Sphere packings of dimension 3
Introduction
The optimal 3-dimensional (3d) lattice in terms of packing density is the face-centered cubic (fcc) lattice. Therefore, spherical subsets of this lattice are asymptotically optimal as the size tends to infinity. The fcc lattice is not unique in this sense: Infinitely other infinite packings exist with the same density as the fcc. The most well-known of these nonlattice packings is the hexagonal close-packing (hcp). The best known packings at finite sizes are, with a few exceptions (M = 1, 2, 3, 4, 6, 37, 38, 39, 40), not lattice subsets.
Conjectured optimal sphere packings were numerically designed in [Sloane95] and described in detail for M ≤ 32, using exact coordinates or contact graphs. The supplementary website [Sloane-web] gives coordinates for the best found packings up to M ≤ 99. These packings are included in this database, in some cases after translation, scaling, and rotation, under the name c3_*.txt.
Subsets of the fcc lattice were designed by enumerating and testing a finite number of possible centroids inside the fundamental simplex [Conway82] of the lattice. This algorithm [Agrell14] straightforwardly generalizes the algorithm in [Chow95] for optimizing 2-dimensional lattice subsets. The obtained packings, which are provenly the optimal fcc subsets although not necessarily the optimal 3-dimensional packings, are included under the name l3_*.txt.
The most power-efficient 3d packing, in the sense of maximizing γ, is the tetrahedron (M = 4). As the size increases, the CFM and γ of good 3d packings both tend to decrease while the gain CFM increases, although these trends are not monotonic. As M → ∞, the gain asymptotically tends to G → 10log10((5/3)(π2/18)1/3) = 1.349 dB, which equals the gain of the fcc lattice over the cubic lattice [Calderbank87, Sec. IV], [Conway99, pp. 2, 73, 112].
Database
| File | N | M | d | E | En | κ | Lat | β | Eb | CFM [dB] | γ [dB] | G [dB] | Comment |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
BPSK3_2 | 3 | 2 | 2 | 1 | 0.25 | 1 | Y | 0.666667 | 1 | 7.78151 | 0 | -2.31065 | |
triangle3_3 | 3 | 3 | 4.24264 | 6 | 0.333333 | 1 | Y | 1.05664 | 3.78558 | 6.53213 | 0.750803 | -0.914813 | |
tetrahedron3_4 | 3 | 4 | 2.82843 | 3 | 0.375 | 1 | Y | 1.33333 | 1.5 | 6.0206 | 1.24939 | 0.0570721 | [Gilbert52]. The tetrahedron, one of the five Platonic solids [Coxeter73], [Agrell11a]. Applied in optical communications in [Dochhan13]. |
l3_5 | 3 | 5 | 1.41421 | 0.96 | 0.48 | 1.02778 | Y | 1.54795 | 0.413449 | 4.9485 | 0.825475 | 0.00907843 | A square pyramid. |
doublesimplex3_5 | 3 | 5 | 4.24264 | 8.4 | 0.466667 | 1.12245 | N | 1.54795 | 3.61768 | 5.07084 | 0.94782 | 0.131423 | Two tetrahedra sharing the same base. A subset of the hcp. |
biortho3_6 | 3 | 6 | 1.41421 | 1 | 0.5 | 1 | Y | 1.72331 | 0.386853 | 4.77121 | 1.11424 | 0.610616 | [Gilbert52]. The octahedron, one of the five Platonic solids [Coxeter73], [Agrell11a]. Used in biorthogonal modulation. |
l3_6 | 3 | 6 | 1.41421 | 1 | 0.5 | 1 | Y | 1.72331 | 0.386853 | 4.77121 | 1.11424 | 0.610616 | Equivalent to biortho3_6. |
l3_7 | 3 | 7 | 9.89949 | 60 | 0.612245 | 1.15333 | Y | 1.87157 | 21.3724 | 3.89166 | 0.593122 | 0.357831 | |
c3_7 | 3 | 7 | 2 | 2.38333 | 0.595831 | 1.11497 | N | 1.87157 | 0.848958 | 4.00968 | 0.711139 | 0.475848 | Two pentagonal pyramids sharing the same base. |
OOK3_8 | 3 | 8 | 1 | 1.5 | 1.5 | 1.33333 | Y | 2 | 0.5 | 0 | -3.0103 | -3.0103 | [Oliver48]. A regular cube with nonnegative coordinates. |
cube3_8 | 3 | 8 | 2 | 3 | 0.75 | 1 | Y | 2 | 1 | 3.0103 | 0 | 0 | [Oliver48], [Gilbert52, p. 517], [Viterbi61]. A regular zero-mean cube, one of the five platonic solids [Coxeter73], [Agrell11a]. |
