Extremal (Maximum Determinant)
points on the sphere S2

Last updated 17-Oct-2007

Extremal systems -- geometric properties
Deg. No. of points Minimum distance Mesh norm Mesh ratio Volume of convex hull Voronoi cells Potential energy Cui & Freeden Disc. Worst case error
n dn = (n+1)2 Degrees Degrees   Tri. Rect. Pent. Hex. Hept. Oct.      
1 4 109.4712 70.5288 1.29 0.51320 4 0 0 0 0 0 3.67e+00 0.09125 1.14669
2 9 69.4899 46.9493 1.35 2.03516 0 3 6 0 0 0 2.58e+01 0.04931 0.62039
3 16 49.6138 32.9618 1.33 2.88596 0 0 12 4 0 0 9.29e+01 0.03192 0.40261
4 25 39.4857 29.7482 1.51 3.30449 0 0 12 13 0 0 2.44e+02 0.02284 0.28760
5 36 32.0586 25.5079 1.59 3.56571 0 0 12 24 0 0 5.29e+02 0.01738 0.21824
6 49 27.1376 20.9026 1.54 3.72631 0 0 12 37 0 0 1.01e+03 0.01376 0.17324
7 64 24.0945 17.7553 1.47 3.83288 0 0 12 52 0 0 1.77e+03 0.01125 0.14138
8 81 21.1536 16.0981 1.52 3.90556 0 0 14 65 2 0 2.88e+03 0.00944 0.11870
9 100 19.2986 13.6100 1.41 3.95978 0 0 12 88 0 0 4.45e+03 0.00804 0.10104
10 121 17.3681 13.8929 1.60 3.99696 0 0 20 93 8 0 6.59e+03 0.00698 0.08797
11 144 16.2763 12.5198 1.54 4.02628 0 0 24 108 12 0 9.42e+03 0.00613 0.07710
12 169 14.6742 11.9941 1.63 4.05131 0 0 24 133 12 0 1.31e+04 0.00543 0.06841
13 196 13.6459 10.6508 1.56 4.07035 0 0 24 160 12 0 1.77e+04 0.00485 0.06113
14 225 12.3918 10.6452 1.72 4.08551 0 0 26 185 14 0 2.35e+04 0.00438 0.05517
15 256 11.9649 9.6383 1.61 4.09772 0 0 30 208 18 0 3.05e+04 0.00398 0.05007
16 289 11.1754 9.0657 1.62 4.10797 0 0 36 229 24 0 3.91e+04 0.00364 0.04582
17 324 10.5062 8.3851 1.60 4.11678 0 0 37 262 25 0 4.93e+04 0.00333 0.04197
18 361 10.0212 8.0303 1.60 4.12416 0 0 41 291 29 0 6.14e+04 0.00307 0.03879
19 400 9.4440 7.6331 1.62 4.13032 0 0 49 314 37 0 7.56e+04 0.00285 0.03587
20 441 8.9486 7.2928 1.63 4.13572 0 0 49 355 37 0 9.21e+04 0.00265 0.03337
21 484 8.6294 6.8728 1.59 4.14045 0 0 54 388 42 0 1.11e+05 0.00247 0.03106
22 529 8.1802 6.9255 1.69 4.14447 0 0 64 413 52 0 1.33e+05 0.00231 0.02911
23 576 7.8139 6.5178 1.67 4.14815 0 0 65 458 53 0 1.58e+05 0.00217 0.02730
24 625 7.4708 6.0143 1.61 4.15128 0 0 79 479 67 0 1.87e+05 0.00204 0.02570
25 676 7.1783 6.0920 1.70 4.15411 0 0 83 522 71 0 2.19e+05 0.00192 0.02421
26 729 6.9689 5.6491 1.62 4.15669 0 0 79 583 67 0 2.55e+05 0.00182 0.02287
27 784 6.7251 5.5553 1.65 4.15891 0 0 90 616 78 0 2.95e+05 0.00172 0.02164
28 841 6.4123 5.3176 1.66 4.16092 0 0 94 665 82 0 3.40e+05 0.00163 0.02054
29 900 6.2493 5.0892 1.63 4.16277 0 0 99 714 87 0 3.90e+05 0.00155 0.01952
30 961 6.0697 5.1176 1.69 4.16445 0 0 103 767 91 0 4.45e+05 0.00147 0.01857
31 1024 5.8045 4.8766 1.68 4.16592 0 0 110 816 98 0 5.06e+05 0.00141 0.01772
32 1089 5.6909 4.6223 1.62 4.16727 0 0 116 869 104 0 5.73e+05 0.00134 0.01692
