Extremal (Maximum Determinant)
points on the sphere S2

Last updated 24-Jan-2003

2601 extremal points and sacled cubature weights

Extremal Systems

  • Unit sphere S2 in R3 has area |S2| = 4 p.
  • The space Pn of polynomials of degree at most n on S2 has dimension dn = (n+1)2.
  • Extremal systems are sets of dn points xj on S2 that maximize the determinant of a basis matrix.
  • The points are independent of the choice of basis.
  • For a global maximum, it follows that the Lebesgue constant is at most dn = (n+1)2.
  • There are many local maxima, often with objective values close to the global maximum.
  • Here the reproducing kernel basis functions gj(x), j = 1,...,dn and associated Gram matrix G: Gij = gj(xi) are used.
  • G is symmetric positive semidefinite and for any fundamental system of points nonsingular (hence positive definite).
  • A lower bound on log det G is

    log (dn!  /  |S2|dn).

  • An upper bound on log det G is

    dn log (dn / |S2|).

  • For n >= 3 the upper bound cannot be achieved.
  • The norm of the gradient with respect to a spherical parametrization of the points gives an idea how close it is to a stationary point (gradient = 0). Keep in mind the size of the objective when thinking about how small it is numerically possible to get this.
log det G

infinity norm of gradient of log det G

Interpolatory Cubature

  • Cubature rule sumj = 1 to dn   wj f(xj) approximates the integral of f(x) over the surface of the unit sphere.
  • Cubature weights w such that the cubature rule is exact for all polynomials of degree <= n satsify G w = e, e = vector of 1s.
  • Cubature rule exact for p(x) = 1 implies that the sum of cubature weights = |S2|.
  • Average, or equal, cubature weight wavg = |S2| / dn
  • Scaled cubature weights are wi / wavg
  • ,
    smallest   wmin / wavg <= 1,
    largest   wmax / wavg >= 1.
  • Conjecture: For extremal systems the cubature weights are always positive.
Scaled minimum and maximum cubature weights

Eigenvalues and condition numbers

  • Many systems have very large condition numbers, making solving a linear system for interpolation or cubature weights unreliable.
  • Extremal systems have excellent conditioning, making the cubature weiights reliable
  • 1-norm condition number
    cond1(G) = || G ||1   || G-1 ||1.
  • As Gii = dn / |S2|, trace(G) = sum of eigenvalues of G = dn2 / |S2|.
  • lmin(G) = smallest eigenvalue.
    lavg(G) = average eigenvalue = dn / |S2|,
    lmax(G) = largest eigenvalue.
  • 2-norm condition number
    cond2(G) = lmax(G) / lmin(G)
  • The 2-norm condition number is generally an order of magnitude less than the 1-norm condition number
  • Maximizing the product of the eigenvalues, while keeping the sum of the eigenvalues fixed, tries to make all eigenvalues equal, improving the condition number.
Eigenvalues of G

Lagrangian norms

  • Lagranians

    Lj(x) = ejT G-1 g(x)

    for x in S2.
  • Lebesgue constant = ||Ln|| = interpolation operator norm (as a map from C(S2) to C(S2)) is

    maxx in S2 ||L(x)||1.

  • Reimer upper bound on the Lebesgue constant is

    R = (n+1) sqrt(lavg / lmin).
  • All calculated extremal point systems have Lebesgue constant closer to (n+1) than the known upper bound of (n+1)2.
  • Lagrangian 2-norm,

    maxx in S2 || Lj(x) ||2

    and infinity norm,

    maxx in S2 maxj | Lj(x) |,

    are calculated by finding global maximum over S2. As Lj(xj) = 1 and Li(xj) = 0 for i not equal to j, the infinity norm of the Lagrangians is >= 1, and should be 1 for extremal systems.
Lagrangian 1-norm (Lebesgue constant) Lagrangian 2-norm

Geometrical properties

  • The geodesic dsitance between two points x and y on the unit sphere S2 is dist(x,y) = cos-1(xT y).
  • The sphere packing problem is to place m points on a sphere to as to maximize the the minimal distance, or minimal separation, between them.
  • For extremal systems the minimum distance between points is known to be asymptotially greater than p/(2n). Numerically the minimum angle is closer to p/n.
  • Mesh norm is

    h = maxx in S2   minj=1,...,dn   dist(x, xj).

    This is also the covering radius for covering the sphere by spherical caps.
  • Potential energy is sum over all distinct points xi and xj of 1 / ||xi - xj||2
  • Cui and Freeden discrepancy is

    D = (1/(2 dn sqrt(p)))
          [ sumj=1,...,dn sumk=1,...,dn
          (1 - 2 log(1 + sqrt((1 - xjT xk)/2)) ]1/2.