l3_8 | 3 | 8 | 5.65685 | 22 | 0.6875 | 1.16529 | Y | 2 | 7.33333 | 3.38819 | 0.377886 | 0.377886 | |
g3_8 | 3 | 8 | 2 | 2.70711 | 0.676777 | 1 | N | 2 | 0.902369 | 3.45646 | 0.446159 | 0.446159 | [Gilbert52] |
c3_8 | 3 | 8 | 2 | 2.64458 | 0.661146 | 1.09465 | N | 2 | 0.881528 | 3.55794 | 0.547638 | 0.547638 | |
l3_9 | 3 | 9 | 12.7279 | 124 | 0.765432 | 1.19095 | Y | 2.11328 | 39.1176 | 2.92185 | 0.150823 | 0.360533 | |
c3_9 | 3 | 9 | 2 | 2.87766 | 0.719416 | 1.07156 | N | 2.11328 | 0.907802 | 3.19111 | 0.420088 | 0.629799 | Three square pyramids, each sharing two points with another. |
l3_10 | 3 | 10 | 7.07107 | 41 | 0.82 | 1.17609 | Y | 2.21462 | 12.3422 | 2.62277 | 0.0551638 | 0.454171 | |
c3_10 | 3 | 10 | 2 | 3.18279 | 0.795697 | 1.08935 | N | 2.21462 | 0.958115 | 2.75343 | 0.185823 | 0.58483 | [Gilbert52]. Two square pyramids placed base-to-base. |
l3_11 | 3 | 11 | 15.5563 | 214 | 0.884298 | 1.11984 | Y | 2.30629 | 61.8599 | 2.29493 | -0.0965365 | 0.475083 | |
c3_11 | 3 | 11 | 66 | 3745.63 | 0.859878 | 1.13616 | N | 2.30629 | 1082.73 | 2.41654 | 0.025078 | 0.596698 | |
icosa3_12 | 3 | 12 | 2.35114 | 5 | 0.904508 | 1 | N | 2.38998 | 1.39471 | 2.19679 | -0.0398803 | 0.690451 | The icosahedron, one of the five Platonic solids [Coxeter73], [Agrell11a]. |
l3_12 | 3 | 12 | 16.9706 | 262 | 0.909722 | 1.09766 | Y | 2.38998 | 73.0831 | 2.17182 | -0.0648418 | 0.665489 | |
c3_12 | 3 | 12 | 2 | 3.56802 | 0.892005 | 1.11139 | N | 2.38998 | 0.995274 | 2.25724 | 0.0205715 | 0.750902 | |
l3_13 | 3 | 13 | 1.41421 | 1.84615 | 0.923077 | 1.08333 | Y | 2.46696 | 0.498901 | 2.10853 | 0.00955449 | 0.886828 | [Gilbert52]. All points with even parity in a 3×3×3 cube. Also, the 12 vertices of the cuboctahedron plus a central point. A “kissing” packing [Agrell09]. |
c3_13 | 3 | 13 | 1482. | 2014756. | 0.917331 | 1.08971 | N | 2.46696 | 544464. | 2.13565 | 0.0366723 | 0.913946 | |
odd3_14 | 3 | 14 | 1.41421 | 2.14286 | 1.07143 | 1.21333 | Y | 2.53824 | 0.56282 | 1.46128 | -0.513999 | 0.500122 | All points with odd parity in a 3x3x3 cube. |
l3_14 | 3 | 14 | 9.89949 | 97 | 0.989796 | 1.12159 | Y | 2.53824 | 25.477 | 1.80546 | -0.169823 | 0.844298 | |
c3_14 | 3 | 14 | 2 | 3.91988 | 0.97997 | 1.10713 | N | 2.53824 | 1.02955 | 1.84879 | -0.126493 | 0.887628 | |
l3_15 | 3 | 15 | 21.2132 | 472 | 1.04889 | 1.14213 | Y | 2.60459 | 120.812 | 1.55362 | -0.309582 | 0.832624 | |
c3_15 | 3 | 15 | 2 | 4.14047 | 1.03512 | 1.12432 | N | 2.60459 | 1.05979 | 1.61101 | -0.252186 | 0.890021 | |
l3_16 | 3 | 16 | 11.3137 | 141 | 1.10156 | 1.15271 | Y | 2.66667 | 35.25 | 1.34082 | -0.420091 | 0.842525 | |
c3_16 | 3 | 16 | 2 | 4.36204 | 1.09051 | 1.10398 | N | 2.66667 | 1.09051 | 1.38462 | -0.376293 | 0.886324 | |
l3_17 | 3 | 17 | 24.0416 | 672 | 1.16263 | 1.17471 | Y | 2.72498 | 164.405 | 1.1065 | -0.560476 | 0.815767 | |
c3_17 | 3 | 17 | 2 | 4.59578 | 1.14894 | 1.11202 | N | 2.72498 | 1.12436 | 1.15792 | -0.509052 | 0.867191 | |
l3_18 | 3 | 18 | 12.7279 | 197 | 1.21605 | 1.18449 | Y | 2.77995 | 47.2431 | 0.9114 | -0.66883 | 0.815 | |
c3_18 | 3 | 18 | 2 | 4.79451 | 1.19863 | 1.12152 | N | 2.77995 | 1.14978 | 0.974067 | -0.606163 | 0.877667 | |
l3_19 | 3 | 19 | 1.41421 | 2.52632 | 1.26316 | 1.1875 | Y | 2.83195 | 0.594717 | 0.746336 | -0.753405 | 0.832599 | |
c3_19 | 3 | 19 | 2 | 5.00676 | 1.25169 | 1.11519 | N | 2.83195 | 1.17864 | 0.785948 | -0.713794 | 0.87221 | |
dodeca3_20 | 3 | 20 | 3.80423 | 28.4164 | 1.96353 | 1 | N | 2.88129 | 6.57494 | -1.16945 | -2.59419 | -0.910892 | The dodecahedron, one of the five Platonic solids [Coxeter73], [Agrell11a]. Possibly useful for ornamentation, but weak as a sphere packing. |