33 1156 5.4752 4.5938 1.68 4.16853 0 0 125 918 113 0 6.47e+05 0.00129 0.01616
34 1225 5.3545 4.4924 1.68 4.16967 0 0 130 977 118 0 7.27e+05 0.00123 0.01548
35 1296 5.2088 4.3896 1.69 4.17069 0 0 134 1040 122 0 8.14e+05 0.00118 0.01484
36 1369 5.0561 4.1741 1.65 4.17168 0 0 145 1091 133 0 9.09e+05 0.00113 0.01425
37 1444 4.9546 4.0767 1.65 4.17257 0 0 142 1172 130 0 1.01e+06 0.00109 0.01369
38 1521 4.8132 4.0067 1.66 4.17340 0 0 143 1247 131 0 1.12e+06 0.00105 0.01316
39 1600 4.6976 3.8334 1.63 4.17415 0 0 155 1302 143 0 1.24e+06 0.00101 0.01267
40 1681 4.5339 3.7680 1.66 4.17485 0 0 170 1353 158 0 1.37e+06 0.00097 0.01221
41 1764 4.4401 3.7812 1.70 4.17548 0 0 190 1396 178 0 1.52e+06 0.00094 0.01178
42 1849 4.3146 3.6194 1.68 4.17611 0 0 192 1477 180 0 1.67e+06 0.00090 0.01137
43 1936 4.2316 3.4584 1.63 4.17667 0 0 201 1546 189 0 1.83e+06 0.00087 0.01098
44 2025 4.1511 3.5645 1.72 4.17721 0 0 210 1617 198 0 2.00e+06 0.00084 0.01063
45 2116 4.0392 3.4071 1.69 4.17771 0 0 219 1690 207 0 2.19e+06 0.00082 0.01028
46 2209 3.9692 3.2811 1.65 4.17818 0 0 227 1767 215 0 2.38e+06 0.00079 0.00995
47 2304 3.8597 3.2030 1.66 4.17861 0 0 240 1836 228 0 2.59e+06 0.00077 0.00964
48 2401 3.7712 3.1579 1.67 4.17902 0 0 247 1919 235 0 2.82e+06 0.00074 0.00936
49 2500 3.7235 3.2606 1.75 4.17940 0 0 277 1958 265 0 3.06e+06 0.00072 0.00908
50 2601 3.6590 3.2693 1.79 4.17977 0 0 266 2081 254 0 3.31e+06 0.00070 0.00881
51 2704 3.5220 2.9531 1.68 4.18011 0 0 284 2148 272 0 3.58e+06 0.00068 0.00856
52 2809 3.4824 3.1025 1.78 4.18043 0 0 301 2219 289 0 3.86e+06 0.00066 0.00832
53 2916 3.4565 2.8995 1.68 4.18074 0 0 296 2336 284 0 4.16e+06 0.00064 0.00808
54 3025 3.3926 2.8743 1.69 4.18103 0 0 309 2419 297 0 4.48e+06 0.00062 0.00786
55 3136 3.3244 2.8382 1.71 4.18130 0 0 333 2482 321 0 4.82e+06 0.00061 0.00766
56 3249 3.2731 2.9065 1.78 4.18156 0 0 350 2561 338 0 5.18e+06 0.00059 0.00746
57 3364 3.2040 2.7182 1.70 4.18181 0 0 356 2664 344 0 5.55e+06 0.00058 0.00726
58 3481 3.1295 2.6215 1.68 4.18205 0 0 360 2773 348 0 5.95e+06 0.00056 0.00708
59 3600 3.0837 2.7330 1.77 4.18227 0 0 372 2868 360 0 6.36e+06 0.00055 0.00691
60 3721 3.0052 2.6070 1.73 4.18248 0 0 407 2919 395 0 6.80e+06 0.00054 0.00674
61 3844 3.0147 2.5178 1.67 4.18269 0 0 393 3070 381 0 7.26e+06 0.00052 0.00657
62 3969 2.9359 2.5143 1.71 4.18287 0 0 412 3157 400 0 7.74e+06 0.00051 0.00642
63 4096 2.9188 2.5096 1.72 4.18305 0 0 434 3240 422 0 8.24e+06 0.00050 0.00627
64 4225 2.8559 2.4498 1.72 4.18323 0 0 441 3355 429 0 8.77e+06 0.00049 0.00612
65 4356 2.7922 2.3898 1.71 4.18340 0 0 473 3422 461 0 9.33e+06 0.00048 0.00599
66 4489 2.7639 2.2984 1.66 4.18356 0 0 496 3509 484 0 9.91e+06 0.00046 0.00586
67 4624 2.7326 2.2912 1.68 4.18371 0 0 480 3676 468 0 1.05e+07 0.00045 0.00573
