The best known packings of unequal spheres with integer radii in a sphere (complete up to N = 100)


Last update: 12-Jul-2026

Overview    Download    Results    History of updates    References

Overview

2-13   14-25   26-37   38-49   50-61   62-73   74-85   86-97   98-100  

Download

You may download ASCII files which contain all the values of radius, ratio etc. by using the links given in the table header below.
All coordinates of all packings are packed as ASCII files here.
All packings are stored as nice PDF files here.

Results

The table below summarizes the current status of the search.
Please use the links in the following table to view a picture for a certain configuration.
Furthermore, note that for certain values of N several distinct optimal configurations exist; however, only one is shown here.
Proven optimal packings are indicated by a radius in bold face type.

Legend:
N
the number of spheres; colors correspond to active researchers in the past, see "References" at the bottom of the page
radius
of the largest sphere in the container sphere, the latter has always a *radius* of 1
ratio
is the radius of the circumsphere if r1=1
density
ratio of total volume occupied by the spheres to container volume, also known as packing fraction ϕ
contacts
number of contacts between spheres and container and mutually between the spheres
loose
number of spheres that have still degrees of freedom for a movement inside the container (so called "rattlers")
boundary
number of spheres that are near to the container boundary (including rattlers if any)
core
number of spheres that are at the container core
reference
for the best known packing so far, see at bottom of the page
records
the sequence of N 's that establish density records

