02-Aug-2000: First complete presentation from N=1 to N=200



04-Aug-2000: Improvements for N = 136, 178, 173, 132, 200, 139, 108, 140, 162, 182, 196, 149, 129, 164, 134, 131, 148, 176, 138, 183, 174, 107, 113.

19-Aug-2000: Improvements for N = 144, 200, 142, 92, 139, 177, 138, 141, 76, 97, 175, 173, 148, 176, 185, 174, 125, 156, 124, 133, 143, 129, 183, 170, 113, 180, 172, 134, 198, 160, 165, 178, 161, 147, 184, 83, 131, 140, 155, 169, 137, 163, 111, 167, 189, 122, 149, 136, 93, 196, 128, 171, 182, 98, 162, 73, 146, 181, 168, 164, 154, 150, 110, 99, 118. 18 yellow entries were gone, 42 remaining...



28-Oct-2002: New packings for N = 201 ... 300.

04-Nov-2003: Improvements for N = 53, 70, 73, 74, 75, 77, 78, 79, 80, 81, 82, 88, 95, 97, 102, 103, 104, 107, 108, 110, 111, 112, 113, 114, 116, 117, 118, 119, 120, and many more.

Many thanks to Rainer Rosenthal pointing me to the fact that the older packings were not in the unit circle, now they are actually.

Symmetry groups added.

02-Mar-2004: Boundary column added.



18-Jun-2004: Antialiased GIFs generated.



02-Sep-2004: James Buddenhagen [16] wrote me that he has found slightly improved packings for N = 64, 103 in January. He made a nice webpage which you can find here. About 70 of his packings have high arithmetic precision (approximately 50 decimal places), so I recalculated all my packings on a 113-mantissa-bit HP-machine marvel to 30 decimal places in order to verify his results. By comparison, our packings match exactly! The key for his investigation was the wrong (i.e. too small) numbers of contacts: 115 instead of 116 for N = 64 and 189 instead of 190 for N = 103. Now they are correct. Furthermore, he made very nice pictures of the contacts which I adopted (see the links in column 'contacts' in the table above). Now you can navigate through these kind of pictures as well. Thank you very much, James, for your ideas!

Added: Table headers are links to simple tables now.



08-Oct-2004: Bug fixed in application form. Thanks to Alex Shainsky!



23-Jan-2006: Bug fixed in coordinate extracting program 'xcci.cpp'. Thanks to Jan Callewaert!



24-Jan-2007: References updated for small values of N. Thanks to David A. Garber!



02-Jul-2007: Improvements for N = 75, 78, 80, 83, 84, 93, 106, 107, 108, 128, 136, 137, 143, 169, 321, 342, 343. Some of these results are encouraged by the article [19] and show higher symmetries than the previous records.

20-Mar-2008: A. Grosso, A.R.M.J.U. Jamali, M. Locatelli and F. Schoen have found new astonishing optima for N = 66, 67, 70, 71, 73, 74, 77, 78, 83, 87, 88, 89, 92, 93, 96, 97, 99.

15-Jul-2008: G. Yaskov and Y. Stoyan from Kharkov wrote me that their program is able to find good packings for very large N. I have got new records for N = 400, 500, 1000 as examples.

15-Sep-2008: David W. Cantrell discovered new records for N = 144, 183, 206 and 207 (see for all [24]).

24-Sep-2008: More records by David W. Cantrell for N = 219, 253, 309.

28-Sep-2008: More records by David W. Cantrell for N = 258, 276, 321, 339, (401), 402, 403.

04-Oct-2008: More records by David W. Cantrell for N = 262, 284, 472.

06-Oct-2008: Late publication of results by A. Grosso, A.R.M.J.U. Jamali, M. Locatelli and F. Schoen for N = 75, 86, 94, 100.

07-Oct-2008: Wenqi Huang and Tao Ye sent me new record packings for N = 101, 105, 118, 124.

20-Oct-2008: More records by David W. Cantrell for N = 340, 372, 399, 421, 433, 499.

25-Oct-2008: Again, new records found by Wenqi Huang and Tao Ye for N = 104, 107, 112, 120 and 126.

09-Nov-2008: More records by David W. Cantrell for N = 108, 295.

14-Nov-2008: David W. Cantrell announced new records for N = 270, 333, 349.

17-Nov-2008: Further new records found by Wenqi Huang and Tao Ye for N = 108, 110, 111, 116, 119, 125 and 128.

18-Nov-2008: Reorganization of this page: new data format for coordinates, they may be downloaded now in a convenient way.

19-Nov-2008: New records by E. Specht for N = 144, 145, 182, 192, 193, 195, 197, 208-218, 220-226, 229-230.

19-Nov-2008: First presentation of packings for N = 501—600.

20-Nov-2008: New records by Wenqi Huang and Tao Ye for N = 113, 119 (again), 127, 134, 136 and 137.

20-Nov-2008: New records by E. Specht for N = 231, 244-246, 256, 258-262, 268, 269, 271-277.

22-Nov-2008: First presentation of packings for N = 601—704.

22-Nov-2008: Further records by G. Yaskov and Y. Stoyan for N = 430, 550, 600.

25-Nov-2008: New records by E. Specht for N = 174, 202, 203, 204, 205, 210-212, 214, 215, 217, 218, 220-222, 227-229, 231-233, 236, 239, 245, 249, 251, 252, 255, 257, 260, 262, 267, 269, 271, 272-280, 282, 283, 285, 286, 290, 294, 296, 300 and several beyond 500.

25-Nov-2008: Again, new records by E. Specht for N = 126, 134, 135, 146, 147, 155, 168, 175, 177, 178, 180, 184, 188, 191, 192, 194, 197, 401 and 471.