  • Volume of the convex hull of a set of points on S2 is always less than the volume of the unit sphere = 4 p / 3 = 4.18879...
Minimum angle between points

Mesh norm (covering radius)

Caveat: All points are only approximate local maximizers of det(G). Attempts to find global extrema have been made, with decreasing reliability as n increases.

Notes on point sets

Extremal systems -- Determinants and cubature weights
Deg. No. of points Files f = log det G Cubature weights
1-norms and condition number
n dn = (n+1)2 Points and weights Lower bound Value f Upper bound Infinity norm grad f wmin
wavg
wmax
wavg
||G||1 ||G-1||1 cond1(G)
1 4 md01.0004 -6.9 -4.58 -4.6 8.9e-16 1.0000 1.0000 0.3 3.142 1.0e+00
2 9 md02.0009 -10.0 -3.21 -3.0 2.6e-15 0.9824 1.0352 1.1 2.427 2.6e+00
3 16 md03.0016 -9.8 3.39 3.9 4.9e-15 0.9422 1.1734 2.3 1.651 3.8e+00
4 25 md04.0025 -5.3 16.14 17.2 1.7e-14 0.8170 1.0582 4.4 1.599 7.1e+00
5 36 md05.0036 4.6 36.17 37.9 1.2e-14 0.8430 1.0872 7.5 1.372 1.0e+01
6 49 md06.0049 20.5 64.09 66.7 4.1e-14 0.7839 1.1799 11.5 1.333 1.5e+01
7 64 md07.0064 43.2 100.69 104.2 4.4e-14 0.8822 1.1326 15.2 1.157 1.8e+01
8 81 md08.0081 73.1 146.19 150.9 4.6e-14 0.7736 1.1407 22.3 1.127 2.5e+01
9 100 md09.0100 110.6 201.56 207.4 5.2e-14 0.8530 1.0804 28.1 1.042 2.9e+01
10 121 md10.0121 156.4 266.32 274.0 9.2e-14 0.7553 1.2428 38.7 1.066 4.1e+01
11 144 md11.0144 210.6 342.16 351.2 7.8e-14 0.8409 1.1251 45.7 0.734 3.4e+01
12 169 md12.0169 273.7 428.03 439.2 2.5e-13 0.7673 1.1901 61.1 0.878 5.4e+01
13 196 md13.0196 346.0 525.16 538.4 1.1e-13 0.8032 1.1526 78.6 0.879 6.9e+01
14 225 md14.0225 427.8 633.52 649.1 1.1e-05 0.7315 1.2343 92.3 0.938 8.7e+01
15 256 md15.0256 519.3 753.48 771.6 7.1e-07 0.7507 1.1720 112.4 0.957 1.1e+02
16 289 md16.0289 620.9 885.59 906.1 6.5e-07 0.7211 1.2224 125.1 0.750 9.4e+01
17 324 md17.0324 732.7 1029.33 1052.9 1.0e-06 0.7695 1.2307 151.3 0.921 1.4e+02
18 361 md18.0361 855.0 1185.65 1212.2 8.8e-07 0.6781 1.2364 172.0 0.815 1.4e+02
19 400 md19.0400 988.1 1354.16 1384.2 1.4e-06 0.7102 1.2226 195.5 0.752 1.5e+02
20 441 md20.0441 1132.1 1535.65 1569.1 3.8e-06 0.6480 1.2252 223.0 0.794 1.8e+02
21 484 md21.0484 1287.1 1730.07 1767.1 2.0e-06 0.7394 1.2282 250.7 0.734 1.8e+02
22 529 md22.0529 1453.5 1937.53 1978.4 4.3e-06 0.6846 1.2596 280.8 0.798 2.2e+02
23 576 md23.0576 1631.3 2158.52 2203.2 2.3e-06 0.7034 1.2583 313.6 0.737 2.3e+02
24 625 md24.0625 1820.8 2392.58 2441.7 8.8e-13 0.6819 1.2744 349.4 0.700 2.4e+02
25 676 md25.0676 2022.2 2640.64 2694.0 3.6e-04 0.7069 1.3050 383.4 0.702 2.7e+02
26 729 md26.0729 2235.4 2902.51 2960.2 1.4e-04 0.6780 1.2447 426.8 0.732 3.1e+02