l3_20 | 3 | 20 | 28.2843 | 1062 | 1.3275 | 1.17632 | Y | 2.88129 | 245.724 | 0.530567 | -0.89417 | 0.789128 | |
c3_20 | 3 | 20 | 2 | 5.25217 | 1.31304 | 1.14767 | N | 2.88129 | 1.21524 | 0.578123 | -0.846614 | 0.836684 | |
l3_21 | 3 | 21 | 29.6985 | 1208 | 1.36961 | 1.17334 | Y | 2.92821 | 275.026 | 0.394929 | -0.959646 | 0.816523 | |
c3_21 | 3 | 21 | 42 | 2398.67 | 1.35979 | 1.15446 | N | 2.92821 | 546.105 | 0.426199 | -0.928376 | 0.847794 | A subset of the hcp. |
l3_22 | 3 | 22 | 31.1127 | 1370 | 1.41529 | 1.17518 | Y | 2.97295 | 307.214 | 0.25246 | -1.03626 | 0.828757 | |
c3_22 | 3 | 22 | 2 | 5.56749 | 1.39187 | 1.15349 | N | 2.97295 | 1.24848 | 0.324916 | -0.963802 | 0.901212 | A subset of the hcp. |
l3_23 | 3 | 23 | 32.5269 | 1546 | 1.46125 | 1.17783 | Y | 3.01571 | 341.766 | 0.113674 | -1.11303 | 0.837141 | |
c3_23 | 3 | 23 | 46 | 3030.67 | 1.43226 | 1.15513 | N | 3.01571 | 669.974 | 0.200688 | -1.02602 | 0.924154 | A subset of the hcp. |
l3_24 | 3 | 24 | 2.82843 | 12 | 1.5 | 1.17593 | Y | 3.05664 | 2.61725 | 0 | -1.16815 | 0.863797 | |
c3_24 | 3 | 24 | 48 | 3390.67 | 1.47164 | 1.15624 | N | 3.05664 | 739.519 | 0.0828864 | -1.08527 | 0.946684 | A subset of the hcp. |
l3_25 | 3 | 25 | 35.3553 | 1932 | 1.5456 | 1.17856 | Y | 3.0959 | 416.034 | -0.130059 | -1.24278 | 0.867829 | |
c3_25 | 3 | 25 | 50 | 3790.67 | 1.51627 | 1.16081 | N | 3.0959 | 816.276 | -0.0468433 | -1.15957 | 0.951045 | A subset of the hcp. |
l3_26 | 3 | 26 | 18.3848 | 535 | 1.58284 | 1.17472 | Y | 3.13363 | 113.819 | -0.233458 | -1.29359 | 0.892806 | |
c3_26 | 3 | 26 | 2 | 6.20513 | 1.55128 | 1.15962 | N | 3.13363 | 1.32012 | -0.145995 | -1.20612 | 0.980269 | A subset of the hcp. |
l3_27 | 3 | 27 | 38.1838 | 2378 | 1.631 | 1.18231 | Y | 3.16993 | 500.117 | -0.363631 | -1.37374 | 0.885757 | |
c3_27 | 3 | 27 | 54 | 4668. | 1.60082 | 1.16766 | N | 3.16993 | 981.727 | -0.282521 | -1.29263 | 0.966867 | A subset of the hcp. |
l3_28 | 3 | 28 | 2.82843 | 13.2857 | 1.66071 | 1.17759 | Y | 3.2049 | 2.76362 | -0.442037 | -1.40449 | 0.92563 | |
c3_28 | 3 | 28 | 56 | 5145.33 | 1.64073 | 1.16705 | N | 3.2049 | 1070.3 | -0.389462 | -1.35191 | 0.978205 | A subset of the hcp. |
l3_29 | 3 | 29 | 41.0122 | 2880 | 1.71225 | 1.17823 | Y | 3.23865 | 592.839 | -0.574752 | -1.49171 | 0.906712 | |
c3_29 | 3 | 29 | 2 | 6.67089 | 1.66772 | 1.14848 | N | 3.23865 | 1.37318 | -0.460328 | -1.37728 | 1.02114 | |
l3_30 | 3 | 30 | 8.48528 | 126 | 1.75 | 1.17657 | Y | 3.27126 | 25.6782 | -0.669468 | -1.54292 | 0.921637 | |
c3_30 | 3 | 30 | 60 | 6171.41 | 1.71428 | 1.15667 | N | 3.27126 | 1257.7 | -0.579905 | -1.45335 | 1.0112 | |
l3_31 | 3 | 31 | 43.8406 | 3440 | 1.7898 | 1.17486 | Y | 3.3028 | 694.361 | -0.767138 | -1.59892 | 0.92974 | |
c3_31 | 3 | 31 | 2 | 7.00998 | 1.7525 | 1.15857 | N | 3.3028 | 1.41496 | -0.675656 | -1.50744 | 1.02122 | |
SP-QAM3_32 | 3 | 32 | 2.82843 | 15 | 1.875 | 1.21333 | Y | 3.33333 | 3 | -0.9691 | -1.76091 | 0.829944 | A cubic subset of fcc. Similar to the 4-dimensional SP-QAM packings. |
l3_32 | 3 | 32 | 22.6274 | 933 | 1.82227 | 1.17163 | Y | 3.33333 | 186.6 | -0.845204 | -1.63702 | 0.95384 | |
c3_32 | 3 | 32 | 576. | 594540. | 1.79199 | 1.16155 | N | 3.33333 | 118908. | -0.772449 | -1.56426 | 1.0266 | |
l3_33 | 3 | 33 | 46.669 | 4040 | 1.85491 | 1.16827 | Y | 3.36293 | 800.889 | -0.922322 | -1.67574 | 0.975517 | |
c3_33 | 3 | 33 | 594. | 643740. | 1.82447 | 1.16099 | N | 3.36293 | 127615. | -0.850463 | -1.60389 | 1.04738 | |
l3_34 | 3 | 34 | 24.0416 | 1089 | 1.88408 | 1.16404 | Y | 3.39164 | 214.056 | -0.990088 | -1.70659 | 1.00339 | |
c3_34 | 3 | 34 | 2 | 7.44637 | 1.86159 | 1.15643 | N | 3.39164 | 1.46367 | -0.937932 | -1.65443 | 1.05554 | |
