## 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: njas@research.att.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://www.research.att.com/~njas/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 buttons here.
• 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.
• Enter number of points:   ( Optional labels i, j : )
Packing
Covering
Max volume arrangement
• Errors: if you entered an invalid number of points then you will simply receive an empty file. Check the number of points on the list below and try again.
• Downloading the coordinates in this way can be slow (the largest web pages are about 4 megs). An alternative method which is faster is described in the instructions that can be found at the end of this file.

## 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)

3     60  26.821267399793
3     72  24.839761973495
3     80  23.296830627411
3     90  20.162622454037
3     92  21.356520221213
3    102  19.348265972666
3    110  20.103610902258
3    120  19.324020013440
3    122  18.712547186976
3    132  18.366515463401
3    140  16.594595766918
3    150  17.107576798500
3    152  16.224875576092
3    162  16.132192103096
3    170  14.845631805825
3    180  15.818759283357
3    182  14.515037787504
3    192  15.178663127905
3    200  14.995766165900
3    210  13.901616468609
3    212  14.468601347933
3    222  13.748848210062
3    230  13.141471244863
3    240  13.558206708313
3    242  12.960811399472
3    252  13.014885731316
3    260  13.118320053486
3    270  12.936992949341
3    272  12.632589254390
3    282  12.441380670551
3    290  12.019802852944
3    300  12.253210095538
3    302  12.002153700457
3    312  11.937356245086
3    320  11.574132455474
3    330  11.250404234692
3    332  11.492736549011
3    342  10.954880074672
3    350  11.186073244454
3    360  11.202475759607
3    362  10.853379524116
3    372  10.923710326921
3    380  10.896774930959
3    390  10.350771168337
3    392  10.588105456497
3    402  10.255441921484
3    410  10.335990747454
3    420  10.340085170221
3    422  10.245952496454
3    432  10.152906319671
3    440   9.800188733332
3    450   9.897924471803
3    452   9.720242172874
3    462   9.832046610472
3    470   9.734951029611
3    480   9.693743499783
3    482   9.475514797790
3    492   9.464377631080
3    500   9.421280529477
3    510   9.403199213488
3    512   9.352484866549
3    522   9.188603995896
3    530   8.923007789083
3    540   9.089173703552
3    542   8.827487501384
3    552   9.023994063134
3    560   8.948738456917
3    570   8.822112489524
3    572   8.868353008392
3    582   8.764641389411
3    590   8.537473123042
3    600   8.656911851651
3    602   8.388566842511
3    612   8.574327579758
3    620   8.526174531901
3    630   8.357932198265
3    632   8.368141298509
3    642   8.272369671428
3    650   8.097757970368
3    660   8.259297848601
3    662   8.036732713478
3    672   8.183104261167
3    680   8.138449779647
3    690   8.078282182133
3    692   8.085160732203
3    702   8.024812398053
3    710   7.867104671877
3    720   7.907534068479
3    722   7.833385905684
3    732   7.852450233754
3    740   7.782608062725
3    750   7.746738474534
3    752   7.719749097598
3    762   7.602619354131
3    770   7.648348244321
3    780   7.603203642241
3    782   7.594138789900
3    792   7.507598931586
3    800   7.446111617381
3    810   7.359568998868
3    812   7.417588583345
3    822   7.267414657010