27 784 md27.0784 2460.8 3178.04 3240.6 2.0e-12 0.6869 1.2694 475.3 0.686 3.3e+02
28 841 md28.0841 2698.5 3467.88 3535.2 4.1e-06 0.6731 1.2604 511.7 0.733 3.7e+02
29 900 md29.0900 2948.6 3771.93 3844.2 2.0e-06 0.7172 1.2589 568.2 0.648 3.7e+02
30 961 md30.0961 3211.2 4090.72 4167.8 5.1e-06 0.6708 1.2573 609.4 0.673 4.1e+02
31 1024 md31.1024 3486.4 4423.34 4506.1 1.1e-05 0.7052 1.3496 658.6 0.720 4.7e+02
32 1089 md32.1089 3774.5 4770.89 4859.1 1.4e-05 0.6467 1.2671 721.2 0.726 5.2e+02
33 1156 md33.1156 4075.5 5132.72 5227.1 1.6e-04 0.6857 1.2446 775.2 0.762 5.9e+02
34 1225 md34.1225 4389.6 5509.82 5610.1 1.5e-05 0.6911 1.2510 841.2 0.640 5.4e+02
35 1296 md35.1296 4716.8 5902.10 6008.3 6.0e-04 0.6960 1.2686 893.9 0.703 6.3e+02
36 1369 md36.1369 5057.3 6309.12 6421.7 2.8e-12 0.6819 1.2383 965.4 0.854 8.2e+02
37 1444 md37.1444 5411.1 6730.76 6850.5 1.0e-03 0.6903 1.2551 1035.8 0.718 7.4e+02
38 1521 md38.1521 5778.4 7168.55 7294.9 2.3e-04 0.6930 1.2759 1123.2 0.693 7.8e+02
39 1600 md39.1600 6159.4 7622.03 7754.8 5.4e-04 0.7059 1.2733 1185.8 0.734 8.7e+02
40 1681 md40.1681 6554.0 8089.44 8230.4 2.0e-03 0.6180 1.2891 1285.9 0.734 9.4e+02
41 1764 md41.1764 6962.4 8573.82 8721.8 3.2e-04 0.7048 1.2713 1330.9 0.665 8.9e+02
42 1849 md42.1849 7384.7 9072.94 9229.1 2.9e-03 0.6955 1.2763 1432.7 0.717 1.0e+03
43 1936 md43.1936 7821.0 9588.84 9752.3 2.1e-04 0.6909 1.2779 1505.4 0.680 1.0e+03
44 2025 md44.2025 8271.4 10120.51 10291.7 4.2e-04 0.6910 1.2888 1601.8 0.704 1.1e+03
45 2116 md45.2116 8735.9 10667.78 10847.2 4.0e-04 0.6730 1.2659 1683.5 0.671 1.1e+03
46 2209 md46.2209 9214.7 11231.12 11418.9 2.3e-05 0.6760 1.2519 1781.9 0.654 1.2e+03
47 2304 md47.2304 9707.8 11810.52 12007.0 4.1e-04 0.7167 1.2587 1903.5 0.606 1.2e+03
48 2401 md48.2401 10215.3 12406.53 12611.5 1.9e-03 0.6646 1.2720 1966.7 0.630 1.2e+03
49 2500 md49.2500 10737.4 13019.21 13232.6 3.8e-04 0.6144 1.3055 2088.4 0.675 1.4e+03
50 2601 md50.2601 11274.0 13647.57 13870.2 3.3e-03 0.6546 1.3126 2214.7 0.627 1.4e+03
51 2704 md51.2704 11825.3 14292.28 14524.4 4.6e-04 0.6598 1.2917 2326.6 0.753 1.8e+03
52 2809 md52.2809 12391.3 14953.51 15195.5 7.8e-04 0.6408 1.2913 2408.9 0.714 1.7e+03
53 2916 md53.2916 12972.2 15632.22 15883.3 6.1e-04 0.6910 1.3396 2559.3 0.652 1.7e+03
54 3025 md54.3025 13567.9 16326.23 16588.0 1.4e-03 0.6102 1.2814 2688.7 0.734 2.0e+03
55 3136 md55.3136 14178.7 17038.36 17309.7 8.7e-04 0.6608 1.2748 2833.6 0.591 1.7e+03
56 3249 md56.3249 14804.4 17767.69 18048.4 2.6e-03 0.6588 1.2854 2955.4 0.657 1.9e+03
57 3364 md57.3364 15445.3 18513.29 18804.3 2.2e-03 0.6330 1.3108 3041.0 0.654 2.0e+03
58 3481 md58.3481 16101.3 19274.65 19577.3 2.5e-03 0.6238 1.3074 3209.5 0.653 2.1e+03