N radius ratio density contacts loose boundary core reference
2 0.666666666667 3.0000000000 0.333333333334 3 2 [1]
3 0.600000000000 5.0000000000 0.288000000000 3 1 3 [1]
4 0.571428571429 7.0000000000 0.291545189505 3 2 4 [1]
5 0.555469288333 9.0013977460 0.308498218929 6 2 4 1 [1]
6 0.542640687119 11.0570403997 0.326228500778 6 3 5 1 [1]
7 0.532113639717 13.1550846991 0.344377775803 6 4 6 1 [1]
8 0.523809523810 15.2727272727 0.363793731779 6 5 7 1 [1]
9 0.517207738534 17.4011317493 0.384319607272 6 6 9 [1]
10 0.511872235096 19.5361250608 0.405704755854 6 7 10 [1]
11 0.507302790421 21.6833027685 0.427279020221 10 7 10 1 [1]
12 0.503093945009 23.8524039477 0.448324774225 13 7 11 1 [1]
13 0.495637320502 26.2288561863 0.458928183324 16 7 11 2 [1]
14 0.488170979342 28.6784765839 0.467423308594 16 8 13 1 [1]
15 0.481608959741 31.1455999657 0.476620138412 19 8 13 2 [1]
16 0.475380296979 33.6572636722 0.485111311844 19 9 13 3 [1]
17 0.469587143447 36.2020132732 0.493383579906 19 10 14 3 [1]
18 0.463377306409 38.8452342207 0.498860350551 25 9 18 [1]
19 0.457343157083 41.5442971120 0.503469255399 25 10 16 3 [1]
20 0.451920196526 44.2556012184 0.508784479008 22 12 17 3 [1]
21 0.446493073822 47.0332043905 0.512873493824 25 12 19 2 [1]
22 0.441177288132 49.8665742590 0.516193391561 28 12 19 3 [1]
23 0.436081790950 52.7423994244 0.519204579014 25 14 19 4 [1]
24 0.431824131008 55.5781816639 0.524239218460 28 14 21 3 [1]
25 0.427595768780 58.4664344816 0.528502313915 28 15 21 4 [1]
26 0.423533837238 61.3882474410 0.532547717920 28 16 23 3 [1]
27 0.419163298512 64.4140364766 0.534616126613 34 15 23 4 [1]
28 0.415568573024 67.3775685111 0.538897370118 31 17 24 4 [1]
29 0.412022908029 70.3844359983 0.542685989095 34 17 25 4 [1]
30 0.408884629416 73.3703295300 0.547450046344 40 16 25 5 [1]
31 0.405327352168 76.4813917299 0.549915557590 40 17 27 4 [1]
32 0.401733548257 79.6547864594 0.551610103124 46 16 27 5 [1]
33 0.398650167107 82.7793457092 0.554829768389 49 16 27 6 [1]
34 0.395483279124 85.9707648710 0.557162091000 37 21 27 7 [1]
35 0.392479325235 89.1766718643 0.559663930884 40 21 30 5 [1]
36 0.389971328885 92.3144788693 0.563818070809 37 23 32 4 [1]
37 0.387600322847 95.4591568145 0.568143272135 37 24 31 6 [1]
38 0.384276654074 98.8870898014 0.567829077170 55 19 36 2 [1]
39 0.382035943380 102.0846354271 0.571884148574 52 21 34 5 [1]
40 0.379528962141 105.3938012381 0.574357768206 49 23 34 6 [1]
41 0.376657208183 108.8522909140 0.574769324712 46 25 35 6 [1]
42 0.374734830628 112.0792532939 0.579161673001 61 21 36 6 [1]
43 0.372114882393 115.5557115143 0.579973332341 52 25 36 7 [1]
44 0.369732868581 119.0048376518 0.581535913861 58 24 37 7 [1]
45 0.366987613806 122.6199422191 0.581028423323 64 23 37 8 [1]
46 0.364891955095 126.0647141098 0.583271502442 67 23 36 10 [1]
47 0.362585380896 129.6246414675 0.584191851209 58 27 39 8 [1]
48 0.360243197225 133.2433210946 0.584626229956 64 26 36 12 [1]
49 0.358059536265 136.8487501021 0.585530701845 67 26 39 10 [1]
50 0.355795177741 140.5302913813 0.585747615482 61 29 42 8 [1]
51 0.353335836288 144.3386001708 0.584708733678 67 28 39 12 [1]
52 0.351293233829 148.0244849387 0.585460715746 70 28 42 10 [1]
53 0.349108498518 151.8152672450 0.585238545483 91 22 39 14 [1]
54 0.347467083098 155.4104046879 0.587506442360 70 30 40 14 [1]
55 0.345615524404 159.1363700888 0.588481885465 61 34 46 9 [1]
56 0.343868227638 162.8530800437 0.589763435807 67 33 43 13 [1]
57 0.342160129843 166.5886672014 0.591029755805 76 31 42 15 [1]
58 0.340600157321 170.2876488848 0.592857845025 88 28 45 13 [1]
59 0.339051615537 174.0148027508 0.594549409378 88 29 45 14 [1]
60 0.337360660081 177.8512052520 0.595294323849 88 30 47 13 [1]
61 0.335911090717 181.5956712528 0.597126825462 70 37 49 12 [1]
62 0.334271309182 185.4780781268 0.597759827597 73 37 47 15 [1]
63 0.332639680367 189.3941213823 0.598248435034 82 35 51 12 [1]
64 0.331154876925 193.2630453590 0.599349591769 97 31 50 14 [1]
65 0.329889250388 197.0358231544 0.601476986690 79 38 52 13 [1]
66 0.328525689205 200.8975315133 0.602911661036 79 39 48 18 [1]
67 0.327433736782 204.6215538401 0.605694046235 85 38 52 15 [1]
68 0.326150201588 208.4928958158 0.607270565494 100 34 52 16 [1]
69 0.324803436372 212.4361760784 0.608343460740 100 35 51 18 [1]
70 0.323386688669 216.4591260330 0.608870712302 88 40 54 16 [1]
71 0.321660995725 220.7292800297 0.607493812417 97 38 53 18 [1]
72 0.320058026626 224.9592074256 0.606651676408 82 44 56 16 [1]
73 0.319289534043 228.6326115223 0.610428268465 97 40 55 18 [1]
74 0.317908199029 232.7715995562 0.610570759614 109 37 58 16 [1]
75 0.316394350951 237.0459515935 0.609806560379 103 40 55 20 [1]
76 0.315261489326 241.0697233033 0.611111751784 100 42 58 18 [1]
77 0.314345493679 244.9534081078 0.613564473884 106 41 58 19 [1]
78 0.312982215293 249.2154384139 0.613279671193 88 48 57 21 [1]
79 0.311772375914 253.3899925175 0.613770186690 115 40 58 21 [1]
80 0.310498325346 257.6503429152 0.613758961014 106 44 61 19 [1]
81 0.309359421909 261.8313659243 0.614430459529 109 44 63 18 [1]
82 0.308293361988 265.9804267962 0.615424614826 127 39 63 19 [1]
83 0.307155343322 270.2215729094 0.615878031154 121 42 60 23 [1]
84 0.306026275484 274.4862344488 0.616275246603 103 49 64 20 [1]
85 0.305207178133 278.4993476233 0.618446613892 118 45 65 20 [1]
86 0.303898554771 282.9891707277 0.617541241968 109 49 64 22 [1]
87 0.303064840426 287.0672819642 0.619430757654 139 40 66 21 [1]
88 0.301719380103 291.6617420133 0.618083243879 118 48 64 24 [1]
89 0.300671667186 296.0039462080 0.618461312858 100 55 66 23 [1]
90 0.299792061746 300.2080824817 0.619784442423 133 45 68 22 [1]
91 0.298918838031 304.4304621262 0.621060812574 136 45 70 21 [1]
92 0.297812750126 308.9189430643 0.620794613020 148 42 66 26 [1]
93 0.297237266647 312.8813592222 0.623767214268 136 47 70 23 [1]
94 0.296122786748 317.4358887822 0.623268027553 109 57 67 27 [1]
95 0.295169331732 321.8491550005 0.623695435354 112 57 73 22 [1]
96 0.294212776578 326.2944631997 0.624017587297 133 51 71 25 [1]
97 0.293428017790 330.5751125287 0.625352895801 121 56 73 24 [1]
98 0.292236180175 335.3451990144 0.624002414137 118 58 68 30 [1]
99 0.291538825194 339.5774128339 0.625740135999 133 54 66 33 [1]
100 0.291058326345 343.5737477631 0.628814922408 139 53 70 30 [1]

Updates

11-Jul-2026: First complete presentation from N=2 to N=100. André Müller is the author of these packings, which seems to be unbeatable. He published his records on github in December 2025 [1].

References

[1]   , github.com/muellan/packing, December 2025.

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©  E. Specht    12-Jul-2026