The best known packings of equal circles in an isosceles right triangle (complete up to N = 299)

Last update: 18-Mar-2011


Overview    Download    Results    History of updates    References

Overview

1-12   13-24   25-36   37-48   49-60   61-72   73-84   85-96   97-108   109-120   121-132   133-144   145-156   157-168   169-180   181-192   193-204   205-216   217-228   229-240   241-252   253-264   265-276   277-288   289-300  


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.
All packings are stored as nice PDF files here.
All contact graphs of all packings are stored as nice PDF files here.
  For industrial applications, for instance if a machine has to do an important job at every circle center,
it is useful to know a tour visiting each of the circle centers once which is of minimal length.
This problem is known as the "Traveling Salesman Problem" (TSP). Thus (very near) optimal tours are provided for every packing.
All optimal TSP tours of 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 circles; colors correspond to active researchers in the past, see "References" at the bottom of the page
radius
of the circles in the isosceles right triangle
distance
packing of circles in a square is equivalent to distributing points in the triangle; the latter are then the circle centers. "distance" is here the greatest distance of these points
ratio
= 1/radius
density
ratio of total area occupied by the circles to container area (for an infinite hexagonal packing you get the well-known value ρ = Pi/(2*sqrt(3))=0.90689968211)
contacts
number of contacts between circles and container and between the circles themselves, respectively
loose
number of circles that have still degrees of freedom for a movement inside the container (so called "rattlers")
boundary
number of circles that have contact to the container (rattlers too if possible)
symmetry group
of the packing (Schönfliess notation); if field is empty then the packing has symmetry element C1
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 symmetry group reference
1 0.292893218813452475599155637895 3.4142135623730950488016887242 0.539012084452647221356168169717 3 1 D1
2 0.207106781186547524400844362105 1.414213562373095048801688724213 4.8284271247461900976033774484 0.539012084452647221356168169718 5 2 D1 [1]
3 0.184699031259064639371307948271 1.000000000000000000000000000004 5.4142135623730950488016887242 0.643028701816816785189812234039 8 3 D1 [1]
4 0.160188620508520367600361869474 0.707106781186547524400844362110 6.2426406871192851464050661726 0.644916045758165562530105410266 10 4 D1 [1]
5 0.139933250189584016010699682098 0.535898384862245412945107316989 7.1462643699419723423291350657 0.615165138079140704708797706019 9 1 5 [1]
6 0.134876071694908969152891508111 0.499999999999999999999999999997 7.4142135623730950488016887242 0.685805455832301883363908754499 15 6 D1 [1]
7 0.122229728867651532414543849327 0.419542091095305994112653020003 8.1813157017044895514201792700 0.657100208776002641852224338399 11 2 7 D1 [1]
8 0.115046593935109792351790738066 0.378937381963011999406093373712 8.6921304299024632446023502474 0.665299789950114100239944951225 16 1 7 D1 [2]
9 0.110240604605771200956705296110 0.353553390593273762200422181054 9.0710678118654752440084436210 0.687235444967990160141763805356 21 8 D1
10 0.106222361897208144374166310900 0.333333333333333333333333333334 9.4142135623730950488016887242 0.708943746755424005131531568449 24 9 D1
11 0.099412103976005362507947914479 0.300981634791721603578444448152 10.0591372680470111791374941975 0.683047780196988340772376936867 21 1 9 [5]
12 0.095943368477780190279390254976 0.285363629059558474163583441908 10.4228152071979104075026842876 0.694050447396894651856705323281 25 9 [5]
13 0.092604027573949621290924338290 0.270839295354589762980617008945 10.7986663884726040681698999193 0.700459406614532423365565735561 26 1 10 [8]
14 0.089753553298429679513887679701 0.258819045102520762348898837623 11.1416201726856413427996343221 0.708616411308056312112059587007 33 8 D1 [3]
15 0.087610065690070458685697053835 0.250000000000000000000000000002 11.4142135623730950488016887242 0.723401057589519737764073319332 35 12 D1
16 0.084090268325901633878526670883 0.235911123893133748627238986663 11.8919825077068784069456103904 0.710871867602807411648147924337 33 11 [5]
17 0.082264484182688609747379051363 0.228788437708031567978739925720 12.1559140610333489720934839182 0.722858925072265284244889852196 37 12 D1 [5]
18 0.079727525625494859159593067446 0.219093900704678792367541899348 12.5427196210417087092350086041 0.718900763870634434009601801220 35 1 13 [6]
19 0.078133140566947791272420642865 0.213118445695386011678756428286 12.7986663884726040681698999192 0.728792729832220021743893192684 41 1 13 [5]
20 0.076378991823013947793545532583 0.206645901063645495488721466896 13.0926053896758488383255557820 0.733090694962788648294113749886 37 2 13 [4]
21 0.074547791814273978529565572365 0.200000000000000000000000000001 13.4142135623730950488016887242 0.733278126867138413075824300159 48 15 D1
22 0.072895080875273247425094690768 0.194096868940093767412479216893 13.7183468073935585308221079380 0.734512165797951773574949634195 41 2 13 [6]
23 0.071581718995599529585444225012 0.189468690981505999703046686857 13.9700472974318314404030117704 0.740477644494823967502186301522 56 12 D1 [6]
24 0.069914330471767597225541240229 0.183671498636988162443379743201 14.3032192864067295597608761157 0.737095086389099129138847517312 45 2 15 [6]
25 0.068552444544352860098178160187 0.179000417606134646468450232529 14.5873718529907177417167200247 0.738185940334435672959955675163 45 3 15 [6]
26 0.067573656554549286526642415752 0.175678184147321129772768935778 14.7986663884726040681698999193 0.745947145954541054610830451870 58 1 16 [6]
27 0.066273724817239233486769019447 0.171310280878153772333688979718 15.0889361169553323070935215886 0.745120295526152838364262854424 49 3 16 [8]
28 0.065186454246851524853291386212 0.167695241234801883228396803956 15.3406104313197788431382464975 0.747571294817113952029887171613 57 16 [6]
29 0.064012298551316766854566681766 0.163829891498776187599737357003 15.6219980008736847777253042795 0.746628752462012498758749340936 54 3 16 [6]
30 0.063062012996332493125050814958 0.160730444812170458519798509723 15.8574068997473510530608982032 0.749612435383815641721784305184 58 3 16 D1 [7]
31 0.062206905110929787982554697853 0.157963256492610792979174847073 16.0753858147541797784595058656 0.753735121369256862162910562291 59 2 17 [6]
32 0.061271965726474658527691524658 0.154961110221743021031194096564 16.3206776238274926598059383309 0.754837510878911758460749542489 62 2 18 [6]
33 0.060341937748463158170313421166 0.151998684295780785001576918436 16.5722221942643677245682383011 0.754974552197990232487828468352 59 4 17 [6]
34 0.059529215264914758075537012723 0.149429245361342254017315174827 16.7984744221780215380063892189 0.757040476665274290115097272902 83 16 D1 [6]
35 0.058702760457182031530456481064 0.146834679638251307279388155544 17.0349740320883713975504186329 0.757818077416705529320183343323 67 2 18 [6]
36 0.057977885816087168807523351764 0.144574041889918151156040843948 17.2479555941746538603835328111 0.760338739233786272657756818154 73 2 20 [8]
37 0.057201813503630322194646488397 0.142169149163683084518644592556 17.4819632237102909491317238982 0.760678581311477360648353382184 69 4 19 [6]
38 0.056524599580478030236429337372 0.140083501498562274734040266922 17.6914123659775772614844737636 0.762848776388717539243404050266 78 2 18 [5]
39 0.055775498265939385768159921378 0.137790336212022595937886483573 17.9290195711379851386758791097 0.762309606655985497474125772542 73 3 19 [6]
40 0.055284785872367966120432533474 0.136295994928642196016911732610 18.0881590517258858324762837785 0.768159000173515421538815978976 80 4 20 [6]
41 0.054567981671521314734928568992 0.134124207638212938081830697545 18.3257648417275746368546954122 0.767077963147849229384865858766 82 2 21 [6]
42 0.053878035839236931707317523765 0.132046099699055734522286915310 18.5604390439145393042724001250 0.766042153382295315754236355492 79 4 21 [6]