59 3600 md59.3600 16772.6 20053.32 20367.6 1.7e-03 0.6558 1.2735 3353.7 0.682 2.3e+03
60 3721 md60.3721 17459.2 20851.73 21175.2 3.0e-03 0.6636 1.2916 3473.0 0.629 2.2e+03
61 3844 md61.3844 18161.2 21663.81 22000.2 8.6e-03 0.6750 1.2980 3598.3 0.726 2.6e+03
62 3969 md62.3969 18878.6 22494.08 22842.6 3.6e-03 0.6760 1.2890 3768.7 0.697 2.6e+03
63 4096 md63.4096 19611.6 23343.22 23702.5 5.2e-04 0.6362 1.2963 3926.6 0.731 2.9e+03
64 4225 md64.4225 20360.1 24208.12 24580.0 1.2e-03 0.6578 1.2859 4069.8 0.705 2.9e+03
65 4356 md65.4356 21124.2 25091.88 25475.1 5.2e-03 0.6108 1.2980 4219.8 0.695 2.9e+03
66 4489 md66.4489 21904.1 25992.26 26388.0 5.8e-03 0.6429 1.3022 4417.6 0.915 4.0e+03
67 4624 md67.4624 22699.7 26909.66 27318.6 1.6e-03 0.6522 1.3033 4548.5 0.719 3.3e+03
68 4761 md68.4761 23511.1 27845.70 28267.0 7.9e-03 0.5913 1.3185 4773.9 0.874 4.2e+03
69 4900 md69.4900 24338.4 28797.95 29233.2 1.4e-03 0.6749 1.3041 4995.2 0.700 3.5e+03
70 5041 md70.5041 25181.6 29769.79 30217.4 7.8e-04 0.6396 1.3303 5116.2 0.802 4.1e+03
71 5184 md71.5184 26040.8 30759.62 31219.6 3.0e-03 0.6263 1.2948 5307.2 0.871 4.6e+03
72 5329 md72.5329 26916.1 31765.56 32239.9 1.5e-02 0.5963 1.3494 5521.0 0.780 4.3e+03
73 5476 md73.5476 27807.5 32788.86 33278.2 7.6e-03 0.6164 1.3144 5671.9 0.791 4.5e+03
74 5625 md74.5625 28715.0 33831.29 34334.7 6.7e-03 0.6259 1.3161 5896.2 0.776 4.6e+03
75 5776 md75.5776 29638.7 34891.14 35409.4 4.1e-03 0.6401 1.3027 6058.9 0.791 4.8e+03
76 5929 md76.5929 30578.7 35969.93 36502.4 1.0e-02 0.6241 1.3797 6316.1 0.841 5.3e+03
77 6084 md77.6084 31535.0 37063.55 37613.7 9.0e-03 0.6374 1.3053 6474.8 0.995 6.4e+03
78 6241 md78.6241 32507.6 38182.27 38743.3 8.4e-04 0.6272 1.3060 6696.6 0.731 4.9e+03
79 6400 md79.6400 33496.7 39313.19 39891.4 1.1e-02 0.5772 1.2862 6858.2 0.855 5.9e+03
80 6561 md80.6561 34502.2 40468.87 41057.9 1.5e-02 0.6220 1.2929 7163.5 0.697 5.0e+03
84 7225 md84.7225 38690.0 45252.94 45909.7 4.3e-02 0.6465 1.3151 8145.0 1.204 9.8e+03
88 7921 md88.7921 43145.1 50340.05 51060.7 9.7e-03 0.5761 1.3187 9154.1 0.859 7.9e+03
92 8649 md92.8649 47870.5 55722.84 56514.1 1.9e-02 0.5804 1.3465 10077.1 1.017 1.0e+04
96 9409 md96.9409 52869.0 61413.13 62272.5 1.3e-02 0.5576 1.3591 11251.7 1.102 1.2e+04
99 10000 md99.0000 56798.7 65874.91 66793.2 4.0e-03 0.6117 1.3285 12123.9 1.064 1.3e+04
127 16384 md127.16384 101144.8 115994.57 117523.0 1.0e+00 0.5566 1.4543 22830.0 1.372 3.1e+04
128 16641 md128.16641 102990.3 118056.64 119625.5 1.4e+00 0.5896 1.3438 23240.0 1.444 3.4e+04
191 36864 md191.36864 257463.1 290802.67 294320.9 1.4e+01 0.5182 1.4264 63210.0 1.962 1.2e+05