68 4761 2.6616 2.2970 1.73 4.18386 0 0 500 3773 488 0 1.12e+07 0.00045 0.00560
69 4900 2.6301 2.3090 1.76 4.18400 0 0 501 3910 489 0 1.18e+07 0.00044 0.00548
70 5041 2.6160 2.1986 1.68 4.18413 0 0 515 4023 503 0 1.25e+07 0.00043 0.00536
71 5184 2.5806 2.2091 1.71 4.18425 0 0 574 4048 562 0 1.32e+07 0.00042 0.00526
72 5329 2.5362 2.1314 1.68 4.18438 0 0 575 4191 563 0 1.40e+07 0.00041 0.00515
73 5476 2.4842 2.1161 1.70 4.18450 0 0 587 4314 575 0 1.48e+07 0.00040 0.00504
74 5625 2.4685 2.1267 1.72 4.18461 0 0 593 4451 581 0 1.56e+07 0.00039 0.00494
75 5776 2.4024 2.1408 1.78 4.18472 0 0 624 4540 612 0 1.64e+07 0.00038 0.00484
76 5929 2.4133 2.1146 1.75 4.18483 0 0 604 4733 592 0 1.73e+07 0.00038 0.00475
77 6084 2.3701 2.0421 1.72 4.18493 0 0 617 4862 605 0 1.82e+07 0.00037 0.00466
78 6241 2.3310 1.9750 1.69 4.18502 0 0 666 4921 654 0 1.92e+07 0.00036 0.00457
79 6400 2.3115 1.9385 1.68 4.18512 0 0 679 5054 667 0 2.02e+07 0.00036 0.00449
80 6561 2.2670 1.9610 1.73 4.18521 0 0 676 5221 664 0 2.12e+07 0.00035 0.00440
81 6724 2.2553 1.9902 1.76 4.18529 0 0 733 5270 721 0 2.23e+07 0.00034 0.00433
82 6889 2.2130 1.9300 1.74 4.18538 0 0 716 5469 704 0 2.34e+07 0.00034 0.00425
83 7056 2.1979 1.8619 1.69 4.18546 0 0 754 5560 742 0 2.46e+07 0.00033 0.00417
84 7225 2.1723 1.8574 1.71 4.18554 0 0 761 5715 749 0 2.58e+07 0.00033 0.00410
85 7396 2.1388 1.9047 1.78 4.18561 0 0 787 5834 775 0 2.70e+07 0.00032 0.00403
86 7569 2.1271 1.8090 1.70 4.18568 0 0 827 5927 815 0 2.83e+07 0.00031 0.00396
87 7744 2.0924 1.7932 1.71 4.18575 0 0 846 6064 834 0 2.96e+07 0.00031 0.00389
88 7921 2.0675 1.8360 1.78 4.18582 0 0 853 6227 841 0 3.10e+07 0.00030 0.00382
89 8100 2.0573 1.7968 1.75 4.18589 0 0 886 6340 874 0 3.24e+07 0.00030 0.00376
90 8281 2.0213 1.7297 1.71 4.18595 0 0 876 6541 864 0 3.39e+07 0.00029 0.00370
91 8464 2.0067 1.7186 1.71 4.18601 0 0 907 6662 895 0 3.54e+07 0.00029 0.00364
92 8649 1.9857 1.6864 1.70 4.18607 0 0 954 6753 942 0 3.70e+07 0.00028 0.00358
93 8836 1.9533 1.7126 1.75 4.18613 0 0 952 6944 940 0 3.86e+07 0.00028 0.00352
94 9025 1.9370 1.6476 1.70 4.18619 0 0 952 7133 940 0 4.03e+07 0.00028 0.00347
95 9216 1.9206 1.6285 1.70 4.18624 0 0 970 7288 958 0 4.20e+07 0.00027 0.00341
96 9409 1.9045 1.6474 1.73 4.18629 0 0 1038 7345 1026 0 4.38e+07 0.00027 0.00336
97 9604 1.8775 1.6109 1.72 4.18634 0 0 1031 7554 1019 0 4.56e+07 0.00026 0.00331
98 9801 1.8439 1.5700 1.70 4.18639 0 0 1028 7757 1016 0 4.75e+07 0.00026 0.00326
99 10000 1.8507 1.5909 1.72 4.18644 0 0 1056 7900 1044 0 4.94e+07 0.00026 0.00321
100 10201 1.8265 1.5700 1.72 4.18649 0 0 1095 8023 1083 0 5.15e+07 0.00025 0.00316
101 10404 1.8098 1.6068 1.78 4.18653 0 0 1127 8162 1115 0 5.35e+07 0.00025 0.00312