l3_35 | 3 | 35 | 49.4975 | 4688 | 1.91347 | 1.15973 | Y | 3.41952 | 913.968 | -1.0573 | -1.73825 | 1.02885 | |
c3_35 | 3 | 35 | 2 | 7.58966 | 1.89741 | 1.15709 | N | 3.41952 | 1.47967 | -1.02071 | -1.70166 | 1.06544 | |
l3_36 | 3 | 36 | 50.9117 | 5022 | 1.9375 | 1.15462 | Y | 3.44662 | 971.387 | -1.1115 | -1.75817 | 1.06453 | |
c3_36 | 3 | 36 | 2 | 7.71811 | 1.92953 | 1.15611 | N | 3.44662 | 1.49289 | -1.0936 | -1.74027 | 1.08244 | |
l3_37 | 3 | 37 | 52.3259 | 5360 | 1.95763 | 1.14956 | Y | 3.47297 | 1028.9 | -1.1564 | -1.76999 | 1.10689 | |
c3_37 | 3 | 37 | 2 | 7.83053 | 1.95763 | 1.14956 | Y | 3.47297 | 1.50314 | -1.1564 | -1.76999 | 1.10689 | A lattice packing, identical to l3_37. |
l3_38 | 3 | 38 | 1.41421 | 3.94737 | 1.97368 | 1.14507 | Y | 3.49862 | 0.752177 | -1.19186 | -1.7735 | 1.15621 | The three points (0,0,1), (1,1,1), and (0,1,2) with all sign combinations and all permutations. |
c3_38 | 3 | 38 | 2 | 7.89474 | 1.97368 | 1.14507 | Y | 3.49862 | 1.50435 | -1.19186 | -1.7735 | 1.15621 | A lattice packing, identical to l3_38. |
l3_39 | 3 | 39 | 18.3848 | 688 | 2.0355 | 1.1665 | Y | 3.5236 | 130.17 | -1.3258 | -1.87654 | 1.10471 | |
c3_39 | 3 | 39 | 2 | 8.14201 | 2.0355 | 1.1665 | Y | 3.5236 | 1.54047 | -1.3258 | -1.87654 | 1.10471 | A lattice packing, identical to l3_39. |
l3_40 | 3 | 40 | 56.5685 | 6686 | 2.08938 | 1.1771 | Y | 3.54795 | 1256.31 | -1.43925 | -1.96007 | 1.07148 | |
c3_40 | 3 | 40 | 80 | 13372. | 2.08938 | 1.1771 | N | 3.54795 | 2512.62 | -1.43925 | -1.96007 | 1.07148 | The best known packing with M = 40 is not unique. Although this packing is not a lattice subset, it has the same En as l3_40. |
l3_41 | 3 | 41 | 57.9828 | 7182 | 2.13623 | 1.18128 | Y | 3.5717 | 1340.54 | -1.53556 | -2.02741 | 1.05328 | |
c3_41 | 3 | 41 | 82 | 14317.3 | 2.12929 | 1.17206 | N | 3.5717 | 2672.36 | -1.52143 | -2.01328 | 1.06742 | |
l3_42 | 3 | 42 | 59.397 | 7698 | 2.18197 | 1.18543 | Y | 3.59488 | 1427.59 | -1.62758 | -2.09134 | 1.03739 | |
c3_42 | 3 | 42 | 84 | 15292. | 2.16723 | 1.17256 | N | 3.59488 | 2835.89 | -1.59814 | -2.0619 | 1.06682 | |
l3_43 | 3 | 43 | 60.8112 | 8220 | 2.22282 | 1.18653 | Y | 3.61751 | 1514.85 | -1.70814 | -2.14464 | 1.03105 | |
c3_43 | 3 | 43 | 2 | 8.8026 | 2.20065 | 1.17285 | N | 3.61751 | 1.62222 | -1.6646 | -2.1011 | 1.07459 | |
l3_44 | 3 | 44 | 31.1127 | 2190 | 2.2624 | 1.18758 | Y | 3.63962 | 401.141 | -1.78477 | -2.19481 | 1.02683 | |
c3_44 | 3 | 44 | 2 | 8.92287 | 2.23072 | 1.17038 | N | 3.63962 | 1.63439 | -1.72353 | -2.13357 | 1.08808 | |
l3_45 | 3 | 45 | 63.6396 | 9316 | 2.30025 | 1.1879 | Y | 3.66124 | 1696.33 | -1.85683 | -2.24116 | 1.02547 | |
c3_45 | 3 | 45 | 471420. | 504614756449. | 2.27062 | 1.17187 | N | 3.66124 | 91884241548. | -1.80053 | -2.18485 | 1.08178 | |
l3_46 | 3 | 46 | 65.0538 | 9874 | 2.33318 | 1.18635 | Y | 3.68237 | 1787.61 | -1.91856 | -2.27788 | 1.03279 | |
c3_46 | 3 | 46 | 2 | 9.19471 | 2.29868 | 1.16953 | N | 3.68237 | 1.66463 | -1.85387 | -2.21319 | 1.09749 | |
l3_47 | 3 | 47 | 66.468 | 10464 | 2.36849 | 1.17819 | Y | 3.70306 | 1883.85 | -1.98381 | -2.3188 | 1.03503 | |
c3_47 | 3 | 47 | 94. | 20680. | 2.34043 | 1.17426 | N | 3.70306 | 3723.05 | -1.93204 | -2.26703 | 1.08681 | |
l3_48 | 3 | 48 | 33.9411 | 2766 | 2.40104 | 1.17841 | Y | 3.72331 | 495.258 | -2.04308 | -2.35439 | 1.04174 | |
c3_48 | 3 | 48 | 2 | 9.50868 | 2.37717 | 1.17589 | N | 3.72331 | 1.70255 | -1.99969 | -2.311 | 1.08514 | |
l3_49 | 3 | 49 | 69.2965 | 11680 | 2.43232 | 1.17824 | Y | 3.74314 | 2080.25 | -2.09929 | -2.38753 | 1.05009 | |
c3_49 | 3 | 49 | 10476. | 264942831. | 2.41413 | 1.17722 | N | 3.74314 | 47187270. | -2.0667 | -2.35494 | 1.08268 | |
l3_50 | 3 | 50 | 35.3553 | 3082 | 2.4656 | 1.17886 | Y | 3.76257 | 546.081 | -2.15831 | -2.42407 | 1.05425 | |