Extremal systems -- Eigenvalues and Lagrangian norms
Deg. No. of points Eigenvalues of G and condition number Lebesgue constant
Lagrangian norms
n
dn = (n+1)2
lmin lavg lmax cond2(G) || Ln|| Reimer
upper
bound
maxx in S2
|| L(x) ||2
maxx in S2
|| L(x) ||inf - 1
1 4 0.3183 0.3183 0.3183 1.00 2.00 2.00 1.00000 0.0e+00
2 9 0.4633 0.7162 0.8950 1.93 3.24 3.73 1.13697 2.2e-16
3 16 0.7967 1.2732 1.9149 2.40 3.60 5.06 1.12051 4.4e-16
4 25 0.8965 1.9894 2.8658 3.20 5.22 7.45 1.27372 8.9e-16
5 36 1.0983 2.8648 4.3354 3.95 6.13 9.69 1.31083 1.1e-15
6 49 1.2027 3.8993 6.1968 5.15 7.46 12.60 1.36945 8.9e-16
7 64 1.3534 5.0930 8.3877 6.20 8.19 15.52 1.35201 1.3e-15
8 81 1.5380 6.4458 10.8123 7.03 8.86 18.42 1.40769 1.3e-15
9 100 1.7882 7.9577 13.5931 7.60 8.60 21.10 1.27795 1.6e-15
10 121 1.8487 9.6289 16.5433 8.95 10.90 25.10 1.43470 1.8e-15
11 144 3.0530 11.4592 19.3570 6.34 11.44 23.25 1.45369 2.4e-15
12 169 2.3608 13.4486 23.5398 9.97 14.38 31.03 1.53730 3.3e-15
13 196 2.5059 15.5972 27.6946 11.05 15.45 34.93 1.55752 3.6e-15
14 225 2.5568 17.9049 31.7788 12.43 20.28 39.69 1.88592 6.0e-13
15 256 2.3949 20.3718 36.4973 15.24 18.23 46.66 1.65131 9.8e-14
16 289 2.9111 22.9979 40.4140 13.88 18.28 47.78 1.61844 5.0e-14
17 324 2.9921 25.7831 46.0495 15.39 21.01 52.84 1.75121 5.8e-14
18 361 3.0030 28.7275 52.0852 17.34 20.51 58.77 1.65585 4.4e-14
19 400 3.3460 31.8310 57.8951 17.30 22.29 61.69 1.69404 1.0e-13
20 441 3.2625 35.0937 63.8296 19.56 23.38 68.87 1.74291 3.0e-13
21 484 3.6911 38.5155 70.6113 19.13 25.73 71.07 1.75259 1.2e-13
22 529 3.5209 42.0965 76.4184 21.70 25.02 79.53 1.72857 1.2e-13
23 576 3.6356 45.8366 84.5816 23.27 27.41 85.22 1.79264 1.4e-13
24 625 4.0385 49.7359 90.9942 22.53 27.71 87.73 1.76238 1.5e-14
25 676 3.9436 53.7944 98.8195 25.06 32.67 96.03 1.98680 3.6e-10
26 729 3.8716 58.0120 107.3719 27.73 29.42 104.52 1.71830 6.7e-10
27 784 4.0445 62.3887 115.0502 28.45 30.78 109.97 1.80435 1.5e-14
28 841 4.0429 66.9247 124.1057 30.70 32.13 117.99 1.86149 1.0e-12
29 900 4.6410 71.6197 133.1534 28.69 32.21 117.85 1.74197 4.6e-13
30 961 4.1986 76.4740 142.7842 34.01 36.28 132.30 1.94961 6.6e-13
31 1024 4.4149 81.4873 151.4716 34.31 36.95 137.48 1.86461 7.4e-11
32 1089 4.0245 86.6599 161.1922 40.05 37.24 153.13 1.84336 2.5e-12
33 1156 4.2848 91.9916 172.3584 40.23 39.28 157.54 1.83171 6.6e-10
34 1225 4.6000 97.4824 181.2699 39.41 39.22 161.12 1.89889 2.6e-12
35 1296 4.6734 103.1324 193.2571 41.35 41.44 169.12 1.88395 9.2e-10
36 1369 4.0042 108.9416 205.1954 51.24 49.03 192.99 2.05478 4.4e-14
37 1444 4.3924 114.9099 216.3410 49.25 46.04 194.36 1.94668 1.5e-09
38 1521 5.1515 121.0373 228.3195 44.32 51.32 189.04 2.12873 5.3e-10
39 1600 4.6368 127.3240 239.2040 51.59 49.60 209.61 1.92896 3.1e-09
40 1681 4.7413 133.7697 251.9551 53.14 48.26 217.78 1.86758 1.2e-07
41 1764 5.0397 140.3747 263.9664 52.38 49.46 221.66 1.92756 3.3e-09
42 1849 4.7133 147.1387 278.6918 59.13 53.05 240.25 1.95924 1.9e-09