3    830   7.359568998868
3    840   7.340258596199
3    842   7.239237657608
3    852   7.295576405790
3    860   7.252094107399
3    870   7.150292367811
3    872   7.207210872170
3    882   7.111161531515
3    890   7.015391293631
3    900   7.081160607937
3    902   6.966982397650
3    912   7.041266996741
3    920   6.982786655916
3    930   6.941592936659
3    932   6.886023195995
3    942   6.893455843106
3    950   6.859923959206
3    960   6.858931148297
3    962   6.840410177168
3    972   6.816029587706
3    980   6.761432728914
3    990   6.727344141830
3    992   6.739280217547
3   1002   6.708734725723
3   1010   6.697001625280
3   1020   6.668139364546
3   1022   6.655189253206
3   1032   6.632234038913
3   1040   6.579086467743
3   1050   6.569631227557
3   1052   6.556441235973
3   1062   6.529318784606
3   1070   6.453193916017
3   1082   6.368519008057
3   1112   6.379569013650
3   1172   6.222544555838
3   1232   6.070933155504
3   1292   5.888182023354
3   1352   5.800030334082
3   1412   5.653254619468
3   1472   5.511195187623
3   1532   5.428426857331
3   1592   5.354357408356
3   1652   5.241367455977
3   1712   5.149671812254
3   1772   5.069036890203
3   1832   4.975849802333
3   1892   4.912180491172
3   1952   4.819386540432
3   2012   4.753743181536
3   2040   4.716958335385
3   2052   4.715185795041
3   2060   4.699424624170
3   2072   4.695563082220
3   4112   3.337626352889
3   8192   2.370035652160
3  32762   1.186027893845
3  33002   1.180639969838

Icosahedral best coverings

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

labels
i,j
3     72  15.144532085111 2,1
3     92  13.676297201544 3,0
3    122  11.685637449452 2,2
3    132  11.216593169250 3,1
3    162  10.193460547112 4,0
3    192   9.246213683185 3,2
3    212   8.838653454591 4,1
3    252   8.127710176122 5,0
3    272   7.760664539139 3,3
3    282   7.617889900096 4,2
3    312   7.272294692156 5,1
3    362   6.758861417535 6,0
3    372   6.622582828824 4,3
3    392   6.458965609790 5,2
3    432   6.170339682766 6,1
3    482   5.814062961918 4,4
3    492   5.752924348142 5,3
3    492   5.784775851139 7,0
3    522   5.595747249333 6,2
3    572   5.355354417598 7,1
3    612   5.154687189818 5,4
3    632   5.074888764183 6,3
3    642   5.056115072666 8,0
3    672   4.929797462352 7,2
3    732   4.729078589132 8,1
3    752   4.648295848279 5,5
3    762   4.617219818603 6,4
3    792   4.533029350096 7,3
3    812   4.490479603994 9,0
3    842   4.402124515864 8,2
3    912   4.218557913621 6,5
3    912   4.233169088336 9,1
3    932   4.174239460090 7,4
3    972   4.091374894291 8,3
3   1002   4.038657206107 10,0
3   1032   3.974509870333 9,2
3   1082   3.871891979419 6,6
3   1092   3.853939041670 7,5
3   1112   3.830953328855 10,1
3   1122   3.804178860903 8,4
3   1172   3.725307578617 9,3
3   1212   3.669436148858 11,0
3   1242   3.621422198143 10,2
3   1272   3.569725534206 7,6
3   1292   3.542654967391 8,5
3   1332   3.491241689736 9,4
3   1332   3.498272262557 11,1
3   1392   3.417577813358 10,3
3   1442   3.362062489397 12,0
3   1472   3.317695364929 7,7
3   1472   3.325210509179 11,2
3   1482   3.306416210909 8,6
3   1512   3.274691399950 9,5
3   1562   3.223730891645 10,4
3   1572   3.218586239818 12,1
3   1632   3.155616147025 11,3
3   1692   3.093711926789 8,7
3   1692   3.102197973127 13,0
3   1712   3.076018937021 9,6
3   1722   3.073309760621 12,2
3   1752   3.042047458937 10,5
3   1812   2.992780255006 11,4