c3_50 | 3 | 50 | 2 | 9.78667 | 2.44667 | 1.17567 | N | 3.76257 | 1.73404 | -2.12484 | -2.39059 | 1.08773 | |
l3_51 | 3 | 51 | 72.1249 | 12992 | 2.4975 | 1.179 | Y | 3.78162 | 2290.38 | -2.21414 | -2.45797 | 1.06028 | |
c3_51 | 3 | 51 | 6 | 89.095 | 2.47486 | 1.16941 | N | 3.78162 | 15.7067 | -2.17459 | -2.41842 | 1.09983 | |
l3_52 | 3 | 52 | 18.3848 | 855 | 2.52959 | 1.17921 | Y | 3.80029 | 149.988 | -2.26958 | -2.49201 | 1.06544 | |
c3_52 | 3 | 52 | 78. | 15229. | 2.50312 | 1.16888 | N | 3.80029 | 2671.55 | -2.22391 | -2.44634 | 1.11111 | |
l3_53 | 3 | 53 | 74.9533 | 14392 | 2.56177 | 1.1795 | Y | 3.81861 | 2512.6 | -2.32448 | -2.52602 | 1.06993 | |
c3_53 | 3 | 53 | 6 | 91.1954 | 2.53321 | 1.16902 | N | 3.81861 | 15.9212 | -2.27579 | -2.47734 | 1.11862 | |
l3_54 | 3 | 54 | 38.1838 | 3778 | 2.59122 | 1.17889 | Y | 3.83659 | 656.485 | -2.37413 | -2.55528 | 1.0785 | |
c3_54 | 3 | 54 | 12 | 369.092 | 2.56314 | 1.16733 | N | 3.83659 | 64.1354 | -2.32681 | -2.50795 | 1.12582 | |
l3_55 | 3 | 55 | 1.41421 | 5.23636 | 2.61818 | 1.17766 | Y | 3.85424 | 0.905732 | -2.41909 | -2.5803 | 1.09064 | |
c3_55 | 3 | 55 | 330 | 281604. | 2.5859 | 1.16528 | N | 3.85424 | 48709. | -2.3652 | -2.52641 | 1.14453 | |
l3_56 | 3 | 56 | 79.196 | 16678 | 2.65912 | 1.18197 | Y | 3.87157 | 2871.88 | -2.48647 | -2.6282 | 1.07928 | |
c3_56 | 3 | 56 | 168 | 73860. | 2.61692 | 1.16579 | N | 3.87157 | 12718.4 | -2.41699 | -2.55872 | 1.14875 | |
l3_57 | 3 | 57 | 80.6102 | 17510 | 2.69468 | 1.18356 | Y | 3.88859 | 3001.94 | -2.54415 | -2.66683 | 1.07657 | |
c3_57 | 3 | 57 | 342 | 309420. | 2.64543 | 1.16543 | N | 3.88859 | 53047.5 | -2.46405 | -2.58672 | 1.15668 | |
l3_58 | 3 | 58 | 82.0244 | 18358 | 2.7286 | 1.18408 | Y | 3.90532 | 3133.84 | -2.59848 | -2.70251 | 1.07622 | |
c3_58 | 3 | 58 | 348 | 323508. | 2.67132 | 1.16286 | N | 3.90532 | 55225.2 | -2.50635 | -2.61039 | 1.16835 | |
l3_59 | 3 | 59 | 83.4386 | 19208 | 2.75898 | 1.18405 | Y | 3.92176 | 3265.2 | -2.64657 | -2.73236 | 1.08115 | |
c3_59 | 3 | 59 | 6 | 96.8136 | 2.68927 | 1.16033 | N | 3.92176 | 16.4575 | -2.53542 | -2.62121 | 1.19229 | |
buckyball3_60 | 3 | 60 | 2.35114 | 33.9443 | 6.14058 | 1 | N | 3.93793 | 5.74656 | -6.12118 | -6.1891 | -2.34138 | The buckyball, or truncated icosahedron, represents a stable carbon molecule (fullerene) and also a classical soccer ball design. |
l3_60 | 3 | 60 | 84.8528 | 20074 | 2.78806 | 1.1825 | Y | 3.93793 | 3398.4 | -2.6921 | -2.76002 | 1.08769 | |
c3_60 | 3 | 60 | 120 | 39364.9 | 2.73367 | 1.16782 | N | 3.93793 | 6664.23 | -2.60655 | -2.67448 | 1.17324 | |
l3_61 | 3 | 61 | 86.267 | 20974 | 2.81833 | 1.18246 | Y | 3.95382 | 3536.49 | -2.739 | -2.78943 | 1.09197 | |
c3_61 | 3 | 61 | 6 | 99.8646 | 2.77402 | 1.17229 | N | 3.95382 | 16.8385 | -2.67018 | -2.7206 | 1.1608 | |
l3_62 | 3 | 62 | 87.6812 | 21890 | 2.84729 | 1.18207 | Y | 3.96946 | 3676.4 | -2.78341 | -2.81669 | 1.09787 | |
c3_62 | 3 | 62 | 372. | 389436. | 2.81417 | 1.17367 | N | 3.96946 | 65405.3 | -2.73259 | -2.76587 | 1.14869 | |
l3_63 | 3 | 63 | 89.0955 | 22822 | 2.87503 | 1.18127 | Y | 3.98485 | 3818.12 | -2.82551 | -2.84199 | 1.10523 | |
c3_63 | 3 | 63 | 378. | 406620. | 2.8458 | 1.17425 | N | 3.98485 | 68027.6 | -2.78114 | -2.79762 | 1.14961 | |
l3_64 | 3 | 64 | 90.5097 | 23794 | 2.90454 | 1.18115 | Y | 4 | 3965.67 | -2.86986 | -2.86986 | 1.10954 | |
c3_64 | 3 | 64 | 48. | 6622.5 | 2.87435 | 1.17382 | N | 4 | 1103.75 | -2.82448 | -2.82448 | 1.15492 | |
l3_65 | 3 | 65 | 91.9239 | 24768 | 2.93112 | 1.18028 | Y | 4.01491 | 4112.67 | -2.90943 | -2.89327 | 1.11784 | |
c3_65 | 3 | 65 | 30. | 2620.62 | 2.91179 | 1.17654 | N | 4.01491 | 435.147 | -2.8807 | -2.86453 | 1.14657 | |
l3_66 | 3 | 66 | 46.669 | 6457 | 2.96465 | 1.18197 | Y | 4.0296 | 1068.26 | -2.95882 | -2.9268 | 1.11555 | |
c3_66 | 3 | 66 | 396. | 462188. | 2.94733 | 1.17868 | N | 4.0296 | 76465.6 | -2.93337 | -2.90136 | 1.141 | |
l3_67 | 3 | 67 | 94.7523 | 26896 | 2.99577 | 1.1827 | Y | 4.04406 | 4433.83 | -3.00417 | -2.95659 | 1.11656 | |
c3_67 | 3 | 67 | 402. | 481976. | 2.98245 | 1.1807 | N | 4.04406 | 79454.2 | -2.98482 | -2.93725 | 1.13591 | |
l3_68 | 3 | 68 | 2.82843 | 24.1765 | 3.02206 | 1.17959 | Y | 4.05831 | 3.97152 | -3.04212 | -2.97927 | 1.12426 | |
c3_68 | 3 | 68 | 1224. | 4517860. | 3.01557 | 1.18274 | N | 4.05831 | 742158. | -3.03279 | -2.96994 | 1.13359 | |
l3_69 | 3 | 69 | 97.5807 | 29090 | 3.05503 | 1.18105 | Y | 4.07235 | 4762.2 | -3.08924 | -3.01139 | 1.12209 | |
c3_69 | 3 | 69 | 2 | 12.1843 | 3.04607 | 1.17996 | N | 4.07235 | 1.99463 | -3.07648 | -2.99863 | 1.13485 | |
l3_70 | 3 | 70 | 98.9949 | 30234 | 3.0851 | 1.1815 | Y | 4.08619 | 4932.71 | -3.13178 | -3.0392 | 1.12383 | |
c3_70 | 3 | 70 | 2 | 12.3136 | 3.07839 | 1.17953 | N | 4.08619 | 2.00897 | -3.12232 | -3.02974 | 1.13329 | |
l3_71 | 3 | 71 | 100.409 | 31380 | 3.11248 | 1.18125 | Y | 4.09983 | 5102.65 | -3.17015 | -3.06309 | 1.12909 | |
c3_71 | 3 | 71 | 2 | 12.4256 | 3.10639 | 1.1795 | N | 4.09983 | 2.0205 | -3.16165 | -3.05459 | 1.13758 | |
l3_72 | 3 | 72 | 25.4558 | 2037 | 3.14352 | 1.18207 | Y | 4.11328 | 330.15 | -3.21325 | -3.09196 | 1.12898 | |
c3_72 | 3 | 72 | 2 | 12.537 | 3.13426 | 1.17947 | N | 4.11328 | 2.03196 | -3.20044 | -3.07915 | 1.14179 | |
l3_73 | 3 | 73 | 103.238 | 33828 | 3.17395 | 1.18265 | Y | 4.12655 | 5465.1 | -3.25509 | -3.11982 | 1.12951 | |
c3_73 | 3 | 73 | 2 | 12.659 | 3.16476 | 1.17806 | N | 4.12655 | 2.04514 | -3.24249 | -3.10722 | 1.14211 | |
l3_74 | 3 | 74 | 104.652 | 35066 | 3.20179 | 1.1825 | Y | 4.13964 | 5647.2 | -3.29302 | -3.14399 | 1.13336 | |
c3_74 | 3 | 74 | 148 | 69868. | 3.18974 | 1.17735 | N | 4.13964 | 11251.9 | -3.27664 | -3.12762 | 1.14974 | |
l3_75 | 3 | 75 | 106.066 | 36370 | 3.23289 | 1.18343 | Y | 4.15255 | 5838.99 | -3.335 | -3.17245 | 1.13257 | |
c3_75 | 3 | 75 | 2 | 12.8555 | 3.21387 | 1.17649 | N | 4.15255 | 2.06387 | -3.30937 | -3.14682 | 1.1582 | |
l3_76 | 3 | 76 | 53.7401 | 9419 | 3.26143 | 1.18351 | Y | 4.16529 | 1507.54 | -3.37316 | -3.19732 | 1.13502 | |
c3_76 | 3 | 76 | 76 | 18740. | 3.24446 | 1.17778 | N | 4.16529 | 2999.39 | -3.35051 | -3.17466 | 1.15767 | |
l3_77 | 3 | 77 | 9.89949 | 322.364 | 3.28942 | 1.18353 | Y | 4.17786 | 51.44 | -3.41029 | -3.22135 | 1.13797 | |
c3_77 | 3 | 77 | 2 | 13.0923 | 3.27306 | 1.17838 | N | 4.17786 | 2.08915 | -3.38863 | -3.1997 | 1.15962 | |
l3_78 | 3 | 78 | 110.309 | 40394 | 3.31969 | 1.18477 | Y | 4.19027 | 6426.64 | -3.45006 | -3.24825 | 1.13773 | |
c3_78 | 3 | 78 | 78 | 20106.4 | 3.3048 | 1.17877 | N | 4.19027 | 3198.91 | -3.43054 | -3.22872 | 1.15725 | |
l3_79 | 3 | 79 | 1.41421 | 6.68354 | 3.34177 | 1.18337 | Y | 4.20252 | 1.06024 | -3.47886 | -3.26436 | 1.14795 | |
c3_79 | 3 | 79 | 158 | 83168. | 3.33152 | 1.17676 | N | 4.20252 | 13193.4 | -3.46551 | -3.25101 | 1.16129 | |
l3_80 | 3 | 80 | 56.5685 | 10797 | 3.37406 | 1.18481 | Y | 4.21462 | 1707.87 | -3.52062 | -3.29364 | 1.14469 | |
c3_80 | 3 | 80 | 2 | 13.4167 | 3.35417 | 1.17539 | N | 4.21462 | 2.12224 | -3.49493 | -3.26795 | 1.17037 | |
l3_81 | 3 | 81 | 114.551 | 44680 | 3.40497 | 1.18575 | Y | 4.22657 | 7047.49 | -3.56022 | -3.32094 | 1.14309 | |
c3_81 | 3 | 81 | 162. | 88646.7 | 3.37779 | 1.17477 | N | 4.22657 | 13982.5 | -3.52541 | -3.28613 | 1.1779 | |
l3_82 | 3 | 82 | 57.9828 | 11551 | 3.43575 | 1.18668 | Y | 4.23837 | 1816.89 | -3.59931 | -3.34792 | 1.14153 | |