43 1936 4.7789 154.0620 291.7093 61.04 51.09 249.83 1.87698 6.2e-09
44 2025 5.3514 161.1444 306.2054 57.22 52.05 246.94 1.95727 9.7e-09
45 2116 4.9449 168.3859 318.8740 64.49 61.11 268.43 2.01183 1.0e-08
46 2209 5.6298 175.7866 332.5665 59.07 61.17 262.63 2.05772 2.0e-11
47 2304 6.0740 183.3465 347.3020 57.18 62.18 263.72 2.07051 5.7e-09
48 2401 6.1597 191.0655 362.0576 58.78 60.23 272.90 2.06775 5.5e-09
49 2500 5.2458 198.9437 376.2948 71.73 61.69 307.91 1.97960 4.5e-09
50 2601 5.9707 206.9810 393.5336 65.91 65.91 300.28 2.20075 2.8e-08
51 2704 5.1253 215.1775 409.8555 79.97 76.19 336.93 2.32610 NaN
52 2809 5.0079 223.5331 423.7830 84.62 68.43 354.09 2.15800 NaN
53 2916 5.1912 232.0479 443.2842 85.39 66.03 361.03 2.09590 NaN
54 3025 5.3777 240.7219 458.4456 85.25 70.13 367.98 2.09612 NaN
55 3136 6.4204 249.5550 475.7938 74.11 70.18 349.13 2.03856 NaN
56 3249 5.8631 258.5472 491.8785 83.89 68.93 378.51 2.03788 NaN
57 3364 5.7613 267.6986 510.4716 88.60 71.07 395.36 2.03044 NaN
58 3481 5.9440 277.0092 527.1784 88.69 72.52 402.77 2.06407 NaN
59 3600 5.3332 286.4789 547.3383 102.63 87.60 439.75 2.29504 NaN
60 3721 6.0939 296.1078 565.5384 92.80 73.94 425.21 2.04889 NaN
61 3844 4.9679 305.8958 583.4588 117.44 79.63 486.51 2.07834 NaN
62 3969 6.1601 315.8430 602.6682 97.83 74.72 451.11 2.01622 NaN
63 4096 5.6710 325.9493 621.4968 109.59 97.34 485.20 2.32874 NaN
64 4225 4.6802 336.2148 642.9337 137.37 87.87 550.92 2.17233 NaN
65 4356 5.2431 346.6395 658.7558 125.64 89.52 536.65 2.20704 NaN
66 4489 5.3227 357.2233 682.6659 128.26 88.69 548.88 2.14265 NaN
67 4624 5.7183 367.9662 702.6606 122.88 84.91 545.48 2.24456 NaN
68 4761 5.2668 378.8683 725.5526 137.76 96.67 585.22 2.18178 NaN
69 4900 5.7200 389.9296 746.0457 130.43 89.94 577.95 2.18318 NaN
70 5041 4.7407 401.1500 768.8082 162.17 96.88 653.11 2.15406 NaN
71 5184 5.2487 412.5296 791.7847 150.85 101.40 638.31 2.24964 NaN
72 5329 5.4375 424.0683 812.7323 149.47 103.23 644.67 2.34210 NaN
73 5476 5.7007 435.7662 837.1908 146.86 99.34 646.98 2.28500 NaN
74 5625 4.9959 447.6233 857.4774 171.64 110.25 709.92 2.28457 NaN
75 5776 5.7219 459.6395 881.3022 154.02 106.87 681.16 2.31223 NaN
76 5929 4.8075 471.8148 904.5354 188.15 121.27 762.81 2.41586 NaN
77 6084 5.2006 484.1493 929.4881 178.73 109.14 752.59 2.28115 NaN
78 6241 5.5989 496.6430 951.8143 170.00 108.07 744.04 2.34106 NaN
79 6400 4.4055 509.2958 980.8990 222.65 115.60 860.16 2.22484 NaN
80 6561 6.2687 522.1078 1002.5664 159.93 107.91 739.23 2.24650 NaN
84 7225 3.6979 574.9472 1106.6575 299.26 129.63 1059.87 2.47896 NaN
88 7921 4.9047 630.3332 1209.4311 246.59 134.55 1008.95 2.35930 NaN
92 8649 4.8429 688.2656 1321.0905 272.79 141.27 1108.68 2.46370 NaN
96 9409 5.2001 748.7444 1439.9782 276.91 137.37 1163.94 2.34955 NaN
99 10000 4.2935 795.7747 1530.8789 356.56 162.00 1361.41 2.55515 NaN
127 16384 NaN 1303.7973 NaN NaN NaN NaN NaN NaN
128 16641 NaN 1324.2487 NaN NaN NaN NaN NaN NaN
191 36864 NaN 2933.5439 NaN NaN NaN NaN NaN NaN