3   1832   2.980200374028 13,1
3   1892   2.930147770804 12,3
3   1922   2.902260274250 8,8
3   1932   2.894701570066 9,7
3   1962   2.873285981385 10,6
3   1962   2.879618518442 14,0
3   1992   2.856557536608 13,2
3   2012   2.838572507835 11,5
3   2082   2.791655507280 12,4
3   2112   2.774616460782 14,1
3   2172   2.729614889549 9,8
3   2172   2.734196931924 13,3
3   2192   2.717406611262 10,7
3   2232   2.693850201666 11,6
3   2252   2.686837347164 15,0
3   2282   2.668138043779 14,2
3   2292   2.659385677717 12,5
3   2372   2.615107796463 13,4
3   2412   2.595513341479 15,1
3   2432   2.579274217401 9,9
3   2442   2.573968426865 10,8
3   2472   2.558841875674 11,7
3   2472   2.562415005674 14,3
3   2522   2.534187181036 12,6
3   2562   2.518246302211 16,0
3   2592   2.500582676677 13,5
3   2592   2.502876199535 15,2
3   2682   2.459012620380 14,4
3   2712   2.442145526900 10,9
3   2732   2.433380291879 11,8
3   2732   2.438092954390 16,1
3   2772   2.416391788693 12,7
3   2792   2.410654819470 15,3
3   2832   2.391387928112 13,6
3   2892   2.369561413480 17,0
3   2912   2.359001520995 14,5
3   2922   2.356776129106 16,2
3   3002   2.320969691511 10,10
3   3012   2.317097689455 11,9
3   3012   2.320099127117 15,4
3   3042   2.306023037552 12,8
3   3072   2.298648556024 17,1
3   3092   2.287905990245 13,7
3   3132   2.275654652283 16,3
3   3162   2.263060249423 14,6
3   3242   2.237453828613 18,0
3   3252   2.232090438109 15,5
3   3272   2.226705320548 17,2
3   3312   2.209428219014 11,10
3   3332   2.202914984094 12,9
3   3362   2.195734837706 16,4
3   3372   2.190267930059 13,8
3   3432   2.171579881476 14,7
3   3432   2.174271529630 18,1
3   3492   2.154813888078 17,3
3   3512   2.147226270662 15,6
3   3612   2.117750682729 16,5
3   3612   2.119297750869 19,0
3   3632   2.109686870986 11,11
3   3642   2.106778060218 12,10
3   3642   2.110175634237 18,2
3   3672   2.098426491729 13,9
3   3722   2.084735990702 14,8
3   3732   2.083793336434 17,4
3   3792   2.065873127601 15,7
3   3812   2.062647863476 19,1
3   3872   2.046038776330 18,3
3   3882   2.042217702056 16,6
3   3972   2.017185890981 12,11
3   3992   2.012218084622 13,10
3   3992   2.014256907819 17,5
3   4002   2.012994074021 20,0
3   4032   2.002553214516 14,9
3   4032   2.005186402927 19,2
3   4092   1.988228662936 15,8
3   4122   1.982533942866 18,4
3   4172   1.969474459217 16,7
3   4212   1.961913799540 20,1
3   4272   1.946647942289 17,6
3   4272   1.947624805253 19,3
3   4322   1.933658161783 12,12
3   4332   1.931415123400 13,11
3   4362   1.924961862026 14,10
3   4392   1.920173984113 18,5
3   4412   1.914368254712 15,9
3   4412   1.916844661950 21,0
3   4442   1.910110821950 20,2
3   4482   1.899723818959 16,8
3   4532   1.890519597739 19,4
3   4572   1.881274549723 17,7
3   4632   1.870551478387 21,1
3   4682   1.859343469967 18,6
3   4692   1.855708148791 13,12
3   4692   1.858170711336 20,3
3   4712   1.851830407685 14,11
3   4752   1.844278308492 15,10
3   4812   1.833066426224 16,9
3   4812   1.834304363912 19,5
3   4842   1.829461011271 22,0
3   4872   1.823612933411 21,2
3   4892   1.818333020904 17,8
3   4962   1.806557632344 20,4
3   4992   1.800316761495 18,7
3   5072   1.784740014606 13,13
3   5072   1.787312335561 22,1
3   5082   1.782975472798 14,12
3   5112   1.777885600118 15,11
3   5112   1.779310310330 19,6
3   5132   1.776515099717 21,3
3   5162   1.769523657985 16,10
3   5232   1.757933964293 17,9