02-Dec-2008: More records by G. Yaskov and Y. Stoyan for N = 550, 600, 650, 700, 750, 800, 850, 900 and 950.

04-Dec-2008: New records by E. Specht for N = 152, 190, 191, 303-308, 316, 318-320, 328-338, 340, 341, 356-360, 362, 366, 368, 369, 371, 372, 382-389, 391, 392, 394, 397-400.

06-Dec-2008: David W. Cantrell gives in [24] an astonishing new conjectured upper bound for R/r, namely R/r <= 1 + (sqrt((4*ρ-1)^2 + 16*ρ*(N-1)) - 1) / (4*ρ) with ρ = Pi/(2*sqrt(3)). On 08-Dec-2008 the only packings below N <= 500 that violate the bound are N = 437 and 473 and for 501 <= N <= 600 the number is still 45. But this is only a question of time ...

08-Dec-2008: More nice records by David W. Cantrell for N = 547, 583, 637, 684 and 721.

09-Dec-2008: Now all packings below N <= 500 meet David's upper bound.

01-Jun-2009: Improvements for N = 106, 108, 114, 124, 128, 131, 133, 134, 136, 137, 141, 142, 144 and 147 by Tao Ye and Wenqi Huang.

23-Nov-2009: Improvements for N = 152, 153, 154, 155, 157, 158, 159, 160, 162, 167, 168, 169, 170, 171, 173, 175, 179, 180, 182, 183, 186, 189, 190, 191, 194, 195 and 196 by Tao Ye and Wenqi Huang.

01-Dec-2009: Improvements for N = 104, 107, 108, 112, 127, 133, 135, 136, 137, 138, 148, 153, 154, 155, 156, 157, 158, 162, 164, 169, 170, 171, 172, 173, 179, 183, 186, 190, 191, 193, 194, 195, 196, 197, 198 and 200 by Tao Ye and Wenqi Huang.



04-Jun-2010: Some new wonderful (since high symmetric) packings for N = 327, 402, 403, 453, 565, 649, 745, 783, 847, 1015, 1027 and 1039 by David W. Cantrell [15].



11-Jun-2010: Colors in the first column indicate the author of the packing, ratios with orange background indicate a violation of Cantrell's conjectured upper bound. For totally 273 packings is this the case, the smallest being N = 507.



17-Jun-2010: Several improvements between N=900 and 1013 by E. Specht [31] ( 269 orange packings remain).



16-Nov-2010: Paul Christophere from NYU has written an applet that shows some packings in dynamic sequence. It can be viewed here. Thank you, Paul!



21-Mar-2012: David W. Cantrell sent me new packings for N=110, 141, 145, 146 and 147 after this section was stale since 2010, thank you for this wake-up!



21-Mar-2012: Further improvements between N=900 and 1100 and extension up to N = 1500 by E. Specht [31].



21-May-2012: Improvements for N=159, 160, 163, 169, 170, 173, 178, 183, 185, 186, 187, 188, 189, 191, 193, 194 and 198 by David W. Cantrell [15].



14-Apr-2014: After nearly two years, some improvements for N= 507, 509, 522, 551, 587, 588, 589, 621, 624, 626, 665, 666, 682, 699, 700, 701, 886, 1093, 1095, 1097, 1098 and 1099 by E. Specht [31].



21-Apr-2014: More improvements for N= 727, 749, 849, 925, 955, 1003, 1135, 1147, 1159, 1231, 1237, 1249, 1255, 1261, 1273, 1339, 1345, 1351, 1357, 1363, 1369, 1483 and 1495 by E. Specht [31]. Among them are possibly the first which were found on a Raspberry Pi, indeed.



21-Apr-2014: Extension up to N = 1660 by E. Specht [31].



23-Apr-2014: Extension up to N = 1820 by E. Specht [31].



30-Apr-2014: Extension up to N = 2060 by E. Specht [31].



07-Apr-2014: Extension up to N = 2200 by E. Specht [31].



01-Jun-2014: Extension up to N = 2600 and improvements for N = 1866, 1869, and 1977 by E. Specht [31].



10-Jun-2014: Extension up to N = 2989 and improvements for N = 1045, 1111, 1123, 1147, 1159, 1165, 1225, 1261, 1285, 1333, 1345, 1369, 1381, 1393, 1483, 1507, 1597, 1639, 1651, 1723, 1735, 1759, 1777, 1867, 1891, 1915, 2077, 2161, 2209, 2221, 2233, 2347, 2371, 2383, 2407, 2431, 2437, 2479, 2491, 2515, 2527 and 2587 by E. Specht [31].



10-Aug-2014: Extension up to N = 4495 and improvements for N = 817, 823, 931, 955, 1111, 1159, 1165, 1261, 1381, 1393, 1597, 1735, 1777, 1891, 1915, 1921, 2065, 2077, 2161, 2197, 2245, 2347, 2371, 2383, 2407, 2437, 2455, 2479, 2491, 2515, 2527, 2611, 2623, 2635, 2647, 2653, 2665, 2677, 2725, 2737, 2749, 2773, 2779, 2791, 2803, 2839, 2857, 2869, 2893, 2905, 2917, 2929, 2941, 2965 and 2989 by E. Specht [31].



12-Aug-2014: I have found some packings on my hard disk by Yuri Stoyan and Georgiy Yaskov from Kharkov, Ukraine, for N = 1077, 1078, 1079, 1080, 1090, 1091, 1092, 1094, 1096, 1099, 1100, 1200, 1300, 1500, 3000, 4000 and 5000 [20].

