## R. H. Hardin, N. J. A. Sloane and W. D. Smith

A library of good packings, coverings and maximal volume arrangements of points on the sphere in 3 dimensions having icosahedral symmetry. The number of points ranges from 60 to 78032.

## Details

• This page is maintained by N. J. A. Sloane, AT&T Shannon Lab, 180 Park Ave, Room C233, Florham Park NJ 07932-0971 USA; Fax: 973 360 8178. Email: njasloane@gmail.com
• We consider three versions of the problem of placing points on a sphere, subject to the constraint that the points are fixed by the rotations in the icosahedral group of order 60 and determinant 1.

1. The packing problem: How should one place n points on a sphere so as to maximize the minimal distance between them? Arrangements (which we hope are close to optimal) are given for various values of n in the range 60 - 33002.

2. The covering problem: How should one place n points on a sphere so as to minimize the maximal distance of any point on the sphere from the closest one of the n points? This maximal distance is called the covering radius. Arrangements (which we hope are close to optimal) are given for various values of n in the range 72 - 78032.

3. The maximal volume problem: How should one place n points on a sphere so as to maximize the volume of their convex hull? Arrangements (which we hope are close to optimal) are given for various values of n in the range 72 - 78032.

• Note that we also have tables of packings, coverings and maximal volume arrangements without the assumption of icosahedral symmetry.
• The icosahedral arrangements are generally quite beautiful.

The plan is to have a button here that will display these packings in postscript or jpeg format. In the mean time here is a link to a postscript file showing the covering with 48002 points.

• If you use any of these arrangements, please acknowledge this source. For example, you might say something like the following.

This arrangement of points is taken from Reference [1], where Reference [1] is:

R. H. Hardin, N. J. A. Sloane and W. D. Smith, Tables of spherical codes with icosahedral symmetry, published electronically at http://NeilSloane.com/icosahedral.codes/.

• These tables are copyright R. H. Hardin, N. J. A. Sloane and W. D. Smith, 1994, 2000.

## The Spherical Codes

• A list of the parameters of the spherical codes is given below.
• Remark. The coverings and maximal volume arrangements are labeled by a pair of numbers (i,j) (this is because we often give more than one spherical code with a given number of points). The labels i and j appear at the end of the lines in the summary tables below.

For example, for coverings with 8192 points we give two solutions, labeled 18,15 and 27,3. (The meaning of these numbers is that, starting at a pentagon, for the 18,15 arrangement we go out 18 and up 15 to get to another pentagon; whereas for the 27,3 arrangement we go out 27 and up 3.)

In general the solution in which i and j are most nearly equal is the nicest.

The number of points in the arrangement, numpts, is related to the labels i,j by

numpts = 2 + 10*(i*i + i*j + j*j).

## To Obtain the Coordinates

• Once you have decided which spherical code you want to see (by consulting the list below), you can obtain the coordinates by clicking the appropriate link.
• If there are n points, you will see a list of 3n numbers, giving the X, Y, Z coordinates for the first point (one coordinate per line), then the X, Y, Z coordinates for the second point, and so on.

## Summary of Results

Copyright R. H. Hardin, N. J. A. Sloane and W. D. Smith, 1994, 2000

Icosahedral best packings

dim numpts  minimum distance (degrees)
file name

3     60  26.821267399793	ipack.3.60.txt
3     72  24.839761973495	ipack.3.72.txt
3     80  23.296830627411	ipack.3.80.txt
3     90  20.162622454037	ipack.3.90.txt
3     92  21.356520221213	ipack.3.92.txt
3    102  19.348265972666	ipack.3.102.txt
3    110  20.103610902258	ipack.3.110.txt
3    120  19.324020013440	ipack.3.120.txt
3    122  18.712547186976	ipack.3.122.txt
3    132  18.366515463401	ipack.3.132.txt
3    140  16.594595766918	ipack.3.140.txt
3    150  17.107576798500	ipack.3.150.txt
3    152  16.224875576092	ipack.3.152.txt
3    162  16.132192103096	ipack.3.162.txt
3    170  14.845631805825	ipack.3.170.txt
3    180  15.818759283357	ipack.3.180.txt
3    182  14.515037787504	ipack.3.182.txt
3    192  15.178663127905	ipack.3.192.txt
3    200  14.995766165900	ipack.3.200.txt
3    210  13.901616468609	ipack.3.210.txt
3    212  14.468601347933	ipack.3.212.txt
3    222  13.748848210062	ipack.3.222.txt
3    230  13.141471244863	ipack.3.230.txt
3    240  13.558206708313	ipack.3.240.txt
3    242  12.960811399472	ipack.3.242.txt
3    252  13.014885731316	ipack.3.252.txt
3    260  13.118320053486	ipack.3.260.txt
3    270  12.936992949341	ipack.3.270.txt
3    272  12.632589254390	ipack.3.272.txt
3    282  12.441380670551	ipack.3.282.txt
3    290  12.019802852944	ipack.3.290.txt
3    300  12.253210095538	ipack.3.300.txt
3    302  12.002153700457	ipack.3.302.txt
3    312  11.937356245086	ipack.3.312.txt
3    320  11.574132455474	ipack.3.320.txt
3    330  11.250404234692	ipack.3.330.txt
3    332  11.492736549011	ipack.3.332.txt
3    342  10.954880074672	ipack.3.342.txt
3    350  11.186073244454	ipack.3.350.txt
3    360  11.202475759607	ipack.3.360.txt
3    362  10.853379524116	ipack.3.362.txt
3    372  10.923710326921	ipack.3.372.txt
3    380  10.896774930959	ipack.3.380.txt
3    390  10.350771168337	ipack.3.390.txt
3    392  10.588105456497	ipack.3.392.txt
3    402  10.255441921484	ipack.3.402.txt
3    410  10.335990747454	ipack.3.410.txt
3    420  10.340085170221	ipack.3.420.txt
3    422  10.245952496454	ipack.3.422.txt
3    432  10.152906319671	ipack.3.432.txt
3    440   9.800188733332	ipack.3.440.txt
3    450   9.897924471803	ipack.3.450.txt
3    452   9.720242172874	ipack.3.452.txt
3    462   9.832046610472	ipack.3.462.txt
3    470   9.734951029611	ipack.3.470.txt
3    480   9.693743499783	ipack.3.480.txt
3    482   9.475514797790	ipack.3.482.txt
3    492   9.464377631080	ipack.3.492.txt
3    500   9.421280529477	ipack.3.500.txt
3    510   9.403199213488	ipack.3.510.txt
3    512   9.352484866549	ipack.3.512.txt
3    522   9.188603995896	ipack.3.522.txt
3    530   8.923007789083	ipack.3.530.txt
3    540   9.089173703552	ipack.3.540.txt
3    542   8.827487501384	ipack.3.542.txt
3    552   9.023994063134	ipack.3.552.txt
3    560   8.948738456917	ipack.3.560.txt
3    570   8.822112489524	ipack.3.570.txt
3    572   8.868353008392	ipack.3.572.txt
3    582   8.764641389411	ipack.3.582.txt
3    590   8.537473123042	ipack.3.590.txt
3    600   8.656911851651	ipack.3.600.txt
3    602   8.388566842511	ipack.3.602.txt
3    612   8.574327579758	ipack.3.612.txt
3    620   8.526174531901	ipack.3.620.txt
3    630   8.357932198265	ipack.3.630.txt
3    632   8.368141298509	ipack.3.632.txt
3    642   8.272369671428	ipack.3.642.txt
3    650   8.097757970368	ipack.3.650.txt
3    660   8.259297848601	ipack.3.660.txt
3    662   8.036732713478	ipack.3.662.txt
3    672   8.183104261167	ipack.3.672.txt
3    680   8.138449779647	ipack.3.680.txt
3    690   8.078282182133	ipack.3.690.txt
3    692   8.085160732203	ipack.3.692.txt
3    702   8.024812398053	ipack.3.702.txt
3    710   7.867104671877	ipack.3.710.txt
3    720   7.907534068479	ipack.3.720.txt
3    722   7.833385905684	ipack.3.722.txt
3    732   7.852450233754	ipack.3.732.txt
3    740   7.782608062725	ipack.3.740.txt
3    750   7.746738474534	ipack.3.750.txt
3    752   7.719749097598	ipack.3.752.txt
3    762   7.602619354131	ipack.3.762.txt
3    770   7.648348244321	ipack.3.770.txt
3    780   7.603203642241	ipack.3.780.txt
3    782   7.594138789900	ipack.3.782.txt
3    792   7.507598931586	ipack.3.792.txt
3    800   7.446111617381	ipack.3.800.txt
3    810   7.359568998868	ipack.3.810.txt
3    812   7.417588583345	ipack.3.812.txt
3    822   7.267414657010	ipack.3.822.txt
3    830   7.359568998868	ipack.3.830.txt
3    840   7.340258596199	ipack.3.840.txt
3    842   7.239237657608	ipack.3.842.txt
3    852   7.295576405790	ipack.3.852.txt
3    860   7.252094107399	ipack.3.860.txt
3    870   7.150292367811	ipack.3.870.txt
3    872   7.207210872170	ipack.3.872.txt
3    882   7.111161531515	ipack.3.882.txt
3    890   7.015391293631	ipack.3.890.txt
3    900   7.081160607937	ipack.3.900.txt
3    902   6.966982397650	ipack.3.902.txt
3    912   7.041266996741	ipack.3.912.txt
3    920   6.982786655916	ipack.3.920.txt
3    930   6.941592936659	ipack.3.930.txt
3    932   6.886023195995	ipack.3.932.txt
3    942   6.893455843106	ipack.3.942.txt
3    950   6.859923959206	ipack.3.950.txt
3    960   6.858931148297	ipack.3.960.txt
3    962   6.840410177168	ipack.3.962.txt
3    972   6.816029587706	ipack.3.972.txt
3    980   6.761432728914	ipack.3.980.txt
3    990   6.727344141830	ipack.3.990.txt
3    992   6.739280217547	ipack.3.992.txt
3   1002   6.708734725723	ipack.3.1002.txt
3   1010   6.697001625280	ipack.3.1010.txt
3   1020   6.668139364546	ipack.3.1020.txt
3   1022   6.655189253206	ipack.3.1022.txt
3   1032   6.632234038913	ipack.3.1032.txt
3   1040   6.579086467743	ipack.3.1040.txt
3   1050   6.569631227557	ipack.3.1050.txt
3   1052   6.556441235973	ipack.3.1052.txt
3   1062   6.529318784606	ipack.3.1062.txt
3   1070   6.453193916017	ipack.3.1070.txt
3   1082   6.368519008057	ipack.3.1082.txt
3   1112   6.379569013650	ipack.3.1112.txt
3   1172   6.222544555838	ipack.3.1172.txt
3   1232   6.070933155504	ipack.3.1232.txt
3   1292   5.888182023354	ipack.3.1292.txt
3   1352   5.800030334082	ipack.3.1352.txt
3   1412   5.653254619468	ipack.3.1412.txt
3   1472   5.511195187623	ipack.3.1472.txt
3   1532   5.428426857331	ipack.3.1532.txt
3   1592   5.354357408356	ipack.3.1592.txt
3   1652   5.241367455977	ipack.3.1652.txt
3   1712   5.149671812254	ipack.3.1712.txt
3   1772   5.069036890203	ipack.3.1772.txt
3   1832   4.975849802333	ipack.3.1832.txt
3   1892   4.912180491172	ipack.3.1892.txt
3   1952   4.819386540432	ipack.3.1952.txt
3   2012   4.753743181536	ipack.3.2012.txt
3   2040   4.716958335385	ipack.3.2040.txt
3   2052   4.715185795041	ipack.3.2052.txt
3   2060   4.699424624170	ipack.3.2060.txt
3   2072   4.695563082220	ipack.3.2072.txt
3   4112   3.337626352889	ipack.3.4112.txt
3   8192   2.370035652160	ipack.3.8192.txt
3  32762   1.186027893845	ipack.3.32762.txt
3  33002   1.180639969838	ipack.3.33002.txt

Icosahedral best coverings

dim numpts  deepest hole (or covering radius, in degrees)

labels        file name
i,j

3     72  15.144532085111 2,1 	icover.3.72.2.1.txt
3     92  13.676297201544 3,0 	icover.3.92.3.0.txt
3    122  11.685637449452 2,2 	icover.3.122.2.2.txt
3    132  11.216593169250 3,1 	icover.3.132.3.1.txt
3    162  10.193460547112 4,0 	icover.3.162.4.0.txt
3    192   9.246213683185 3,2 	icover.3.192.3.2.txt
3    212   8.838653454591 4,1 	icover.3.212.4.1.txt
3    252   8.127710176122 5,0 	icover.3.252.5.0.txt
3    272   7.760664539139 3,3 	icover.3.272.3.3.txt
3    282   7.617889900096 4,2 	icover.3.282.4.2.txt
3    312   7.272294692156 5,1 	icover.3.312.5.1.txt
3    362   6.758861417535 6,0 	icover.3.362.6.0.txt
3    372   6.622582828824 4,3 	icover.3.372.4.3.txt
3    392   6.458965609790 5,2 	icover.3.392.5.2.txt
3    432   6.170339682766 6,1 	icover.3.432.6.1.txt
3    482   5.814062961918 4,4 	icover.3.482.4.4.txt
3    492   5.752924348142 5,3 	icover.3.492.5.3.txt
3    492   5.784775851139 7,0 	icover.3.492.7.0.txt
3    522   5.595747249333 6,2 	icover.3.522.6.2.txt
3    572   5.355354417598 7,1 	icover.3.572.7.1.txt
3    612   5.154687189818 5,4 	icover.3.612.5.4.txt
3    632   5.074888764183 6,3 	icover.3.632.6.3.txt
3    642   5.056115072666 8,0 	icover.3.642.8.0.txt
3    672   4.929797462352 7,2 	icover.3.672.7.2.txt
3    732   4.729078589132 8,1 	icover.3.732.8.1.txt
3    752   4.648295848279 5,5 	icover.3.752.5.5.txt
3    762   4.617219818603 6,4 	icover.3.762.6.4.txt
3    792   4.533029350096 7,3 	icover.3.792.7.3.txt
3    812   4.490479603994 9,0 	icover.3.812.9.0.txt
3    842   4.402124515864 8,2 	icover.3.842.8.2.txt
3    912   4.218557913621 6,5 	icover.3.912.6.5.txt
3    912   4.233169088336 9,1 	icover.3.912.9.1.txt
3    932   4.174239460090 7,4 	icover.3.932.7.4.txt
3    972   4.091374894291 8,3 	icover.3.972.8.3.txt
3   1002   4.038657206107 10,0 	icover.3.1002.10.0.txt
3   1032   3.974509870333 9,2 	icover.3.1032.9.2.txt
3   1082   3.871891979419 6,6 	icover.3.1082.6.6.txt
3   1092   3.853939041670 7,5 	icover.3.1092.7.5.txt
3   1112   3.830953328855 10,1 	icover.3.1112.10.1.txt
3   1122   3.804178860903 8,4 	icover.3.1122.8.4.txt
3   1172   3.725307578617 9,3 	icover.3.1172.9.3.txt
3   1212   3.669436148858 11,0 	icover.3.1212.11.0.txt
3   1242   3.621422198143 10,2 	icover.3.1242.10.2.txt
3   1272   3.569725534206 7,6 	icover.3.1272.7.6.txt
3   1292   3.542654967391 8,5 	icover.3.1292.8.5.txt
3   1332   3.491241689736 9,4 	icover.3.1332.9.4.txt
3   1332   3.498272262557 11,1 	icover.3.1332.11.1.txt
3   1392   3.417577813358 10,3 	icover.3.1392.10.3.txt
3   1442   3.362062489397 12,0 	icover.3.1442.12.0.txt
3   1472   3.317695364929 7,7 	icover.3.1472.7.7.txt
3   1472   3.325210509179 11,2 	icover.3.1472.11.2.txt
3   1482   3.306416210909 8,6 	icover.3.1482.8.6.txt
3   1512   3.274691399950 9,5 	icover.3.1512.9.5.txt
3   1562   3.223730891645 10,4 	icover.3.1562.10.4.txt
3   1572   3.218586239818 12,1 	icover.3.1572.12.1.txt
3   1632   3.155616147025 11,3 	icover.3.1632.11.3.txt
3   1692   3.093711926789 8,7 	icover.3.1692.8.7.txt
3   1692   3.102197973127 13,0 	icover.3.1692.13.0.txt
3   1712   3.076018937021 9,6 	icover.3.1712.9.6.txt
3   1722   3.073309760621 12,2 	icover.3.1722.12.2.txt
3   1752   3.042047458937 10,5 	icover.3.1752.10.5.txt
3   1812   2.992780255006 11,4 	icover.3.1812.11.4.txt
3   1832   2.980200374028 13,1 	icover.3.1832.13.1.txt
3   1892   2.930147770804 12,3 	icover.3.1892.12.3.txt
3   1922   2.902260274250 8,8 	icover.3.1922.8.8.txt
3   1932   2.894701570066 9,7 	icover.3.1932.9.7.txt
3   1962   2.873285981385 10,6 	icover.3.1962.10.6.txt
3   1962   2.879618518442 14,0 	icover.3.1962.14.0.txt
3   1992   2.856557536608 13,2 	icover.3.1992.13.2.txt
3   2012   2.838572507835 11,5 	icover.3.2012.11.5.txt
3   2082   2.791655507280 12,4 	icover.3.2082.12.4.txt
3   2112   2.774616460782 14,1 	icover.3.2112.14.1.txt
3   2172   2.729614889549 9,8 	icover.3.2172.9.8.txt
3   2172   2.734196931924 13,3 	icover.3.2172.13.3.txt
3   2192   2.717406611262 10,7 	icover.3.2192.10.7.txt
3   2232   2.693850201666 11,6 	icover.3.2232.11.6.txt
3   2252   2.686837347164 15,0 	icover.3.2252.15.0.txt
3   2282   2.668138043779 14,2 	icover.3.2282.14.2.txt
3   2292   2.659385677717 12,5 	icover.3.2292.12.5.txt
3   2372   2.615107796463 13,4 	icover.3.2372.13.4.txt
3   2412   2.595513341479 15,1 	icover.3.2412.15.1.txt
3   2432   2.579274217401 9,9 	icover.3.2432.9.9.txt
3   2442   2.573968426865 10,8 	icover.3.2442.10.8.txt
3   2472   2.558841875674 11,7 	icover.3.2472.11.7.txt
3   2472   2.562415005674 14,3 	icover.3.2472.14.3.txt
3   2522   2.534187181036 12,6 	icover.3.2522.12.6.txt
3   2562   2.518246302211 16,0 	icover.3.2562.16.0.txt
3   2592   2.500582676677 13,5 	icover.3.2592.13.5.txt
3   2592   2.502876199535 15,2 	icover.3.2592.15.2.txt
3   2682   2.459012620380 14,4 	icover.3.2682.14.4.txt
3   2712   2.442145526900 10,9 	icover.3.2712.10.9.txt
3   2732   2.433380291879 11,8 	icover.3.2732.11.8.txt
3   2732   2.438092954390 16,1 	icover.3.2732.16.1.txt
3   2772   2.416391788693 12,7 	icover.3.2772.12.7.txt
3   2792   2.410654819470 15,3 	icover.3.2792.15.3.txt
3   2832   2.391387928112 13,6 	icover.3.2832.13.6.txt
3   2892   2.369561413480 17,0 	icover.3.2892.17.0.txt
3   2912   2.359001520995 14,5 	icover.3.2912.14.5.txt
3   2922   2.356776129106 16,2 	icover.3.2922.16.2.txt
3   3002   2.320969691511 10,10 	icover.3.3002.10.10.txt
3   3012   2.317097689455 11,9 	icover.3.3012.11.9.txt
3   3012   2.320099127117 15,4 	icover.3.3012.15.4.txt
3   3042   2.306023037552 12,8 	icover.3.3042.12.8.txt
3   3072   2.298648556024 17,1 	icover.3.3072.17.1.txt
3   3092   2.287905990245 13,7 	icover.3.3092.13.7.txt
3   3132   2.275654652283 16,3 	icover.3.3132.16.3.txt
3   3162   2.263060249423 14,6 	icover.3.3162.14.6.txt
3   3242   2.237453828613 18,0 	icover.3.3242.18.0.txt
3   3252   2.232090438109 15,5 	icover.3.3252.15.5.txt
3   3272   2.226705320548 17,2 	icover.3.3272.17.2.txt
3   3312   2.209428219014 11,10 	icover.3.3312.11.10.txt
3   3332   2.202914984094 12,9 	icover.3.3332.12.9.txt
3   3362   2.195734837706 16,4 	icover.3.3362.16.4.txt
3   3372   2.190267930059 13,8 	icover.3.3372.13.8.txt
3   3432   2.171579881476 14,7 	icover.3.3432.14.7.txt
3   3432   2.174271529630 18,1 	icover.3.3432.18.1.txt
3   3492   2.154813888078 17,3 	icover.3.3492.17.3.txt
3   3512   2.147226270662 15,6 	icover.3.3512.15.6.txt
3   3612   2.117750682729 16,5 	icover.3.3612.16.5.txt
3   3612   2.119297750869 19,0 	icover.3.3612.19.0.txt
3   3632   2.109686870986 11,11 	icover.3.3632.11.11.txt
3   3642   2.106778060218 12,10 	icover.3.3642.12.10.txt
3   3642   2.110175634237 18,2 	icover.3.3642.18.2.txt
3   3672   2.098426491729 13,9 	icover.3.3672.13.9.txt
3   3722   2.084735990702 14,8 	icover.3.3722.14.8.txt
3   3732   2.083793336434 17,4 	icover.3.3732.17.4.txt
3   3792   2.065873127601 15,7 	icover.3.3792.15.7.txt
3   3812   2.062647863476 19,1 	icover.3.3812.19.1.txt
3   3872   2.046038776330 18,3 	icover.3.3872.18.3.txt
3   3882   2.042217702056 16,6 	icover.3.3882.16.6.txt
3   3972   2.017185890981 12,11 	icover.3.3972.12.11.txt
3   3992   2.012218084622 13,10 	icover.3.3992.13.10.txt
3   3992   2.014256907819 17,5 	icover.3.3992.17.5.txt
3   4002   2.012994074021 20,0 	icover.3.4002.20.0.txt
3   4032   2.002553214516 14,9 	icover.3.4032.14.9.txt
3   4032   2.005186402927 19,2 	icover.3.4032.19.2.txt
3   4092   1.988228662936 15,8 	icover.3.4092.15.8.txt
3   4122   1.982533942866 18,4 	icover.3.4122.18.4.txt
3   4172   1.969474459217 16,7 	icover.3.4172.16.7.txt
3   4212   1.961913799540 20,1 	icover.3.4212.20.1.txt
3   4272   1.946647942289 17,6 	icover.3.4272.17.6.txt
3   4272   1.947624805253 19,3 	icover.3.4272.19.3.txt
3   4322   1.933658161783 12,12 	icover.3.4322.12.12.txt
3   4332   1.931415123400 13,11 	icover.3.4332.13.11.txt
3   4362   1.924961862026 14,10 	icover.3.4362.14.10.txt
3   4392   1.920173984113 18,5 	icover.3.4392.18.5.txt
3   4412   1.914368254712 15,9 	icover.3.4412.15.9.txt
3   4412   1.916844661950 21,0 	icover.3.4412.21.0.txt
3   4442   1.910110821950 20,2 	icover.3.4442.20.2.txt
3   4482   1.899723818959 16,8 	icover.3.4482.16.8.txt
3   4532   1.890519597739 19,4 	icover.3.4532.19.4.txt
3   4572   1.881274549723 17,7 	icover.3.4572.17.7.txt
3   4632   1.870551478387 21,1 	icover.3.4632.21.1.txt
3   4682   1.859343469967 18,6 	icover.3.4682.18.6.txt
3   4692   1.855708148791 13,12 	icover.3.4692.13.12.txt
3   4692   1.858170711336 20,3 	icover.3.4692.20.3.txt
3   4712   1.851830407685 14,11 	icover.3.4712.14.11.txt
3   4752   1.844278308492 15,10 	icover.3.4752.15.10.txt
3   4812   1.833066426224 16,9 	icover.3.4812.16.9.txt
3   4812   1.834304363912 19,5 	icover.3.4812.19.5.txt
3   4842   1.829461011271 22,0 	icover.3.4842.22.0.txt
3   4872   1.823612933411 21,2 	icover.3.4872.21.2.txt
3   4892   1.818333020904 17,8 	icover.3.4892.17.8.txt
3   4962   1.806557632344 20,4 	icover.3.4962.20.4.txt
3   4992   1.800316761495 18,7 	icover.3.4992.18.7.txt
3   5072   1.784740014606 13,13 	icover.3.5072.13.13.txt
3   5072   1.787312335561 22,1 	icover.3.5072.22.1.txt
3   5082   1.782975472798 14,12 	icover.3.5082.14.12.txt
3   5112   1.777885600118 15,11 	icover.3.5112.15.11.txt
3   5112   1.779310310330 19,6 	icover.3.5112.19.6.txt
3   5132   1.776515099717 21,3 	icover.3.5132.21.3.txt
3   5162   1.769523657985 16,10 	icover.3.5162.16.10.txt
3   5232   1.757933964293 17,9 	icover.3.5232.17.9.txt
3   5252   1.755639722914 20,5 	icover.3.5252.20.5.txt
3   5292   1.749696685504 23,0 	icover.3.5292.23.0.txt
3   5322   1.743279127099 18,8 	icover.3.5322.18.8.txt
3   5322   1.744585656368 22,2 	icover.3.5322.22.2.txt
3   5412   1.729648442054 21,4 	icover.3.5412.21.4.txt
3   5432   1.725781837879 19,7 	icover.3.5432.19.7.txt
3   5472   1.718159484815 14,13 	icover.3.5472.14.13.txt
3   5492   1.715075232571 15,12 	icover.3.5492.15.12.txt
3   5532   1.709063208448 16,11 	icover.3.5532.16.11.txt
3   5532   1.711160016297 23,1 	icover.3.5532.23.1.txt
3   5562   1.705703050336 20,6 	icover.3.5562.20.6.txt
3   5592   1.700124308904 17,10 	icover.3.5592.17.10.txt
3   5592   1.701687873242 22,3 	icover.3.5592.22.3.txt
3   5672   1.688345875312 18,9 	icover.3.5672.18.9.txt
3   5712   1.683326439481 21,5 	icover.3.5712.21.5.txt
3   5762   1.676596846619 24,0 	icover.3.5762.24.0.txt
3   5772   1.673891352565 19,8 	icover.3.5772.19.8.txt
3   5792   1.672104094657 23,2 	icover.3.5792.23.2.txt
3   5882   1.657117551926 14,14 	icover.3.5882.14.14.txt
3   5882   1.658949422434 22,4 	icover.3.5882.22.4.txt
3   5892   1.655703736863 15,13 	icover.3.5892.15.13.txt
3   5892   1.656964997043 20,7 	icover.3.5892.20.7.txt
3   5922   1.651617711575 16,12 	icover.3.5922.16.12.txt
3   5972   1.644902119021 17,11 	icover.3.5972.17.11.txt
3   6012   1.641227096458 24,1 	icover.3.6012.24.1.txt
3   6032   1.637797846484 21,6 	icover.3.6032.21.6.txt
3   6042   1.635577744593 18,10 	icover.3.6042.18.10.txt
3   6072   1.632872144303 23,3 	icover.3.6072.23.3.txt
3   6132   1.623753111604 19,9 	icover.3.6132.19.9.txt
3   6192   1.616639079196 22,5 	icover.3.6192.22.5.txt
3   6242   1.609583562972 20,8 	icover.3.6242.20.8.txt
3   6252   1.609359748756 25,0 	icover.3.6252.25.0.txt
3   6282   1.605389514175 24,2 	icover.3.6282.24.2.txt
3   6312   1.599589033694 15,14 	icover.3.6312.15.14.txt
3   6332   1.597095309698 16,13 	icover.3.6332.16.13.txt
3   6372   1.592230474801 17,12 	icover.3.6372.17.12.txt
3   6372   1.593254747857 21,7 	icover.3.6372.21.7.txt
3   6372   1.593745581155 23,4 	icover.3.6372.23.4.txt
3   6432   1.584992376426 18,11 	icover.3.6432.18.11.txt
3   6512   1.575433582261 19,10 	icover.3.6512.19.10.txt
3   6512   1.576782082446 25,1 	icover.3.6512.25.1.txt
3   6522   1.574973101230 22,6 	icover.3.6522.22.6.txt
3   6572   1.569375530683 24,3 	icover.3.6572.24.3.txt
3   6612   1.563667917592 20,9 	icover.3.6612.20.9.txt
3   6692   1.554955846809 23,5 	icover.3.6692.23.5.txt
3   6732   1.549841115865 21,8 	icover.3.6732.21.8.txt
3   6752   1.546527850635 15,15 	icover.3.6752.15.15.txt
3   6762   1.545377924235 16,14 	icover.3.6762.16.14.txt
3   6762   1.547307332801 26,0 	icover.3.6762.26.0.txt
3   6792   1.542048496457 17,13 	icover.3.6792.17.13.txt
3   6792   1.543781689508 25,2 	icover.3.6792.25.2.txt
3   6842   1.536574621638 18,12 	icover.3.6842.18.12.txt
3   6872   1.534121761809 22,7 	icover.3.6872.22.7.txt
3   6882   1.533426267038 24,4 	icover.3.6882.24.4.txt
3   6912   1.528963079847 19,11 	icover.3.6912.19.11.txt
3   7002   1.519288003357 20,10 	icover.3.7002.20.10.txt
3   7682   1.449774552065 16,16 	icover.3.7682.16.16.txt
3   8192   1.403881943807 18,15 	icover.3.8192.18.15.txt
3   8192   1.405300351294 27,3 	icover.3.8192.27.3.txt
3   8672   1.364413944909 17,17 	icover.3.8672.17.17.txt
3   9722   1.288545767384 18,18 	icover.3.9722.18.18.txt
3  10832   1.220670083820 19,19 	icover.3.10832.19.19.txt
3  12002   1.159587155975 20,20 	icover.3.12002.20.20.txt
3  13232   1.104325930573 21,21 	icover.3.13232.21.21.txt
3  14522   1.054091991326 22,22 	icover.3.14522.22.22.txt
3  15872   1.008229191699 23,23 	icover.3.15872.23.23.txt
3  17282   0.966190795787 24,24 	icover.3.17282.24.24.txt
3  18752   0.927517598993 25,25 	icover.3.18752.25.25.txt
3  20282   0.891821076004 26,26 	icover.3.20282.26.26.txt
3  21872   0.858770306968 27,27 	icover.3.21872.27.27.txt
3  23522   0.828081659327 28,28 	icover.3.23522.28.28.txt
3  25232   0.799510654132 29,29 	icover.3.25232.29.29.txt
3  27002   0.772845388796 30,30 	icover.3.27002.30.30.txt
3  28832   0.747901370396 31,31 	icover.3.28832.31.31.txt
3  30722   0.724517158473 32,32 	icover.3.30722.32.32.txt
3  32672   0.702550866516 33,33 	icover.3.32672.33.33.txt
3  34682   0.681877319450 34,34 	icover.3.34682.34.34.txt
3  36752   0.662385678203 35,35 	icover.3.36752.35.35.txt
3  38882   0.643977395866 36,36 	icover.3.38882.36.36.txt
3  41072   0.626564612737 37,37 	icover.3.41072.37.37.txt
3  43322   0.610068684751 38,38 	icover.3.43322.38.38.txt
3  45632   0.594419060597 39,39 	icover.3.45632.39.39.txt
3  48002   0.579552240743 40,40 	icover.3.48002.40.40.txt
3  50432   0.565410925600 41,41 	icover.3.50432.41.41.txt
3  52922   0.551943280335 42,42 	icover.3.52922.42.42.txt
3  55472   0.539102268802 43,43 	icover.3.55472.43.43.txt
3  58082   0.526845169006 44,44 	icover.3.58082.44.44.txt
3  60752   0.515133033200 45,45 	icover.3.60752.45.45.txt
3  63482   0.503930303949 46,46 	icover.3.63482.46.46.txt
3  66272   0.493204458195 47,47 	icover.3.66272.47.47.txt
3  69122   0.482936940333 48,48 	icover.3.69122.48.48.txt
3  72032   0.473076997075 49,49 	icover.3.72032.49.49.txt
3  75002   0.463611644179 50,50 	icover.3.75002.50.50.txt
3  78032   0.454517612602 51,51 	icover.3.78032.51.51.txt

Icosahedral maximal volume arrangements

dim numpts  volume         labels       file name
i,j

3     72   3.875747022839 2,1  	ivol.3.72.2.1.txt
3     92   3.942450828068 3,0  	ivol.3.92.3.0.txt
3    122   4.002559429605 2,2  	ivol.3.122.2.2.txt
3    132   4.016465892972 3,1  	ivol.3.132.3.1.txt
3    162   4.048080191657 4,0  	ivol.3.162.4.0.txt
3    192   4.069982183555 3,2  	ivol.3.192.3.2.txt
3    212   4.081099952679 4,1  	ivol.3.212.4.1.txt
3    252   4.098103594953 5,0  	ivol.3.252.5.0.txt
3    272   4.104769803539 3,3  	ivol.3.272.3.3.txt
3    282   4.107730653999 4,2  	ivol.3.282.4.2.txt
3    312   4.115486214613 5,1  	ivol.3.312.5.1.txt
3    362   4.125577157078 6,0  	ivol.3.362.6.0.txt
3    372   4.127283783967 4,3  	ivol.3.372.4.3.txt
3    392   4.130408635626 5,2  	ivol.3.392.5.2.txt
3    432   4.135796663594 6,1  	ivol.3.432.6.1.txt
3    482   4.141288207632 4,4  	ivol.3.482.4.4.txt
3    492   4.142244400030 7,0  	ivol.3.492.5.3.txt
3    492   4.142250389453 5,3  	ivol.3.492.7.0.txt
3    522   4.144917182979 6,2  	ivol.3.522.6.2.txt
3    572   4.148743468148 7,1  	ivol.3.572.7.1.txt
3    612   4.151360421220 5,4  	ivol.3.612.5.4.txt
3    632   4.152541890396 6,3  	ivol.3.632.6.3.txt
3    642   4.153102625819 8,0  	ivol.3.642.8.0.txt
3    672   4.154694891950 7,2  	ivol.3.672.7.2.txt
3    732   4.157484909377 8,1  	ivol.3.732.8.1.txt
3    752   4.158319096801 5,5  	ivol.3.752.5.5.txt
3    762   4.158718124654 6,4  	ivol.3.762.6.4.txt
3    792   4.159855017314 7,3  	ivol.3.792.7.3.txt
3    812   4.160565270208 9,0  	ivol.3.812.9.0.txt
3    842   4.161570488526 8,2  	ivol.3.842.8.2.txt
3    912   4.163657063456 9,1  	ivol.3.912.6.5.txt
3    912   4.163658855168 6,5  	ivol.3.912.9.1.txt
3    932   4.164197243230 7,4  	ivol.3.932.7.4.txt
3    972   4.165207785894 8,3  	ivol.3.972.8.3.txt
3   1002   4.165912301638 10,0  	ivol.3.1002.10.0.txt
3   1032   4.166577151269 9,2  	ivol.3.1032.9.2.txt
3   1082   4.167603798408 6,6  	ivol.3.1082.6.6.txt
3   1092   4.167797531105 7,5  	ivol.3.1092.7.5.txt
3   1112   4.168173622895 10,1  	ivol.3.1112.10.1.txt
3   1122   4.168358075216 8,4  	ivol.3.1122.8.4.txt
3   1172   4.169228724984 9,3  	ivol.3.1172.9.3.txt
3   1212   4.169873317662 11,0  	ivol.3.1212.11.0.txt
3   1242   4.170330140249 10,2  	ivol.3.1242.10.2.txt
3   1272   4.170765988844 7,6  	ivol.3.1272.7.6.txt
3   1292   4.171044667943 8,5  	ivol.3.1292.8.5.txt
3   1332   4.171576448259 11,1  	ivol.3.1332.9.4.txt
3   1332   4.171576985908 9,4  	ivol.3.1332.11.1.txt
3   1392   4.172318232391 10,3  	ivol.3.1392.10.3.txt
3   1442   4.172888718212 12,0  	ivol.3.1442.12.0.txt
3   1472   4.173212738761 11,2  	ivol.3.1472.7.7.txt
3   1472   4.173213355837 7,7  	ivol.3.1472.11.2.txt
3   1482   4.173318352699 8,6  	ivol.3.1482.8.6.txt
3   1512   4.173625030156 9,5  	ivol.3.1512.9.5.txt
3   1562   4.174110038095 10,4  	ivol.3.1562.10.4.txt
3   1572   4.174203027057 12,1  	ivol.3.1572.12.1.txt
3   1632   4.174739215607 11,3  	ivol.3.1632.11.3.txt
3   1692   4.175237025705 13,0  	ivol.3.1692.8.7.txt
3   1692   4.175237584812 8,7  	ivol.3.1692.13.0.txt
3   1712   4.175395769446 9,6  	ivol.3.1712.9.6.txt
3   1722   4.175473110247 12,2  	ivol.3.1722.12.2.txt
3   1752   4.175701326114 10,5  	ivol.3.1752.10.5.txt
3   1812   4.176134412679 11,4  	ivol.3.1812.11.4.txt
3   1832   4.176272285613 13,1  	ivol.3.1832.13.1.txt
3   1892   4.176669199073 12,3  	ivol.3.1892.12.3.txt
3   1922   4.176858615759 8,8  	ivol.3.1922.8.8.txt
3   1932   4.176920324800 9,7  	ivol.3.1932.9.7.txt
3   1962   4.177101336378 14,0  	ivol.3.1962.10.6.txt
3   1962   4.177101684683 10,6  	ivol.3.1962.14.0.txt
3   1992   4.177277353637 13,2  	ivol.3.1992.13.2.txt
3   2012   4.177391953311 11,5  	ivol.3.2012.11.5.txt
3   2082   4.177774942948 12,4  	ivol.3.2082.12.4.txt
3   2112   4.177931195503 14,1  	ivol.3.2112.14.1.txt
3   2172   4.178231131700 13,3  	ivol.3.2172.9.8.txt
3   2172   4.178231380961 9,8  	ivol.3.2172.13.3.txt
3   2192   4.178327654924 10,7  	ivol.3.2192.10.7.txt
3   2232   4.178515035143 11,6  	ivol.3.2232.11.6.txt
3   2252   4.178606008529 15,0  	ivol.3.2252.15.0.txt
3   2282   4.178739881475 14,2  	ivol.3.2282.14.2.txt
3   2292   4.178783860405 12,5  	ivol.3.2292.12.5.txt
3   2372   4.179121166052 13,4  	ivol.3.2372.13.4.txt
3   2412   4.179281355733 15,1  	ivol.3.2412.15.1.txt
3   2432   4.179359791899 9,9  	ivol.3.2432.9.9.txt
3   2442   4.179398385807 10,8  	ivol.3.2442.10.8.txt
3   2472   4.179512130515 14,3  	ivol.3.2472.11.7.txt
3   2472   4.179512296815 11,7  	ivol.3.2472.14.3.txt
3   2522   4.179696134619 12,6  	ivol.3.2522.12.6.txt
3   2562   4.179837896345 16,0  	ivol.3.2562.16.0.txt
3   2592   4.179941503356 15,2  	ivol.3.2592.13.5.txt
3   2592   4.179941607173 13,5  	ivol.3.2592.15.2.txt
3   2682   4.180238407384 14,4  	ivol.3.2682.14.4.txt
3   2712   4.180333116208 10,9  	ivol.3.2712.10.9.txt
3   2732   4.180394799486 16,1  	ivol.3.2732.11.8.txt
3   2732   4.180394994162 11,8  	ivol.3.2732.16.1.txt
3   2772   4.180516074855 12,7  	ivol.3.2772.12.7.txt
3   2792   4.180575202215 15,3  	ivol.3.2792.15.3.txt
3   2832   4.180691290482 13,6  	ivol.3.2832.13.6.txt
3   2892   4.180859142520 17,0  	ivol.3.2892.17.0.txt
3   2912   4.180913691787 14,5  	ivol.3.2912.14.5.txt
3   2922   4.180940565325 16,2  	ivol.3.2922.16.2.txt
3   3002   4.181149856000 10,10  	ivol.3.3002.10.10.txt
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