102 10609 1.7839 1.5813 1.77 4.18657 0 0 1143 8335 1131 0 5.57e+07 0.00024 0.00307
103 10816 1.7733 1.5382 1.73 4.18662 0 0 1138 8552 1126 0 5.79e+07 0.00024 0.00303
104 11025 1.7534 1.4902 1.70 4.18666 0 0 1178 8681 1166 0 6.01e+07 0.00024 0.00299
105 11236 1.7373 1.4746 1.70 4.18670 0 0 1200 8848 1188 0 6.25e+07 0.00023 0.00294
106 11449 1.7037 1.4749 1.73 4.18674 0 0 1234 8993 1222 0 6.49e+07 0.00023 0.00290
107 11664 1.6823 1.4838 1.76 4.18677 0 0 1249 9178 1237 0 6.73e+07 0.00023 0.00286
108 11881 1.6964 1.4682 1.73 4.18681 0 0 1294 9305 1282 0 6.99e+07 0.00022 0.00282
109 12100 1.6711 1.5141 1.81 4.18685 0 0 1335 9442 1323 0 7.25e+07 0.00022 0.00279
110 12321 1.6437 1.4565 1.77 4.18688 0 0 1318 9697 1306 0 7.52e+07 0.00022 0.00275
111 12544 1.6430 1.4167 1.72 4.18691 0 0 1365 9826 1353 0 7.79e+07 0.00022 0.00271
112 12769 1.6352 1.4646 1.79 4.18695 0 0 1384 10013 1372 0 8.07e+07 0.00021 0.00268
113 12996 1.6102 1.4252 1.77 4.18698 0 0 1403 10202 1391 0 8.36e+07 0.00021 0.00264
114 13225 1.5828 1.4392 1.82 4.18701 0 0 1437 10363 1425 0 8.66e+07 0.00021 0.00260
115 13456 1.5923 1.3783 1.73 4.18704 0 0 1444 10580 1432 0 8.97e+07 0.00020 0.00257
116 13689 1.5494 1.3631 1.76 4.18707 0 0 1487 10727 1475 0 9.28e+07 0.00020 0.00254
117 13924 1.5455 1.3602 1.76 4.18710 0 0 1530 10876 1518 0 9.60e+07 0.00020 0.00251
118 14161 1.5399 1.3373 1.74 4.18713 0 0 1565 11043 1553 0 9.93e+07 0.00020 0.00248
119 14400 1.5310 1.3670 1.79 4.18716 0 0 1594 11224 1582 0 1.03e+08 0.00019 0.00245
120 14641 1.5196 1.3565 1.79 4.18718 0 0 1597 11459 1585 0 1.06e+08 0.00019 0.00242
121 14884 1.5164 1.3045 1.72 4.18721 0 0 1618 11660 1606 0 1.10e+08 0.00019 0.00239
122 15129 1.4904 1.2902 1.73 4.18723 0 0 1647 11847 1635 0 1.13e+08 0.00019 0.00236
123 15376 1.4817 1.3443 1.81 4.18726 0 0 1643 12102 1631 0 1.17e+08 0.00018 0.00233
124 15625 1.4718 1.2974 1.76 4.18728 0 0 1716 12205 1704 0 1.21e+08 0.00018 0.00230
125 15876 1.4571 1.2517 1.72 4.18731 0 0 1683 12522 1671 0 1.25e+08 0.00018 0.00227
126 16129 1.4434 1.3082 1.81 4.18733 0 0 1778 12586 1764 1 1.29e+08 0.00018 0.00225
127 16384 1.4151 1.2771 1.80 4.18735 0 0 1778 12840 1766 0 1.33e+08 0.00018 0.00222
128 16641 1.4289 1.2657 1.77 4.18738 0 0 1802 13050 1788 1 1.37e+08 0.00017 0.00219
129 16900 1.4125 1.2280 1.74 4.18740 0 0 1868 13176 1856 0 1.42e+08 0.00017 0.00217
130 17161 1.4090 1.2393 1.76 4.18742 0 0 1860 13453 1848 0 1.46e+08 0.00017 0.00214
131 17424 1.4035 1.1954 1.70 4.18744 0 0 1884 13668 1872 0 1.51e+08 0.00017 0.00212
132 17689 1.3804 1.2011 1.74 4.18746 0 0 1894 13914 1880 1 1.55e+08 0.00017 0.00210
133 17956 1.3506 1.2170 1.80 4.18748 0 0 2004 13960 1992 0 1.60e+08 0.00016 0.00207
134 18225 1.3456 1.2137 1.80 4.18750 0 0 2046 14145 2034 0 1.65e+08 0.00016 0.00205
135 18496 1.3287 1.1854 1.78 4.18752 0 0 2016 14476 2004 0 1.70e+08 0.00016 0.00203
136 18769 1.3387 1.1876 1.77 4.18754 0 0 2080 14622 2066 1 1.75e+08 0.00016 0.00201
137 19044 1.3294 1.1512 1.73 4.18755 0 0 2063 14930 2051 0 1.80e+08 0.00016 0.00198
138 19321 1.3177 1.1717 1.78 4.18757 0 0 2114 15105 2102 0 1.85e+08 0.00016 0.00196
139 19600 1.3044 1.1679 1.79 4.18759 0 0 2089 15435 2075 1 1.91e+08 0.00015 0.00194
140 19881 1.2945 1.1659 1.80 4.18761 0 0 2184 15525 2172 0 1.96e+08 0.00015 0.00192
141 20164 1.2939 1.1420 1.77 4.18762 0 0 2224 15729 2210 1 2.02e+08 0.00015 0.00190
142 20449 1.2760 1.1157 1.75 4.18764 0 0 2261 15940 2247 1 2.07e+08 0.00015 0.00188
143 20736 1.2633 1.1391 1.80 4.18766 0 0 2270 16208 2258 0 2.13e+08 0.00015 0.00186
144 21025 1.2448 1.0969 1.76 4.18767 0 0 2337 16363 2325 0 2.19e+08 0.00015 0.00184
145 21316 1.2595 1.1006 1.75 4.18769 0 0 2299 16730 2287 0 2.25e+08 0.00014 0.00182
146 21609 1.2445 1.1489 1.85 4.18770 0 0 2374 16874 2360 1 2.32e+08 0.00014 0.00180
147 21904 1.2351 1.0815 1.75 4.18772 0 0 2373 17170 2361 0 2.38e+08 0.00014 0.00179
148 22201 1.2279 1.0913 1.78 4.18773 0 0 2391 17431 2379 0 2.45e+08 0.00014 0.00177
149 22500 1.2303 1.0845 1.76 4.18774 0 0 2463 17586 2451 0 2.51e+08 0.00014 0.00175
150 22801 1.2110 1.0742 1.77 4.18776 0 0 2508 17797 2496 0 2.58e+08 0.00014 0.00173
151 23104 1.1984 1.0592 1.77 4.18777 0 0 2541 18034 2529 0 2.65e+08 0.00014 0.00172
152 23409 1.1875 1.0291 1.73 4.18778 0 0 2572 18277 2560 0 2.72e+08 0.00013 0.00170
153 23716 1.1739 1.0170 1.73 4.18780 0 0 2557 18614 2545 0 2.79e+08 0.00013 0.00168
154 24025 1.1807 1.0401 1.76 4.18781 0 0 2588 18861 2576 0 2.87e+08 0.00013 0.00167
155 24336 1.1646 1.0172 1.75 4.18782 0 0 2662 19024 2650 0 2.94e+08 0.00013 0.00165
156 24649 1.1709 1.0230 1.75 4.18784 0 0 2664 19333 2652 0 3.02e+08 0.00013 0.00164
157 24964 1.1597 1.0106 1.74 4.18785 0 0 2737 19503 2723 1 3.09e+08 0.00013 0.00162
158 25281 1.1483 1.0109 1.76 4.18786 0 0 2777 19739 2765 0 3.17e+08 0.00013 0.00160
159 25600 1.1366 1.0350 1.82 4.18787 0 0 2809 19994 2797 0 3.25e+08 0.00013 0.00159
160 25921 1.1291 0.9998 1.77 4.18788 0 0 2830 20274 2816 1 3.34e+08 0.00013 0.00157
161 26244 1.1333 1.0139 1.79 4.18789 0 0 2798 20661 2784 1 3.42e+08 0.00012 0.00156
162 26569 1.1234 1.0048 1.79 4.18790 0 0 2940 20701 2928 0 3.51e+08 0.00012 0.00155
163 26896 1.1001 0.9948 1.81 4.18791 0 0 2981 20947 2967 1 3.59e+08 0.00012 0.00153
164 27225 1.1050 0.9790 1.77 4.18793 0 0 2983 21272 2969 1 3.68e+08 0.00012 0.00152
165 27556 1.1072 0.9623 1.74 4.18794 0 0 2983 21602 2971 0 3.77e+08 0.00012 0.00150


Last updated 17-Oct-2007   |   Page contact: R.Womersley (AT) unsw.edu.au   |   Top   |   Contents