c3_82 | 3 | 82 | 2 | 13.6066 | 3.40165 | 1.1742 | N | 4.23837 | 2.14022 | -3.55598 | -3.30459 | 1.18485 | |
l3_83 | 3 | 83 | 117.38 | 47744 | 3.46523 | 1.18718 | Y | 4.25003 | 7489.21 | -3.63641 | -3.3731 | 1.14147 | |
c3_83 | 3 | 83 | 2 | 13.7028 | 3.4257 | 1.17369 | N | 4.25003 | 2.14945 | -3.58658 | -3.32327 | 1.1913 | |
l3_84 | 3 | 84 | 59.397 | 12337 | 3.49688 | 1.18846 | Y | 4.26154 | 1929.97 | -3.6759 | -3.40083 | 1.13857 | |
c3_84 | 3 | 84 | 2 | 13.822 | 3.4555 | 1.1748 | N | 4.26154 | 2.16228 | -3.62419 | -3.34912 | 1.19028 | |
l3_85 | 3 | 85 | 120.208 | 50968 | 3.5272 | 1.18928 | Y | 4.27293 | 7952.08 | -3.71338 | -3.42673 | 1.13723 | |
c3_85 | 3 | 85 | 2 | 13.9451 | 3.48627 | 1.17629 | N | 4.27293 | 2.17573 | -3.6627 | -3.37605 | 1.18791 | |
l3_86 | 3 | 86 | 60.8112 | 13155 | 3.55733 | 1.19006 | Y | 4.28418 | 2047.07 | -3.75033 | -3.45225 | 1.13599 | |
c3_86 | 3 | 86 | 172 | 104092. | 3.51852 | 1.17814 | N | 4.28418 | 16197.9 | -3.70269 | -3.40462 | 1.18362 | |
l3_87 | 3 | 87 | 1.41421 | 7.17241 | 3.58621 | 1.19046 | Y | 4.2953 | 1.11322 | -3.78544 | -3.47611 | 1.13615 | |
c3_87 | 3 | 87 | 2 | 14.1956 | 3.54891 | 1.17929 | N | 4.2953 | 2.20328 | -3.74003 | -3.4307 | 1.18156 | |
l3_88 | 3 | 88 | 31.1127 | 3500 | 3.6157 | 1.1864 | Y | 4.30629 | 541.843 | -3.82101 | -3.50058 | 1.13543 | |
c3_88 | 3 | 88 | 176. | 110897. | 3.58011 | 1.18075 | N | 4.30629 | 17168.3 | -3.77805 | -3.45761 | 1.1784 | |
l3_89 | 3 | 89 | 125.865 | 57684 | 3.64121 | 1.18588 | Y | 4.31716 | 8907.72 | -3.85154 | -3.52016 | 1.13936 | |
c3_89 | 3 | 89 | 178. | 114428. | 3.61154 | 1.1823 | N | 4.31716 | 17670.3 | -3.81601 | -3.48463 | 1.17489 | |
l3_90 | 3 | 90 | 63.6396 | 14863 | 3.66988 | 1.18605 | Y | 4.3279 | 2289.48 | -3.8856 | -3.54343 | 1.13935 | |
c3_90 | 3 | 90 | 684 | 1702049. | 3.63798 | 1.17553 | N | 4.3279 | 262182. | -3.84768 | -3.50551 | 1.17727 | |
l3_91 | 3 | 91 | 128.693 | 61224 | 3.69665 | 1.18595 | Y | 4.33853 | 9407.8 | -3.91718 | -3.56435 | 1.14144 | |
c3_91 | 3 | 91 | 10374. | 394686088. | 3.66741 | 1.17636 | N | 4.33853 | 60648209. | -3.88268 | -3.52985 | 1.17593 | |
l3_92 | 3 | 92 | 1.41421 | 7.43478 | 3.71739 | 1.18221 | Y | 4.34904 | 1.13968 | -3.94147 | -3.57813 | 1.15043 | |
c3_92 | 3 | 92 | 10488. | 406602761. | 3.69645 | 1.17752 | N | 4.34904 | 62328336. | -3.91694 | -3.5536 | 1.17496 | |
l3_93 | 3 | 93 | 131.522 | 64808 | 3.74656 | 1.18301 | Y | 4.35944 | 9910.75 | -3.97541 | -3.60171 | 1.14939 | |
c3_93 | 3 | 93 | 10602. | 418656412. | 3.72462 | 1.17826 | N | 4.35944 | 64022977. | -3.94991 | -3.5762 | 1.1749 | |
l3_94 | 3 | 94 | 66.468 | 16669 | 3.77297 | 1.18471 | Y | 4.36973 | 2543.1 | -4.00593 | -3.62198 | 1.15143 | |
c3_94 | 3 | 94 | 10716. | 431139615. | 3.7545 | 1.17942 | N | 4.36973 | 65776760. | -3.98461 | -3.60067 | 1.17274 | |
l3_95 | 3 | 95 | 134.35 | 68536 | 3.79701 | 1.18403 | Y | 4.3799 | 10431.9 | -4.0335 | -3.63946 | 1.15604 | |
c3_95 | 3 | 95 | 2 | 15.1241 | 3.78104 | 1.17575 | N | 4.3799 | 2.30205 | -4.0152 | -3.62115 | 1.17435 | |
l3_96 | 3 | 96 | 4.24264 | 68.75 | 3.81944 | 1.1831 | Y | 4.38998 | 10.4405 | -4.05909 | -3.65507 | 1.16229 | |
c3_96 | 3 | 96 | 2 | 15.2201 | 3.80503 | 1.17529 | N | 4.38998 | 2.31134 | -4.04266 | -3.63864 | 1.17872 | |
l3_97 | 3 | 97 | 137.179 | 72392 | 3.84696 | 1.18351 | Y | 4.39994 | 10968.6 | -4.09026 | -3.67639 | 1.16263 | |
c3_97 | 3 | 97 | 1746. | 11668136. | 3.82748 | 1.17454 | N | 4.39994 | 1767923. | -4.06822 | -3.65435 | 1.18467 | |
l3_98 | 3 | 98 | 69.2965 | 18594 | 3.87214 | 1.1823 | Y | 4.40981 | 2811.01 | -4.11859 | -3.695 | 1.16546 | |
c3_98 | 3 | 98 | 294. | 332692. | 3.84899 | 1.17371 | N | 4.40981 | 50295.7 | -4.09256 | -3.66896 | 1.19149 | |