Extremal systems -- geometric properties
Deg. No. of points Minimum distance Mesh norm 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 0.51320 4 0 0 0 0 0 3.67e+00 0.0913 1.1467
2 9 69.4899 46.9493 2.03516 0 3 6 0 0 0 2.58e+01 0.0493 0.6204
3 16 49.6138 32.9618 2.88596 0 0 12 4 0 0 9.29e+01 0.0319 0.4026
4 25 39.4857 29.7482 3.30449 0 0 12 13 0 0 2.44e+02 0.0228 0.2876
5 36 32.0586 25.5079 3.56571 0 0 12 24 0 0 5.29e+02 0.0174 0.2182
6 49 27.1376 20.9026 3.72631 0 0 12 37 0 0 1.01e+03 0.0138 0.1732
7 64 24.0945 17.7553 3.83288 0 0 12 52 0 0 1.77e+03 0.0112 0.1414
8 81 21.1536 16.0981 3.90556 0 0 14 65 2 0 2.88e+03 0.0094 0.1187
9 100 19.2986 13.6100 3.95978 0 0 12 88 0 0 4.45e+03 0.0080 0.1010
10 121 17.3681 13.8929 3.99696 0 0 20 93 8 0 6.59e+03 0.0070 0.0880
11 144 16.2763 12.5198 4.02628 0 0 24 108 12 0 9.42e+03 0.0061 0.0771
12 169 14.6742 11.9941 4.05131 0 0 24 133 12 0 1.31e+04 0.0054 0.0684
13 196 13.6459 10.6508 4.07035 0 0 24 160 12 0 1.77e+04 0.0049 0.0611
14 225 12.3918 10.6452 4.08551 0 0 26 185 14 0 2.35e+04 0.0044 0.0552
15 256 11.9649 9.6383 4.09772 0 0 30 208 18 0 3.05e+04 0.0040 0.0501
16 289 11.1754 9.0657 4.10797 0 0 36 229 24 0 3.91e+04 0.0036 0.0458
17 324 10.5062 8.3851 4.11678 0 0 37 262 25 0 4.93e+04 0.0033 0.0420
18 361 10.0212 8.0303 4.12416 0 0 41 291 29 0 6.14e+04 0.0031 0.0388
19 400 9.4440 7.6331 4.13032 0 0 49 314 37 0 7.56e+04 0.0028 0.0359
20 441 8.9486 7.2928 4.13572 0 0 49 355 37 0 9.21e+04 0.0026 0.0334
21 484 8.6294 6.8728 4.14045 0 0 54 388 42 0 1.11e+05 0.0025 0.0311
22 529 8.1802 6.9255 4.14447 0 0 64 413 52 0 1.33e+05 0.0023 0.0291
23 576 7.8139 6.5178 4.14815 0 0 65 458 53 0 1.58e+05 0.0022 0.0273
24 625 7.4708 6.0143 4.15128 0 0 79 479 67 0 1.87e+05 0.0020 0.0257
25 676 7.2496 6.0655 4.15409 0 0 83 522 71 0 2.19e+05 0.0019 0.0242
26 729 6.9428 5.7202 4.15667 0 0 83 575 71 0 2.55e+05 0.0018 0.0229
27 784 6.7251 5.5553 4.15891 0 0 90 616 78 0 2.95e+05 0.0017 0.0216
28 841 6.4123 5.3176 4.16092 0 0 94 665 82 0 3.40e+05 0.0016 0.0205
29 900 6.2493 5.0892 4.16277 0 0 99 714 87 0 3.90e+05 0.0015 0.0195
30 961 6.0697 5.1176 4.16445 0 0 103 767 91 0 4.45e+05 0.0015 0.0186
31 1024 5.8045 4.8766 4.16592 0 0 110 816 98 0 5.06e+05 0.0014 0.0177
32 1089 5.7029 4.6194 4.16725 0 0 119 863 107 0 5.73e+05 0.0013 0.0169
33 1156 5.4771 4.5535 4.16852 0 0 126 916 114 0 6.47e+05 0.0013 0.0162
34 1225 5.3378 4.4484 4.16966 0 0 131 975 119 0 7.27e+05 0.0012 0.0155
35 1296 5.2088 4.3897 4.17069 0 0 134 1040 122 0 8.14e+05 0.0012 0.0148
36 1369 5.0561 4.1741 4.17168 0 0 145 1091 133 0 9.09e+05 0.0011 0.0142
37 1444 4.8945 4.0648 4.17257 0 0 141 1174 129 0 1.01e+06 0.0011 0.0137
38 1521 4.7883 4.1858 4.17339 0 0 145 1243 133 0 1.12e+06 0.0010 0.0132
39 1600 4.6934 3.8214 4.17415 0 0 157 1298 145 0 1.24e+06 0.0010 0.0127
40 1681 4.5325 3.8650 4.17484 0 0 176 1341 164 0 1.37e+06 0.0010 0.0122
41 1764 4.4401 3.7811 4.17548 0 0 190 1396 178 0 1.52e+06 0.0009 0.0118
42 1849 4.3098 3.6003 4.17609 0 0 204 1453 192 0 1.67e+06 0.0009 0.0114
43 1936 4.2338 3.4436 4.17667 0 0 204 1540 192 0 1.83e+06 0.0009 0.0110
44 2025 4.1445 3.4577 4.17720 0 0 221 1597 205 2 2.00e+06 0.0008 0.0106
45 2116 4.0389 3.4181 4.17771 0 0 214 1700 202 0 2.19e+06 0.0008 0.0103
