The best known packings of equal 4d hyperspheres in a larger 4d container hypersphere (complete up to N = 300)

Last update: 06-Jul-2025

Download    Results    History of updates    References

Radii plots

2-50   50-100   100-150   150-200   200-250   250-300  

Density plots

2-50   50-100   100-150   150-200   200-250   250-300  

Contacts plots

2-50   50-100   100-150   150-200   200-250   250-300  

Rattlers plots

2-50   50-100   100-150   150-200   200-250   250-300  

Boundary plots

2-50   50-100   100-150   150-200   200-250   250-300  

Core plots

2-50   50-100   100-150   150-200   200-250   250-300  

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.

Results

The table below summarizes the current status of the search.

Legend:
N
the number of hyperspheres (called objects here); colors correspond to active researchers in the past, see "References" at the bottom of the page
radius
of the objects in the container hypersphere (which has always unit radius)
distance
packing of objects in a hypersphere is equivalent to distributing points in a hypersphere; the latter are then the centers of the objects. "distance" is here the largest distance between these points
ratio
= 1/radius
density
ratio of total volume occupied by the objects to container volume
contacts
number of mutual contacts between objects plus number of contacts between objects and container
loose
number of objects that have still degrees of freedom for a movement inside the container (so called "rattlers")
boundary
number of objects that have contact to the container (possible contacts of rattlers with the container are suppressed)
core
number of objects that don't have contact to the container (possible contacts of rattlers with the container are suppressed)
reference
for the best known packing so far
records
the sequence of N 's that establish density records

N radius distance ratio density contacts loose boundary core reference
1 1.000000000000 1.0000000000 1.000000000000
2 0.500000000000 2.000000000000 2.0000000000 0.125000000000 3 2
3 0.464101615138 1.732050807571 2.1547005384 0.139178955858 6 3
4 0.449489742783 1.632993161854 2.2247448714 0.163282309383 10 4
5 0.441518440112 1.581138830083 2.2649110641 0.190005158161 15 5
6 0.414213562373 1.414213562373 2.4142135624 0.176623509137 18 6
7 0.414213562373 1.414213562373 2.4142135624 0.206060760660 23 7
8 0.414213562373 1.414213562373 2.4142135624 0.235498012183 32 8
9 0.392944640987 1.294592445822 2.5448877417 0.214569500526 31 9 [1]
10 0.392280956059 1.290994448734 2.5491933385 0.236803927091 40 10
11 0.382838054889 1.240640509097 2.6120705275 0.236294214963 44 11 [1]
12 0.379795897113 1.224744871390 2.6329931619 0.249679175341 48 12
13 0.370481999157 1.177033853395 2.6991864713 0.244912977520 52 13 [1]
14 0.368408375855 1.166603107993 2.7143791117 0.257896837991 56 14 [1]
15 0.362916293358 1.139304896906 2.7554563361 0.260205891393 57 15 [1]
16 0.356226222001 1.106681365955 2.8072049115 0.257646070068 64 16 [1]
17 0.351532811847 1.084196142131 2.8446846675 0.259604605901 69 17 [1]
18 0.349469104305 1.074412012151 2.8614832833 0.268477350099 76 18 [1]
19 0.347197094453 1.063711853923 2.8802084349 0.276094610648 85 19 [1]
20 0.347197094453 1.063711853923 2.8802084349 0.290625905945 92 20 [1]
21 0.339546206978 1.028220931019 2.9451072621 0.279135342800 87 21 [2]
22 0.333802699406 1.002113635430 2.9957816452 0.273137958157 85 22 [2]
23 0.333333333333 1.000000000000 3.0000000000 0.283950617283 111 23 [2]
24 0.333333333333 1.000000000000 3.0000000000 0.296296296295 134 23 1
25 0.333333333333 1.000000000000 3.0000000000 0.308641975307 144 24 1 [1]
26 0.324771535266 0.961960439254 3.0790875782 0.289257868250 95 1 25 1 [2]
27 0.323945806542 0.958342717719 3.0869360856 0.297339915569 104 1 26 1 [2]
28 0.319877628168 0.940647275891 3.1261954946 0.293152430969 99 4 27 1 [2]
29 0.317459037864 0.930227064675 3.1500126968 0.294543050677 107 1 28 1 [2]
30 0.315990168743 0.923934581941 3.1646554194 0.299099397374 116 1 29 1 [2]
31 0.313159704035 0.911885065203 3.1932588616 0.298143392997 115 1 30 1 [2]
32 0.311458932449 0.904692391281 3.2106961651 0.301129387689 111 3 31 1 [31]
33 0.310154401219 0.899199478165 3.2242005790 0.305369552975 117 3 32 1 [2]
34 0.308815216672 0.893582220329 3.2381824017 0.309224349176 133 1 33 1 [2]
35 0.306285269828 0.883029454347 3.2649301109 0.308015440310 123 3 34 1 [2]
36 0.303143148415 0.870029899901 3.2987715712 0.304013961199 139 1 35 1 [2]
37 0.301235817121 0.862195929619 3.3196583645 0.304668923436 136 2 36 1 [2]
38 0.300651321630 0.859803788664 3.3261121041 0.310481741567 138 4 37 1 [2]
39 0.297669990199 0.847664163698 3.3594249771 0.306199741467 148 1 38 1 [2]
40 0.295854844085 0.840323452060 3.3800359196 0.306460657593 146 1 39 1 [2]
41 0.294797502608 0.836064828750 3.3921589944 0.309655684475 145 4 40 1 [31]
42 0.293368844389 0.830330907602 3.4086782531 0.311103753447 159 1 41 1 [2]
43 0.292419588808 0.826533872851 3.4197435407 0.314408518302 156 3 42 1 [2]
44 0.291127113403 0.821380303599 3.4349256870 0.316070012447 149 6 43 1 [2]
45 0.289236012044 0.813873569694 3.4573841374 0.314935757899 144 9 44 1 [2]
46 0.286691165360 0.803834612548 3.4880740003 0.310752815127 171 2 45 1 [2]
47 0.284970492933 0.797087365253 3.5091352431 0.309954133514 167 4 46 1 [2]
48 0.283980525334 0.793220115882 3.5213682305 0.312173095154 183 1 47 1 [2]
49 0.282540848919 0.787615151311 3.5393112317 0.312263384716 183 2 48 1 [2]
50 0.281364857635 0.783053432954 3.5541041209 0.313364227426 191 1 49 1 [2]
51 0.281261778192 0.782654295147 3.5554066622 0.319163374806 200 1 50 1 [2]
52 0.278712353673 0.772818874945 3.5879285106 0.313782138887 187 4 51 1 [2]
53 0.277952002667 0.769898964317 3.5977434608 0.316340720655 203 1 52 1 [2]
54 0.276798004181 0.765479093756 3.6127428121 0.316990011264 211 52 2 [31]
55 0.275874520804 0.761952254768 3.6248363824 0.318573067043 212 1 54 1 [2]
56 0.273814710292 0.754118030681 3.6521047351 0.314785799234 211 2 54 2 [31]
57 0.272711368137 0.749939862083 3.6668805075 0.315273758553 223 55 2 [31]
58 0.272039211500 0.747400727615 3.6759406649 0.317653776632 227 56 2 [31]
59 0.271059814521 0.743709346585 3.6892226233 0.318502289770 231 57 2 [31]
60 0.270294660358 0.740832348823 3.6996661298 0.320258832245 231 1 58 2 [31]
61 0.269622572869 0.738310256734 3.7088882780 0.322370156092 223 4 59 2 [31]
62 0.268830626376 0.735344329436 3.7198142692 0.323822234533 231 3 60 2 [31]
63 0.268114739680 0.732668778062 3.7297464555 0.325554205434 231 4 61 2 [31]
64 0.266800405039 0.727770192108 3.7481202469 0.324284291875 235 4 62 2 [31]
65 0.265294765188 0.722180141417 3.7693921299 0.321979394115 251 1 63 2 [31]
66 0.264968466676 0.720971698936 3.7740339918 0.325327447539 242 6 64 2 [31]
67 0.264026543809 0.717489310485 3.7874979749 0.325585582624 255 3 65 2 [31]
68 0.262885925177 0.713284236881 3.8039313034 0.324771754665 262 1 65 3 [31]
69 0.261916266401 0.709719655042 3.8180141071 0.324712476417 267 1 66 3 [31]
70 0.261266627691 0.707336740113 3.8275075881 0.326162328224 263 3 67 3 [31]
71 0.260636725181 0.705030217371 3.8367578449 0.327642921392 275 1 68 3 [31]
72 0.260127117620 0.703167054274 3.8442743269 0.329666647200 267 4 68 4 [31]
73 0.259369077971 0.700400348558 3.8555097154 0.330366234111 279 2 70 3 [31]
74 0.258522851648 0.697318460111 3.8681300072 0.330542624349 283 2 70 4 [31]
75 0.258082894946 0.695718950777 3.8747240502 0.332734742279 279 4 71 4 [31]
76 0.257688781287 0.694287718657 3.8806501199 0.335116366490 278 5 72 4 [31]
77 0.257434674091 0.693365728533 3.8844806106 0.338188544925 287 4 73 4 [31]
78 0.256958022526 0.691637970171 3.8916862380 0.340050429973 296 2 75 3 [31]
79 0.256468324828 0.689865229396 3.8991169793 0.341792108949 295 4 76 3 [31]
80 0.255463163131 0.686233777781 3.9144586943 0.340724315812 305 3 76 4 [31]
81 0.254058669351 0.681176009191 3.9360987073 0.337459062243 309 3 76 5 [31]
82 0.252894034506 0.676996426708 3.9542253417 0.335403964825 310 3 77 5 [31]
83 0.252008330509 0.673826569968 3.9681227917 0.334763177506 287 10 78 5 [31]
84 0.251296923021 0.671285936302 3.9793563247 0.334987012632 310 5 79 5 [31]
85 0.250610150755 0.668837857912 3.9902613561 0.335284561756 331 1 80 5 [31]
86 0.249710499519 0.665637728794 4.0046373778 0.334384135718 330 2 81 5 [31]
87 0.249149531018 0.663646202035 4.0136539528 0.335242869019 337 82 5 [31]
88 0.248505598356 0.661363804740 4.0240542129 0.335604194507 339 1 83 5 [31]
89 0.247998099891 0.659567748047 4.0322889588 0.336653721827 344 1 84 5 [31]
90 0.247386955229 0.657408098219 4.0422503243 0.337092966691 342 3 84 6 [31]
91 0.246960900415 0.655904588623 4.0492239797 0.338496507358 353 1 85 6 [31]
92 0.246689832003 0.654948897501 4.0536733593 0.340716232023 337 5 87 5 [31]
93 0.246502468889 0.654288723483 4.0567545003 0.343374501636 322 11 88 5 [31]
94 0.245932827006 0.652283604999 4.0661509575 0.343869662019 354 3 88 6 [31]
95 0.245100274372 0.649358493721 4.0799627930 0.342845769522 371 1 88 7 [31]
96 0.244513498515 0.647300773831 4.0897537603 0.343148884535 373 1 89 7 [31]
97 0.243874998759 0.645065295708 4.1004613228 0.343115908925 371 3 90 7 [31]
98 0.243524972021 0.643841403917 4.1063550555 0.344667301676 379 2 90 8 [31]
99 0.243259915388 0.642915369054 4.1108293506 0.346670909608 387 92 7 [31]
100 0.243022037871 0.642084842701 4.1148531580 0.348804945053 375 4 94 6 [31]
101 0.242327122009 0.639661598160 4.1266532269 0.348280750469 387 2 94 7 [31]
102 0.241661468889 0.637344560443 4.1380200352 0.347880283611 403 94 8 [31]
103 0.241051212866 0.635223922753 4.1484960317 0.347755901443 399 1 95 8 [31]
104 0.240535464233 0.633434355141 4.1573911073 0.348136699955 399 1 96 8 [31]
105 0.240195408090 0.632255742193 4.1632769250 0.349500741388 410 97 8 [31]
106 0.239472470586 0.629753588987 4.1758453385 0.348600691488 402 2 98 8 [31]
107 0.238986214814 0.628073287150 4.1843417654 0.349039984420 421 98 9 [31]
108 0.238490717810 0.626363258828 4.1930353063 0.349389362546 409 3 100 8 [31]
109 0.238116740820 0.625074085697 4.1996207262 0.350417846486 421 1 101 8 [31]
110 0.237733257728 0.623753456748 4.2063950562 0.351360107918 429 1 101 9 [31]
111 0.237355353009 0.622453337725 4.2130922573 0.352305239060 421 4 102 9 [31]
112 0.237131194237 0.621682764967 4.2170748695 0.354138200903 415 6 103 9 [31]
113 0.236792514357 0.620519370712 4.2231064724 0.355263278839 433 2 104 9 [31]
114 0.236630043629 0.619961636305 4.2260060670 0.357424556145 432 1 104 10 [31]
115 0.236364128573 0.619049307182 4.2307604205 0.358941860157 437 1 106 9 [31]
116 0.236222422236 0.618563385764 4.2332983911 0.361195609913 408 7 107 9 [31]
117 0.236108284019 0.618172128534 4.2353448298 0.363605765679 425 1 108 9 [31]
118 0.236077896536 0.618067982234 4.2358899951 0.366524757443 420 2 109 9 [31]
119 0.236068068183 0.618034299526 4.2360663503 0.369569349667 343 5 107 12 [31]
120 0.236067977500 0.618033988751 4.2360679775 0.372674401818 827 1 119 1 [31]
121 0.236067977500 0.618033988751 4.2360679775 0.375780021834 840 1 120 1
122 0.236067977500 0.618033988751 4.2360679775 0.378885641849 794 6 111 11 [31]
123 0.236067977500 0.618033988751 4.2360679775 0.381991261864 836 1 113 10 [31]
124 0.236067977500 0.618033988751 4.2360679775 0.385096881879 858 113 11 [31]
125 0.236067977500 0.618033988751 4.2360679775 0.388202501894 874 114 11 [31]
126 0.236067977500 0.618033988751 4.2360679775 0.391308121909 878 117 9 [31]
127 0.236067977500 0.618033988751 4.2360679775 0.394413741924 883 119 8 [31]
128 0.236067977500 0.618033988751 4.2360679775 0.397519361940 896 120 8 [31]
129 0.231651340790 0.602984954846 4.3168323420 0.371474479806 501 2 119 10 [31]
130 0.231357014792 0.601988229241 4.3223240968 0.372455198304 492 4 120 10 [31]
131 0.229728590394 0.596487387508 4.3529627648 0.364864396347 495 4 120 11 [31]
132 0.228922953625 0.593774525389 4.3682819227 0.362519434475 516 1 120 12 [31]
133 0.227953914564 0.590518931095 4.3868516227 0.359120220670 500 6 121 12 [31]
134 0.227351412942 0.588498877110 4.3984771727 0.358010223530 517 3 122 12 [31]
135 0.226922653844 0.587063260804 4.4067878771 0.357968806996 535 123 12 [31]
136 0.226201706521 0.584652895793 4.4208331377 0.356059367863 522 3 122 14 [31]
137 0.225752991090 0.583154958281 4.4296201577 0.355839880569 526 4 123 14 [31]
138 0.225246166112 0.581465121590 4.4395872181 0.355229248410 525 4 124 14 [31]
139 0.224938240752 0.580439527736 4.4456647152 0.355850824989 549 125 14 [31]
140 0.224518599354 0.579043157365 4.4539739820 0.355743794103 548 1 126 14 [31]
141 0.224016561889 0.577374595608 4.4639556628 0.355090965656 555 1 126 15 [31]
142 0.224009237740 0.577350269190 4.4641016151 0.357562575906 672 22 118 24 [31]
143 0.224009237740 0.577350269190 4.4641016151 0.360080622215 672 23 119 24 [31]
144 0.224009237740 0.577350269190 4.4641016151 0.362598668525 672 24 120 24 [31]
145 0.224009237740 0.577350269190 4.4641016151 0.365116714834 696 24 120 25 [31]
146 0.222552396380 0.572520631214 4.4933238926 0.358164019923 569 1 132 14 [31]
147 0.222230775041 0.571456848406 4.4998268121 0.358537130474 572 2 132 15 [31]
148 0.221806081573 0.570053495204 4.5084426581 0.358224688730 568 3 133 15 [31]
149 0.221581652790 0.569312513211 4.5130090303 0.359187701661 575 4 134 15 [31]
150 0.221322459749 0.568457283814 4.5182942623 0.359909418674 586 2 135 15 [31]
151 0.221156600901 0.567910317163 4.5216828072 0.361223979451 593 1 136 15 [31]
152 0.220960306508 0.567263281586 4.5256997322 0.362326953696 593 1 137 15 [31]
153 0.220858357788 0.566927361657 4.5277888055 0.364038054927 603 1 138 15 [31]
154 0.220677273610 0.566330907947 4.5315042353 0.365217147301 550 12 139 15 [31]
155 0.220570761438 0.565980208404 4.5336924689 0.366879518909 609 1 140 15 [31]
156 0.220308265077 0.565116327926 4.5390943442 0.367491890671 614 1 141 15 [31]
157 0.219922039177 0.563846308247 4.5470658773 0.367260877387 621 142 15 [31]
158 0.219265888196 0.561691579453 4.5606729265 0.365208920605 625 142 16 [31]
159 0.218842806513 0.560304144512 4.5694899272 0.364691991970 627 1 143 16 [31]
160 0.218358476129 0.558717697206 4.5796252920 0.363747654737 631 1 144 16 [31]
161 0.217891658736 0.557190474107 4.5894368137 0.362901109654 635 142 19 [31]
162 0.217540823657 0.556043894006 4.5968383460 0.362809030789 631 1 144 18 [31]
163 0.217115168538 0.554654170863 4.6058504651 0.362199852143 642 144 19 [31]
164 0.216915247633 0.554001969718 4.6100954677 0.363081542591 638 2 144 20 [31]
165 0.216544096525 0.552792047554 4.6179970549 0.362801720373 635 4 145 20 [31]
166 0.216234104242 0.551782376376 4.6246173956 0.362914946000 651 2 146 20 [31]
167 0.216116299209 0.551398884786 4.6271382754 0.364306197400 656 1 148 19 [31]
168 0.215990389718 0.550989138107 4.6298356205 0.365634353929 658 1 149 19 [31]
169 0.215785895566 0.550323934104 4.6342231839 0.366419790765 670 150 19 [31]
170 0.215589695168 0.549686035076 4.6384406232 0.367249248002 674 150 20 [31]
171 0.215441346347 0.549203925911 4.6416345653 0.368393813242 666 3 151 20 [31]
172 0.215283327160 0.548690590149 4.6450415515 0.369462218642 674 2 152 20 [31]
173 0.215123800036 0.548172565421 4.6484861267 0.370510010653 683 1 153 20 [31]
174 0.214963144967 0.547651090745 4.6519602239 0.371539744012 686 1 154 20 [31]
175 0.214479938085 0.546083921936 4.6624407342 0.370326466312 691 1 155 20 [31]
176 0.213572614060 0.543146431262 4.6822482573 0.366180253563 692 1 155 21 [31]
177 0.213176991646 0.541867712007 4.6909377615 0.365539726845 616 21 157 20 [31]
178 0.212946462894 0.541123196465 4.6960160146 0.366017394471 700 1 158 20 [31]
179 0.212786720718 0.540607548979 4.6995413841 0.366970469563 628 20 159 20 [31]
180 0.212694219639 0.540309051310 4.7015852227 0.368379330549 699 4 160 20 [31]
181 0.212260374152 0.538909982911 4.7111949369 0.367412795182 719 160 21 [31]
182 0.211886963127 0.537707037477 4.7194975342 0.366849839935 720 161 21 [31]
183 0.211388144925 0.536101869544 4.7306342574 0.365404254702 715 2 158 25 [31]
184 0.211219475545 0.535559560605 4.7344119069 0.366229785638 725 1 163 21 [31]
185 0.211098797707 0.535171697276 4.7371184055 0.367379372251 715 4 164 21 [31]
186 0.210974452576 0.534772171230 4.7399103910 0.368495695342 723 3 165 21 [31]
187 0.210842684779 0.534348933254 4.7428726353 0.369552170451 734 2 166 21 [31]
188 0.210577303068 0.533496956411 4.7488498781 0.369661384884 730 3 167 21 [31]
189 0.210238549209 0.532410258815 4.7565016205 0.369242096371 745 1 165 24 [31]
190 0.209897197672 0.531316170639 4.7642370222 0.368790871345 743 2 165 25 [31]
191 0.209705533764 0.530702270415 4.7685913769 0.369379620541 750 2 167 24 [31]
192 0.209246565087 0.529233401585 4.7790509707 0.368073523599 752 2 167 25 [31]
193 0.208883620176 0.528073050952 4.7873547919 0.367430202113 738 7 168 25 [31]
194 0.208471720939 0.526757480317 4.7968136661 0.366429420729 763 2 167 27 [31]
195 0.208201733458 0.525895906207 4.8030339776 0.366413929734 750 6 168 27 [31]
196 0.207769000003 0.524516208033 4.8130375561 0.365240618391 752 5 170 26 [31]
197 0.207577856839 0.523907259863 4.8174695280 0.365755041460 762 4 171 26 [31]
198 0.207363289662 0.523224036833 4.8224543584 0.366094063285 767 5 171 27 [31]
199 0.207237988509 0.522825224986 4.8253701322 0.367054497286 779 3 172 27 [31]
200 0.207171250658 0.522612861427 4.8269245700 0.368424028438 781 2 174 26 [31]
201 0.207033547283 0.522174794591 4.8301350826 0.369282690008 775 3 175 26 [31]
202 0.206950551663 0.521910839474 4.8320721639 0.370525176780 777 5 175 27 [31]
203 0.206826275050 0.521515699636 4.8349756324 0.371465837563 786 3 177 26 [31]
204 0.206669089615 0.521016102888 4.8386529493 0.372162210840 803 2 177 27 [31]
205 0.206513135301 0.520520614741 4.8423069968 0.372858958402 788 4 178 27 [31]
206 0.205971347322 0.518800792962 4.8550442234 0.370761354247 808 2 178 28 [31]
207 0.205680470429 0.517878417367 4.8619103112 0.370461067000 816 1 179 28 [31]
208 0.205326174942 0.516755852446 4.8702996599 0.369692467316 813 3 180 28 [31]
209 0.204873955169 0.515324473399 4.8810499079 0.368208061841 816 2 179 30 [31]
210 0.204754919947 0.514947970337 4.8838875289 0.369110737161 819 3 181 29 [31]
211 0.204554129122 0.514313133328 4.8886815646 0.369415792532 819 4 182 29 [31]
212 0.204282473989 0.513454755768 4.8951825405 0.369198813031 821 4 183 29 [31]
213 0.204053178162 0.512730681406 4.9006832876 0.369277678505 835 3 181 32 [31]
214 0.203884464476 0.512198180736 4.9047385860 0.369785870327 835 3 182 32 [31]
215 0.203682998566 0.511562601826 4.9095899365 0.370047589548 837 4 183 32 [31]
216 0.203582040585 0.511244223410 4.9120246419 0.371032201990 848 2 186 30 [31]
217 0.203438871092 0.510792866257 4.9154814644 0.371702500547 837 4 187 30 [31]
218 0.203249074120 0.510194761044 4.9200716133 0.372023862239 837 6 188 30 [31]
219 0.203077482827 0.509654272406 4.9242288514 0.372469916301 861 2 188 31 [31]
220 0.202684437558 0.508417111380 4.9337779064 0.371282344105 858 1 185 35 [31]
221 0.202442735831 0.507656929291 4.9396684741 0.371194100120 848 5 187 34 [31]
222 0.202231971063 0.506994423763 4.9448165626 0.371323327379 871 2 191 31 [31]
223 0.202080362450 0.506518082624 4.9485263579 0.371878706498 876 2 192 31 [31]
224 0.201963483182 0.506150981630 4.9513901436 0.372682866388 869 5 193 31 [31]
225 0.201799219541 0.505635234846 4.9554205525 0.373130239760 877 3 193 32 [31]
226 0.201601959620 0.505016168437 4.9602692448 0.373325311477 883 4 193 33 [31]
227 0.201442633079 0.504516372706 4.9641924588 0.373793216142 889 3 195 32 [31]
228 0.201229175588 0.503847084641 4.9694583158 0.373851079301 894 2 195 33 [31]
229 0.200948660213 0.502968082793 4.9763954581 0.373401400227 900 2 196 33 [31]
230 0.200703953344 0.502201791648 4.9824628929 0.373208515233 897 4 196 34 [31]
231 0.200469707763 0.501468699085 4.9882848195 0.373084330421 903 3 197 34 [31]
232 0.200198460569 0.500620343170 4.9950434042 0.372675565761 906 4 197 35 [31]
233 0.199881165280 0.499628696656 5.0029726342 0.371914757703 899 5 198 35 [31]
234 0.199518174987 0.498495203145 5.0120747148 0.370805111219 902 6 199 35 [31]
235 0.199132267633 0.497291274414 5.0217878392 0.369516996814 907 6 200 35 [31]
236 0.199043512610 0.497014546343 5.0240270928 0.370428260351 918 4 201 35 [31]
237 0.198816768626 0.496307862772 5.0297568304 0.370305693714 921 5 202 35 [31]
238 0.198618169475 0.495689225559 5.0347861056 0.370384545314 921 5 203 35 [31]
239 0.198409763193 0.495040368714 5.0400745604 0.370382156538 913 8 204 35 [31]
240 0.198296277128 0.494687180490 5.0429590232 0.371081654744 936 2 205 35 [31]
241 0.198168862783 0.494290764191 5.0462014363 0.371671029911 808 37 206 35 [31]
242 0.198080781303 0.494016795420 5.0484453536 0.372550136984 933 2 207 35 [31]
243 0.197943065870 0.493588565715 5.0519577213 0.373050343358 957 2 208 35 [31]
244 0.197626510067 0.492604784546 5.0600498873 0.372195083430 945 5 206 38 [31]
245 0.197471541826 0.492123462577 5.0640208242 0.372549643978 956 3 208 37 [31]
246 0.197281067718 0.491532116122 5.0689101168 0.372629081524 968 2 208 38 [31]
247 0.197007638164 0.490683716377 5.0759453254 0.372073904016 970 2 211 36 [31]
248 0.196957374644 0.490527821126 5.0772407066 0.373199168542 970 2 212 36 [31]
249 0.196789909070 0.490008557641 5.0815613703 0.373431240479 971 4 213 36 [31]
250 0.196647342526 0.489566669623 5.0852454305 0.373845653539 970 6 214 36 [31]
251 0.196483284240 0.489058361539 5.0894914744 0.374090049694 974 4 214 37 [31]
252 0.196419092207 0.488859529394 5.0911547791 0.375089873028 955 9 215 37 [31]
253 0.196090977289 0.487843703079 5.0996737016 0.374068350961 988 1 215 38 [31]
254 0.195925475109 0.487331631693 5.1039814983 0.374280629247 990 3 215 39 [31]
255 0.195739634189 0.486756882495 5.1088273673 0.374330547805 991 4 215 40 [31]
256 0.195457675033 0.485885375991 5.1161971502 0.373637862974 1005 3 216 40 [31]
257 0.195348467651 0.485547991390 5.1190573032 0.374259780792 1004 2 217 40 [31]
258 0.195241428378 0.485217393795 5.1218637781 0.374893241458 1012 2 217 41 [31]
259 0.195134753841 0.484888010191 5.1246637532 0.375524488847 1002 3 218 41 [31]
260 0.194923256104 0.484235217528 5.1302241712 0.375342703522 1011 5 218 42 [31]
261 0.194780312043 0.483794211583 5.1339891055 0.375682301954 1016 5 219 42 [31]
262 0.194569235607 0.483143292282 5.1395586609 0.375489660000 1020 4 219 43 [31]
263 0.194252794272 0.482168086693 5.1479310954 0.374476739987 1028 2 219 44 [31]
264 0.193945031204 0.481220360178 5.1561001269 0.373524039247 1027 3 221 43 [31]
265 0.193821526678 0.480840243424 5.1593856324 0.373984767983 1026 3 221 44 [31]
266 0.193616836431 0.480210513260 5.1648400957 0.373812754722 1030 5 222 44 [31]
267 0.193334818158 0.479343406681 5.1723740686 0.373036698173 1038 4 223 44 [31]
268 0.193195426079 0.478915049130 5.1761059788 0.373355157462 1038 4 224 44 [31]
269 0.193044690391 0.478451998748 5.1801476538 0.373580091546 1037 7 225 44 [31]
270 0.192897916042 0.478001283545 5.1840891831 0.373829790510 1052 4 226 44 [31]
271 0.192672648784 0.477309850815 5.1901502694 0.373464703549 1049 5 227 44 [31]
272 0.192419764302 0.476534109668 5.1969713383 0.372878735699 1061 5 228 44 [31]
273 0.192204193016 0.475873212893 5.2028001279 0.372575315642 1063 4 228 45 [31]
274 0.192131554148 0.475650596665 5.2047671421 0.373375094810 1060 4 230 44 [31]
275 0.191931092256 0.475036449037 5.2102032466 0.373176281724 1080 2 231 44 [31]
276 0.191793782372 0.474615953672 5.2139333592 0.373462652949 1085 1 232 44 [31]
277 0.191693753237 0.474309716162 5.2166540803 0.374034456467 1088 2 232 45 [31]
278 0.191568585500 0.473926623988 5.2200625556 0.374405281115 1083 5 233 45 [31]
279 0.191418109175 0.473466228584 5.2241661163 0.374572846303 1083 6 234 45 [31]
280 0.191286022130 0.473062237984 5.2277735135 0.374878881125 1099 3 235 45 [31]
281 0.191164050267 0.472689302027 5.2311090846 0.375259083849 1100 3 236 45 [31]
282 0.190918796673 0.471939765472 5.2378289484 0.374665636167 1101 4 237 45 [31]
283 0.190682901353 0.471219257994 5.2443087078 0.374139396119 1111 3 237 46 [31]
284 0.190411059207 0.470389476962 5.2517957947 0.373324950062 1100 6 236 48 [31]
285 0.190162824429 0.469632242544 5.2586513847 0.372689654083 1097 4 237 48 [31]
286 0.190015795234 0.469183952271 5.2627203900 0.372842015859 1124 3 236 50 [31]
287 0.189854910552 0.468693603220 5.2671800645 0.372880124484 1118 4 237 50 [31]
288 0.189757878142 0.468397959135 5.2698734292 0.373414991518 1113 6 239 49 [31]
289 0.189698103066 0.468215868144 5.2715339997 0.374239647199 1123 3 241 48 [31]
290 0.189544054026 0.467746717061 5.2758183586 0.374316230534 1140 3 242 48 [31]
291 0.189311206202 0.467037925404 5.2823074770 0.373764697562 1127 6 243 48 [31]
292 0.189189285138 0.466666958564 5.2857116050 0.374083881939 1139 5 244 48 [31]
293 0.189107605443 0.466418495751 5.2879946190 0.374717177513 1144 4 245 48 [31]
294 0.188826715133 0.465564432794 5.2958608071 0.373767110323 1158 2 244 50 [31]
295 0.188587142981 0.464836467280 5.3025884172 0.373138739991 1149 5 244 51 [31]
296 0.188377529662 0.464199887378 5.3084887661 0.372741802055 1140 7 244 52 [31]
297 0.188258501287 0.463838553494 5.3118451128 0.373056694001 1163 2 245 52 [31]
298 0.188158854076 0.463536136400 5.3146582174 0.373520895341 1155 6 243 55 [31]
299 0.187991586199 0.463028665722 5.3193870014 0.373443442240 1163 1 244 55 [31]
300 0.187648382196 0.461988080244 5.3291160217 0.371963694591 1167 5 245 55 [31]


History of updates

Please note that the results are taken from a running search. For updates look at the list below.

04-Jul-2025: First presentation for N=2–100 by E. Specht [31].
The packings for N=2–23 can be easily derived from spherical codes in [1], where maximum separation angles φ between points on the surface of a 4-sphere are given. The radius r of a corresponding hypersphere for our packing problem can be calculated by r = sin(φ/2) / (1 + sin(φ/2)). For the range N=24–53, 55, we can take N-1 points from the corresponding spherical code and place one central hypersphere at the origin.
05-Jul-2025: Extension to N=200 by E. Specht [31].
06-Jul-2025: Extension to N=300 by E. Specht [31].

References

[1]   A. L. Mackay, The packing of three-dimensional spheres on the surface of a four-dimensional hypersphere, J. Phys. A: Math. Gen. 13 (1980) 3373–3379.
[2]   N. J. A. Sloane, with collaboration of R. H. Hardin, W. D. Smith, and others, Tables of Spherical Codes, http://neilsloane.com/packings/dim4, 1994.
[31]   , program hsp4, September 2010–July 2025.


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©  E. Specht    06-Jul-2025