l3_99 | 3 | 99 | 140.007 | 76392 | 3.89715 | 1.18246 | Y | 4.41957 | 11523.3 | -4.14656 | -3.71336 | 1.16833 | |
c3_99 | 3 | 99 | 1782. | 12286952. | 3.86927 | 1.17276 | N | 4.41957 | 1853415. | -4.11538 | -3.68217 | 1.19952 | |
l3_100 | 3 | 100 | 7.07107 | 196 | 3.92 | 1.1817 | Y | 4.42924 | 29.5009 | -4.17195 | -3.72926 | 1.17346 | |
l3_101 | 3 | 101 | 142.836 | 80488 | 3.9451 | 1.18166 | Y | 4.43881 | 12088.5 | -4.19967 | -3.74761 | 1.17595 | |
l3_102 | 3 | 102 | 144.25 | 82574 | 3.96838 | 1.18126 | Y | 4.44828 | 12375.4 | -4.22522 | -3.76389 | 1.1803 | |
l3_103 | 3 | 103 | 145.664 | 84688 | 3.99133 | 1.18083 | Y | 4.45767 | 12665.5 | -4.25026 | -3.77979 | 1.18485 | |
l3_104 | 3 | 104 | 18.3848 | 1357 | 4.01479 | 1.18053 | Y | 4.46696 | 202.524 | -4.27572 | -3.7962 | 1.1887 | |
l3_105 | 3 | 105 | 148.492 | 89072 | 4.03955 | 1.1805 | Y | 4.47616 | 13266.1 | -4.30241 | -3.81395 | 1.19102 | |
l3_106 | 3 | 106 | 149.907 | 91342 | 4.0647 | 1.18055 | Y | 4.48528 | 13576.6 | -4.32938 | -3.83208 | 1.19279 | |
l3_107 | 3 | 107 | 151.321 | 93640 | 4.08944 | 1.18052 | Y | 4.49431 | 13890.2 | -4.35573 | -3.84969 | 1.19489 | |
l3_108 | 3 | 108 | 38.1838 | 5996 | 4.11248 | 1.18018 | Y | 4.50326 | 887.654 | -4.38013 | -3.86546 | 1.19867 | |
l3_109 | 3 | 109 | 154.149 | 98312 | 4.13736 | 1.18021 | Y | 4.51212 | 14525.6 | -4.40632 | -3.88311 | 1.20038 | |
l3_110 | 3 | 110 | 15.5563 | 1006.8 | 4.16033 | 1.17986 | Y | 4.52091 | 148.466 | -4.43037 | -3.89871 | 1.20399 | |
l3_111 | 3 | 111 | 156.978 | 103128 | 4.18505 | 1.17988 | Y | 4.52961 | 15178.3 | -4.45609 | -3.91609 | 1.20565 | |
l3_112 | 3 | 112 | 158.392 | 105578 | 4.20831 | 1.1796 | Y | 4.53824 | 15509.4 | -4.48016 | -3.93189 | 1.20873 | |
l3_113 | 3 | 113 | 159.806 | 108048 | 4.23087 | 1.17921 | Y | 4.54679 | 15842.4 | -4.50339 | -3.94694 | 1.21239 | |
l3_114 | 3 | 114 | 161.22 | 110530 | 4.25246 | 1.17867 | Y | 4.55526 | 16176.2 | -4.52549 | -3.96096 | 1.21693 | |
l3_115 | 3 | 115 | 162.635 | 113032 | 4.27342 | 1.17805 | Y | 4.56366 | 16511.9 | -4.54684 | -3.97431 | 1.22198 | |
l3_116 | 3 | 116 | 1.41421 | 8.58621 | 4.2931 | 1.17729 | Y | 4.57199 | 1.252 | -4.5668 | -3.98635 | 1.2282 | |
l3_117 | 3 | 117 | 165.463 | 118504 | 4.32844 | 1.18056 | Y | 4.58024 | 17248.6 | -4.6024 | -4.01411 | 1.21854 | |
l3_118 | 3 | 118 | 166.877 | 121474 | 4.36204 | 1.18302 | Y | 4.58843 | 17649.3 | -4.63598 | -4.03994 | 1.21067 | |
l3_119 | 3 | 119 | 24.0416 | 2539.71 | 4.39397 | 1.18481 | Y | 4.59655 | 368.351 | -4.66766 | -4.06394 | 1.20449 | |
l3_120 | 3 | 120 | 21.2132 | 1991 | 4.42444 | 1.18608 | Y | 4.60459 | 288.263 | -4.69767 | -4.08636 | 1.19974 | |
l3_121 | 3 | 121 | 171.12 | 130432 | 4.45434 | 1.1872 | Y | 4.61258 | 18851.7 | -4.72692 | -4.10809 | 1.19555 | |
l3_122 | 3 | 122 | 86.267 | 33364 | 4.4832 | 1.18802 | Y | 4.62049 | 4813.92 | -4.75497 | -4.12869 | 1.19234 | |
l3_123 | 3 | 123 | 173.948 | 136512 | 4.5116 | 1.18875 | Y | 4.62834 | 19663.2 | -4.78239 | -4.14874 | 1.18955 | |
l3_124 | 3 | 124 | 43.8406 | 8724 | 4.53902 | 1.18923 | Y | 4.63613 | 1254.49 | -4.80871 | -4.16775 | 1.18766 | |
l3_125 | 3 | 125 | 176.777 | 142686 | 4.56595 | 1.18959 | Y | 4.64386 | 20483.8 | -4.8344 | -4.18621 | 1.1862 | |
l3_126 | 3 | 126 | 178.191 | 145810 | 4.59215 | 1.18979 | Y | 4.65152 | 20897.8 | -4.85925 | -4.2039 | 1.18538 | |
l3_127 | 3 | 127 | 179.605 | 148956 | 4.61765 | 1.18987 | Y | 4.65912 | 21313.9 | -4.88329 | -4.22085 | 1.18517 | |
l3_128 | 3 | 128 | 181.019 | 152142 | 4.64301 | 1.18994 | Y | 4.66667 | 21734.6 | -4.90708 | -4.23761 | 1.18502 |
References
- [Agrell14]
- E. Agrell, unpublished, 2014.
- [Sloane-web]
- http://neilsloane.com/cluster (previously http://www2.research.att.com/~njas/cluster/), accessed April 2010.