3   5252   1.755639722914 20,5
3   5292   1.749696685504 23,0
3   5322   1.743279127099 18,8
3   5322   1.744585656368 22,2
3   5412   1.729648442054 21,4
3   5432   1.725781837879 19,7
3   5472   1.718159484815 14,13
3   5492   1.715075232571 15,12
3   5532   1.709063208448 16,11
3   5532   1.711160016297 23,1
3   5562   1.705703050336 20,6
3   5592   1.700124308904 17,10
3   5592   1.701687873242 22,3
3   5672   1.688345875312 18,9
3   5712   1.683326439481 21,5
3   5762   1.676596846619 24,0
3   5772   1.673891352565 19,8
3   5792   1.672104094657 23,2
3   5882   1.657117551926 14,14
3   5882   1.658949422434 22,4
3   5892   1.655703736863 15,13
3   5892   1.656964997043 20,7
3   5922   1.651617711575 16,12
3   5972   1.644902119021 17,11
3   6012   1.641227096458 24,1
3   6032   1.637797846484 21,6
3   6042   1.635577744593 18,10
3   6072   1.632872144303 23,3
3   6132   1.623753111604 19,9
3   6192   1.616639079196 22,5
3   6242   1.609583562972 20,8
3   6252   1.609359748756 25,0
3   6282   1.605389514175 24,2
3   6312   1.599589033694 15,14
3   6332   1.597095309698 16,13
3   6372   1.592230474801 17,12
3   6372   1.593254747857 21,7
3   6372   1.593745581155 23,4
3   6432   1.584992376426 18,11
3   6512   1.575433582261 19,10
3   6512   1.576782082446 25,1
3   6522   1.574973101230 22,6
3   6572   1.569375530683 24,3
3   6612   1.563667917592 20,9
3   6692   1.554955846809 23,5
3   6732   1.549841115865 21,8
3   6752   1.546527850635 15,15
3   6762   1.545377924235 16,14
3   6762   1.547307332801 26,0
3   6792   1.542048496457 17,13
3   6792   1.543781689508 25,2
3   6842   1.536574621638 18,12
3   6872   1.534121761809 22,7
3   6882   1.533426267038 24,4
3   6912   1.528963079847 19,11
3   7002   1.519288003357 20,10
3   7682   1.449774552065 16,16
3   8192   1.403881943807 18,15
3   8192   1.405300351294 27,3
3   8672   1.364413944909 17,17
3   9722   1.288545767384 18,18
3  10832   1.220670083820 19,19
3  12002   1.159587155975 20,20
3  13232   1.104325930573 21,21
3  14522   1.054091991326 22,22
3  15872   1.008229191699 23,23
3  17282   0.966190795787 24,24
3  18752   0.927517598993 25,25
3  20282   0.891821076004 26,26
3  21872   0.858770306968 27,27
3  23522   0.828081659327 28,28
3  25232   0.799510654132 29,29
3  27002   0.772845388796 30,30
3  28832   0.747901370396 31,31
3  30722   0.724517158473 32,32
3  32672   0.702550866516 33,33
3  34682   0.681877319450 34,34
3  36752   0.662385678203 35,35
3  38882   0.643977395866 36,36
3  41072   0.626564612737 37,37
3  43322   0.610068684751 38,38
3  45632   0.594419060597 39,39
3  48002   0.579552240743 40,40
3  50432   0.565410925600 41,41
3  52922   0.551943280335 42,42
3  55472   0.539102268802 43,43
3  58082   0.526845169006 44,44
3  60752   0.515133033200 45,45
3  63482   0.503930303949 46,46
3  66272   0.493204458195 47,47
3  69122   0.482936940333 48,48
3  72032   0.473076997075 49,49
3  75002   0.463611644179 50,50
3  78032   0.454517612602 51,51

Icosahedral maximal volume arrangements

dim numpts  volume         labels
i,j

3     72   3.875747022839 2,1
3     92   3.942450828068 3,0
3    122   4.002559429605 2,2
3    132   4.016465892972 3,1
3    162   4.048080191657 4,0
3    192   4.069982183555 3,2
3    212   4.081099952679 4,1
3    252   4.098103594953 5,0
3    272   4.104769803539 3,3
3    282   4.107730653999 4,2
3    312   4.115486214613 5,1
3    362   4.125577157078 6,0
3    372   4.127283783967 4,3
3    392   4.130408635626 5,2
3    432   4.135796663594 6,1
3    482   4.141288207632 4,4
3    492   4.142244400030 7,0
3    492   4.142250389453 5,3
3    522   4.144917182979 6,2
3    572   4.148743468148 7,1
3    612   4.151360421220 5,4
3    632   4.152541890396 6,3
3    642   4.153102625819 8,0
3    672   4.154694891950 7,2
3    732   4.157484909377 8,1
3    752   4.158319096801 5,5
3    762   4.158718124654 6,4
3    792   4.159855017314 7,3
3    812   4.160565270208 9,0
3    842   4.161570488526 8,2
3    912   4.163657063456 9,1
3    912   4.163658855168 6,5
3    932   4.164197243230 7,4
3    972   4.165207785894 8,3
3   1002   4.165912301638 10,0
3   1032   4.166577151269 9,2
3   1082   4.167603798408 6,6
3   1092   4.167797531105 7,5
3   1112   4.168173622895 10,1
3   1122   4.168358075216 8,4
3   1172   4.169228724984 9,3
3   1212   4.169873317662 11,0
3   1242   4.170330140249 10,2
3   1272   4.170765988844 7,6
3   1292   4.171044667943 8,5
3   1332   4.171576448259 11,1
3   1332   4.171576985908 9,4
3   1392   4.172318232391 10,3
3   1442   4.172888718212 12,0
3   1472   4.173212738761 11,2
3   1472   4.173213355837 7,7
3   1482   4.173318352699 8,6
3   1512   4.173625030156 9,5
3   1562   4.174110038095 10,4
3   1572   4.174203027057 12,1
3   1632   4.174739215607 11,3
3   1692   4.175237025705 13,0
3   1692   4.175237584812 8,7
3   1712   4.175395769446 9,6
3   1722   4.175473110247 12,2
3   1752   4.175701326114 10,5
3   1812   4.176134412679 11,4
3   1832   4.176272285613 13,1
3   1892   4.176669199073 12,3
3   1922   4.176858615759 8,8
3   1932   4.176920324800 9,7
3   1962   4.177101336378 14,0
3   1962   4.177101684683 10,6
3   1992   4.177277353637 13,2
3   2012   4.177391953311 11,5
3   2082   4.177774942948 12,4
3   2112   4.177931195503 14,1
3   2172   4.178231131700 13,3
3   2172   4.178231380961 9,8
3   2192   4.178327654924 10,7
3   2232   4.178515035143 11,6
3   2252   4.178606008529 15,0
3   2282   4.178739881475 14,2
3   2292   4.178783860405 12,5
3   2372   4.179121166052 13,4
3   2412   4.179281355733 15,1
3   2432   4.179359791899 9,9
3   2442   4.179398385807 10,8
3   2472   4.179512130515 14,3
3   2472   4.179512296815 11,7
3   2522   4.179696134619 12,6
3   2562   4.179837896345 16,0
3   2592   4.179941503356 15,2
3   2592   4.179941607173 13,5
3   2682   4.180238407384 14,4
3   2712   4.180333116208 10,9
3   2732   4.180394799486 16,1
3   2732   4.180394994162 11,8
3   2772   4.180516074855 12,7
3   2792   4.180575202215 15,3
3   2832   4.180691290482 13,6
3   2892   4.180859142520 17,0
3   2912   4.180913691787 14,5
3   2922   4.180940565325 16,2
3   3002   4.181149856000 10,10
3   3012   4.181175095592 15,4
3   3012   4.181175209946 11,9
3   3042   4.181250272836 12,8
3   3072   4.181323737426 17,1
3   3092   4.181372144493 13,7
3   3132   4.181466763925 16,3
3   3162   4.181536296497 14,6
3   3242   4.181715156605 18,0
3   3252   4.181736975009 15,5
3   3272   4.181780022382 17,2
3   3312   4.181864799712 11,10
3   3332   4.181906350629 12,9
3   3362   4.181967669839 16,4
3   3372   4.181987975370 13,8
3   3432   4.182106754446 18,1
3   3432   4.182106848570 14,7
3   3492   4.182221584224 17,3
3   3512   4.182259033776 15,6
3   3612   4.182439743027 19,0
3   3612   4.182439793140 16,5
3   3632   4.182474826643 11,11
3   3642   4.182492051083 18,2
3   3642   4.182492160130 12,10
3   3672   4.182543594703 13,9
3   3722   4.182627478212 14,8
3   3732   4.182643926743 17,4
3   3792   4.182741202088 15,7
3   3812   4.182772860711 19,1
3   3872   4.182866100544 18,3
3   3882   4.182881396973 16,6
3   3972   4.183015303840 12,11
3   3992   4.183044167869 17,5
3   3992   4.183044225795 13,10
3   4002   4.183058485556 20,0
3   4032   4.183101130800 19,2
3   4032   4.183101209708 14,9
3   4092   4.183184598508 15,8
3   4122   4.183225340223 18,4
3   4172   4.183292055493 16,7
3   4212   4.183344205256 20,1
3   4272   4.183420691186 19,3
3   4272   4.183420721197 17,6
3   4322   4.183482872552 12,12
3   4332   4.183495119949 13,11
3   4362   4.183531525491 14,10
3   4392   4.183567390990 18,5
3   4412   4.183591034054 21,0
3   4412   4.183591102089 15,9
3   4442   4.183626146751 20,2
3   4482   4.183672277909 16,8
3   4532   4.183728694403 19,4
3   4572   4.183772996944 17,7
3   4632   4.183837933599 21,1
3   4682   4.183890842869 18,6
3   4692   4.183901259914 20,3
3   4692   4.183901323100 13,12
3   4712   4.183922067612 14,11
3   4752   4.183963033186 15,10
3   4812   4.184023173014 19,5
3   4812   4.184023205498 16,9
3   4842   4.184052682249 22,0
3   4872   4.184081853477 21,2
3   4892   4.184101140796 17,8
3   4962   4.184167249866 20,4
3   4992   4.184195049426 18,7
3   5072   4.184267484391 22,1
3   5072   4.184267545705 13,13
3   5082   4.184276442585 14,12
3   5112   4.184302892806 19,6
3   5112   4.184302924531 15,11
3   5132   4.184320359716 21,3
3   5162   4.184346377559 16,10
3   5232   4.184405817439 17,9
3   5252   4.184422484658 20,5
3   5292   4.184455477006 23,0
3   5322   4.184479911273 22,2
3   5322   4.184479943996 18,8
3   5412   4.184551588114 21,4
3   5432   4.184567209253 19,7
3   5472   4.184598100950 14,13
3   5492   4.184613363259 15,12
3   5532   4.184643510569 23,1
3   5532   4.184643557024 16,11
3   5562   4.184665893066 20,6
3   5592   4.184688002022 22,3
3   5592   4.184688038266 17,10
3   5672   4.184745883576 18,9
3   5712   4.184774179871 21,5
3   5762   4.184809011555 24,0
3   5772   4.184815936583 19,8
3   5792   4.184829632021 23,2
3   5882   4.184890230922 22,4
3   5882   4.184890269015 14,14
3   5892   4.184896862957 20,7
3   5892   4.184896886472 15,13
3   5922   4.184916604831 16,12
3   5972   4.184949028788 17,11
3   6012   4.184974544246 24,1
3   6032   4.184987209356 21,6
3   6042   4.184993521286 18,10
3   6072   4.185012247705 23,3
3   6132   4.185049234117 19,9
3   6192   4.185085461852 22,5
3   6242   4.185115146891 20,8
3   6252   4.185121001550 25,0
3   6282   4.185138523756 24,2
3   6312   4.185155915915 15,14
3   6332   4.185167392506 16,13
3   6372   4.185190100093 23,4
3   6372   4.185190111183 21,7
3   6372   4.185190129682 17,12
3   6432   4.185223705486 18,11
3   6512   4.185267483660 25,1
3   6512   4.185267511279 19,10
3   6522   4.185272896514 22,6
3   6572   4.185299644337 24,3
3   6612   4.185320778585 20,9
3   6692   4.185362235066 23,5
3   6732   4.185382611392 21,8
3   6752   4.185392720420 15,15
3   6762   4.185397709315 26,0
3   6762   4.185397743763 16,14
3   6792   4.185412693653 25,2
3   6792   4.185412725089 17,13
3   6842   4.185437402171 18,12
3   6872   4.185452021472 22,7
3   6882   4.185456862588 24,4
3   6912   4.185471350599 19,11
3   7002   4.185514001674 20,10
3   7682   4.185803986362 16,16
3   8192   4.185989856737 27,3
3   8192   4.185989878189 18,15
3   8672   4.186144864935 17,17
3   9722   4.186430546331 18,18
3  10832   4.186672333976 19,19
3  12002   4.186878779543 20,20
3  13232   4.187056449309 21,21
3  14522   4.187210452619 22,22
3  15872   4.187344813248 23,23
3  17282   4.187462734835 24,24
3  18752   4.187566793602 25,25
3  20282   4.187659080207 26,26
3  21872   4.187741305574 27,27
3  23522   4.187814880741 28,28
3  25232   4.187880977793 29,29
3  27002   4.187940576826 30,30
3  28832   4.187994502480 31,31
3  30722   4.188043452618 32,32
3  32672   4.188088021008 33,33
3  34682   4.188128715428 34,34
3  36752   4.188165972169 35,35
3  38882   4.188200167793 36,36
3  41072   4.188231628684 37,37
3  43322   4.188260638854 38,38
3  45632   4.188287446378 39,39
3  48002   4.188312268728 40,40
3  50432   4.188335297178 41,41
3  52922   4.188356700512 42,42
3  55472   4.188376628115 43,43
3  58082   4.188395212601 44,44
3  60752   4.188412572014 45,45
3  63482   4.188428811707 46,46
3  66272   4.188444025943 47,47
3  69122   4.188458299278 48,48
3  72032   4.188471707724 49,49
3  75002   4.188484319775 50,50
3  78032   4.188496197274 51,51

```

• Downloading the coordinates in the above way can be slow (the largest web pages are about 4 megs). The following instructions describe an alternative, faster but more complicated procedure, which requires a Unix machine with a C compiler and a shell.
• Unpacking: The spherical codes are stored in a highly compressed format. To uncompress them, you need to retrieve four files, namely:   creconstruct.c.txt,   cdot.c.txt,   cicosagen.c.txt,   makefile.txt,   and rename them   creconstruct.c,   cdot.c,   cicosagen.c,   makefile,   respectively.

Then type "make" to produce the executable program "creconstruct" from them.

• To recover the icosahedral packings two further files are needed, namely:
codes.icos.txt and icos.sh.txt, which should be renamed "codes.icos" and "icos.sh", respectively.

• To recover the icosahedral coverings two further files are needed, namely:
codes.icover.txt and icover.sh.txt, which should be renamed "codes.icover" and "icover.sh", respectively.

• To recover the icosahedral maxvolume arrangements two further files are needed, namely: codes.ivol.txt and ivol.sh.txt, which should be renamed "codes.ivol" and "ivol.sh", respectively.

• Instructions for unpacking: If you fetch the pair of files you need into the same directory as you make "creconstruct" in, say the coverings for example, then executing

icover.sh 8192 > cover.3.8192

gives an 8192-point icosahedral covering, one coordinate to a line.

The available numbers can be found with

awk NF==5 codes.icover

(look at the second field) or from the catalog reproduced 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.

These can be obtained with

icover.sh 18,15 > cover.3.8192
icover.sh 27,3 > cover.3.8192

Lines with these labels precede the NF==5 lines in codes.icover

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).

• Notes on the compression: Only 1/60 of the solution is stored, since it has icosahedral symmetry. For the packings, the fact that the distance between points is likely to be a constant gives additional compression. Fixed points of the icosahedral group are deduced from the number of points in the solution.

The text from the compressed file from the NF==5 line through the end of the consecutive text (not the blank line) can be piped into "creconstruct" and it will write the desired code on standard output. Then the *.sh file is not needed.

• To repeat, that is the alternative way to download these codes. The recommended way is described above.