46 2209 3.9915 3.3276 4.17817 0 0 230 1761 218 0 2.38e+06 0.0008 0.0100
47 2304 3.8597 3.2030 4.17861 0 0 240 1836 228 0 2.59e+06 0.0008 0.0096
48 2401 3.7710 3.1771 4.17902 0 0 251 1911 239 0 2.82e+06 0.0007 0.0094
49 2500 3.7245 3.3069 4.17940 0 0 280 1952 268 0 3.06e+06 0.0007 0.0091
50 2601 3.6292 3.2215 4.17978 0 0 245 2123 233 0 3.31e+06 0.0007 0.0088
51 2704 3.5393 2.9878 4.18011 0 0 285 2146 273 0 3.58e+06 0.0007 0.0086
52 2809 3.4586 3.0287 4.18042 0 0 312 2197 300 0 3.86e+06 0.0007 0.0083
53 2916 3.4531 2.9216 4.18074 0 0 296 2336 284 0 4.16e+06 0.0006 0.0081
54 3025 3.3778 2.8511 4.18103 0 0 303 2431 291 0 4.48e+06 0.0006 0.0079
55 3136 3.3299 2.7913 4.18130 0 0 342 2464 330 0 4.82e+06 0.0006 0.0077
56 3249 3.2623 2.9055 4.18156 0 0 348 2565 336 0 5.18e+06 0.0006 0.0075
57 3364 3.2016 2.7544 4.18181 0 0 356 2664 344 0 5.55e+06 0.0006 0.0073
58 3481 3.1223 2.6623 4.18205 0 0 362 2769 350 0 5.95e+06 0.0006 0.0071
59 3600 3.0348 2.5938 4.18227 0 0 383 2846 371 0 6.36e+06 0.0005 0.0069
60 3721 3.0079 2.6119 4.18247 0 0 407 2919 395 0 6.80e+06 0.0005 0.0067
61 3844 2.9840 2.5314 4.18268 0 0 396 3064 384 0 7.26e+06 0.0005 0.0066
62 3969 2.9264 2.4943 4.18287 0 0 421 3139 409 0 7.74e+06 0.0005 0.0064
63 4096 2.9127 2.4278 4.18305 0 0 436 3236 424 0 8.24e+06 0.0005 0.0063
64 4225 2.8548 2.3933 4.18323 0 0 459 3319 447 0 8.77e+06 0.0005 0.0061
65 4356 2.7822 2.4488 4.18339 0 0 482 3404 470 0 9.33e+06 0.0005 0.0060
66 4489 2.7605 2.3149 4.18355 0 0 509 3484 495 1 9.91e+06 0.0005 0.0059
67 4624 2.7191 2.3345 4.18371 0 0 483 3670 471 0 1.05e+07 0.0005 0.0057
68 4761 2.6829 2.2359 4.18386 0 0 500 3773 488 0 1.12e+07 0.0004 0.0056
69 4900 2.6410 2.2702 4.18399 0 0 516 3880 504 0 1.18e+07 0.0004 0.0055
70 5041 2.5739 2.1720 4.18413 0 0 524 4005 512 0 1.25e+07 0.0004 0.0054
71 5184 2.5696 2.2412 4.18426 0 0 549 4098 537 0 1.32e+07 0.0004 0.0053
72 5329 2.5468 2.1501 4.18438 0 0 591 4159 579 0 1.40e+07 0.0004 0.0052
73 5476 2.4939 2.1234 4.18450 0 0 578 4332 566 0 1.48e+07 0.0004 0.0050
74 5625 2.4466 2.1450 4.18461 0 0 614 4409 602 0 1.56e+07 0.0004 0.0049
75 5776 2.4271 2.0621 4.18472 0 0 615 4558 603 0 1.64e+07 0.0004 0.0048
76 5929 2.4069 2.0271 4.18483 0 0 621 4699 609 0 1.73e+07 0.0004 0.0048
77 6084 2.3691 2.0784 4.18493 0 0 632 4832 620 0 1.82e+07 0.0004 0.0047
78 6241 2.3273 2.0349 4.18502 0 0 682 4889 670 0 1.92e+07 0.0004 0.0046
79 6400 2.2909 1.9662 4.18511 0 0 693 5026 681 0 2.02e+07 0.0004 0.0045
80 6561 2.2953 1.9958 4.18521 0 0 709 5155 697 0 2.12e+07 0.0003 0.0044
84 7225 2.1534 1.8640 4.18553 0 0 792 5653 780 0 2.58e+07 0.0003 0.0041
88 7921 2.0597 1.7860 4.18582 0 0 871 6191 859 0 3.10e+07 0.0003 0.0038
92 8649 1.9884 1.6880 4.18607 0 0 977 6708 963 1 3.70e+07 0.0003 0.0036
96 9409 1.8866 1.6873 4.18629 0 0 1022 7377 1010 0 4.38e+07 0.0003 0.0034
99 10000 1.8153 1.5863 4.18644 0 0 1039 7934 1027 0 4.94e+07 0.0003 0.0032
127 16384 1.4220 1.2288 4.18735 0 0 1768 12861 1754 1 1.33e+08 0.0002 0.0022
128 16641 1.4221 1.2618 4.18738 0 0 1824 13005 1812 0 1.37e+08 0.0002 0.0022
191 36864 0.9526 0.8576 4.18815 0 0 4006 28864 3994 0 6.76e+08 0.0001 0.0012

References

Acknowlegements

The use of the high performance computing facilities of is gratefully acknowledged. Last updated: