sqrt[6 (-3 + 5 Sqrt[3])]), 17.4305050545} This packing is known: <http://hydra.nat.uni-magdeburg.de/packing/csq/csq80.html> {208, 1/17 (124 + 150 Sqrt[3] + 15 Sqrt[6 (-29 + 20 Sqrt[3])]), 27.7102187539} This packing is known: <http://hydra.nat.uni-magdeburg.de/packing/csq/csq208.html> {396, 1/17 (169 + 225 Sqrt[3] + 15 Sqrt[-434 + 270 Sqrt[3]]), 37.9840687497} This is the only member of the new subfamily which is known to be suboptimal. All of the following packings are new. {644, 1/17 (214 + 300 Sqrt[3] + 15 Sqrt[-798 + 480 Sqrt[3]]), 48.2520046326} <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/circs644sq.gif> [The best packing previously known had also used parallelogramic hexagonal-lattice groups: <http://hydra.nat.uni-magdeburg.de/packing/csq/csq644.html>, but those parallelograms were not all equilateral.] {952, 1/17 (259 + 375 Sqrt[3] + 15 Sqrt[-1266 + 750 Sqrt[3]]), 58.5139546050} <http://i403.photobucket.com/albums/pp113/DWCantrell_photos/Sq/circs952sq.gif> {1320, 1/17 (304 + 450 Sqrt[3] + 15 Sqrt[2 (-919 + 540 Sqrt[3])]), 68.7698241958} {1748, 1/17 (349 + 525 Sqrt[3] + 15 Sqrt[6 (-419 + 245 Sqrt[3])]), 79.0194945178} {2236, 1/17 (394 + 600 Sqrt[3] + 15 Sqrt[6 (-549 + 320 Sqrt[3])]), 89.2628199887} {2784, 2 + 15/17 (27 + 45 Sqrt[3] + Sqrt[-4178 + 2430 Sqrt[3]]), 99.4996254087} {3392, 1/17 (484 + 750 Sqrt[3] + 15 Sqrt[6 (-861 + 500 Sqrt[3])]), 109.729702250} {4060, 1/17 (529 + 825 Sqrt[3] + 15 Sqrt[-6258 + 3630 Sqrt[3]]), 119.952803960} {4788, 1/17 (574 + 900 Sqrt[3] + 15 Sqrt[-7454 + 4320 Sqrt[3]]), 130.168640000} {5576, 1/17 (619 + 975 Sqrt[3] + 15 Sqrt[-8754 + 5070 Sqrt[3]]), 140.376868259} {6424, 1/17 (664 + 1050 Sqrt[3] + 15 Sqrt[6 (-1693 + 980 Sqrt[3])]), 150.577085300} {7332, 1/17 (709 + 1125 Sqrt[3] + 15 Sqrt[-11666 + 6750 Sqrt[3]]), 160.768813701} {8300, 1/17 (754 + 1200 Sqrt[3] + 15 Sqrt[-13278 + 7680 Sqrt[3]]), 170.951485401} {9328, 47 + 75 Sqrt[3] + 15 Sqrt[-(882/17) + 30 Sqrt[3]], 181.124419449} ------------------------------------------------------------------------- David W. Cantrell |