43 0.053337740416613861875143736503 0.130427094211380614083309404024 18.7484507628020145497928537209 0.768630388674890596969264872270 85 3 19 [6]
44 0.052739546815925135884153722541 0.128643093375962429169552304440 18.9611033915452957964392890595 0.768962802849917813386719124959 81 5 19 [8]
45 0.052151704852573534064757542009 0.126898601321927510962326280610 19.1748285665229393837959486631 0.769005423134607725961422248862 88 4 21 [6]
46 0.051730124673871185786780817652 0.125652747817059801009124882779 19.3310958808707185043611081699 0.773436654572145936380643767322 89 5 22 [6]
47 0.051230682535404290746659075563 0.124182421825949378377867464546 19.5195525515187912287004513627 0.775064793820534598826964084684 89 3 22 [6]
48 0.050659107957245864162268168619 0.122507182323498491803671493855 19.7397869864577468325519190570 0.773991483519431576070081110368 98 2 24 [8]
49 0.050148758538649490901737888561 0.121018055705622258820397167842 19.9406730922222757867001956497 0.774276932726294917396142303279 93 3 23 [6]
50 0.049782889215332344842427198878 0.119954349025118093054811098764 20.0872230551861134084426034794 0.778592235490629276597279891647 93 5 22 [5]
51 0.049210643899436240853141091882 0.118297041933952656377415456044 20.3208070604306000729218833009 0.776011468594735083280519881684 96 5 24 [6]
52 0.048843851530028133627914182530 0.117238844362682935991648687327 20.4734059390304597630729736653 0.779476482909589510514531418763 99 6 24 [6]
53 0.048400723345548409963977099342 0.115964652628269197203697744259 20.6608482451941211780275466014 0.780116461780540638015062231818 106 5 24 [5]
54 0.047987323959451553136362699298 0.114780101843661777760004140266 20.8388365403534995232605017902 0.781315953289781108631252019968 107 4 24 [6]
55 0.047519033968221680040031495221 0.113443089562473589340875610645 21.0441988502701735133144102773 0.780329030297894828661661174953 111 2 23 [6]
56 0.047111702421810535957614208254 0.112284262614101534201126692131 21.2261486763222192763174061451 0.780954068217553079330428142327 110 3 25 [8]
57 0.046745895733448388429067091576 0.111246839464816218305222213050 21.3922523958496667011341327400 0.782603338305746579331040278103 115 4 26 [8]
58 0.046421284638022819702349500743 0.110328825264206380622212143914 21.5418424500238498024314545449 0.785311887169001338739714054955 116 4 24 [6]
59 0.045950456511985541273636177626 0.109001592015179821038726850137 21.7625694260330977544022337719 0.782729209321531585837676411263 117 4 26 [5]
60 0.045601392571645883160832003974 0.108020866323540715451200258564 21.9291548701909998111243097817 0.783948131578674519800025888348 120 5 25 [7]
61 0.045294150990393085251273677778 0.107159932334472510666007271179 22.0779058252378007724102790329 0.786310274247114615647657656971 124 4 25 [5]
62 0.044918749375195635798626985710 0.106110900201777571974771442406 22.2624185648455550742879115783 0.786007825999668278239075052311 124 6 27 [6]
63 0.044570982561404562440413676600 0.105141921602523263642068781790 22.4361219459840210847290290587 0.786366192701570570094479832223 123 6 28 [6]
64 0.044224566623236258846364834541 0.104179401422913080565102119096 22.6118665790290597256008985487 0.786478788035896997646418703708 124 6 27 [5]
65 0.043931051031698020223651763542 0.103365961641301707028534334475 22.7629427595178585656640774941 0.788199967099329858900138922454 126 7 27 [7]
66 0.043635317229970822854502642176 0.102548311898965464847962544015 22.9172162248691564961538247256 0.789587163319288599106976419171 125 7 29 [6]
67 0.043311925364952404881383052691 0.101656410685815891681288042758 23.0883294052125057677335661368 0.789713661663124726965474850495 125 8 30 [6]
68 0.043008487573184573962470605529 0.100821641235024471939076071659 23.2512245007074252683256708183 0.790309353784900965489376761078 158 2 25 [8]
69 0.042717378651785596354539992142 0.100022692235264315883355374503 23.4096761449616657976880478109 0.791112320086862692409518453076 142 4 28 [6]
70 0.042484939929021566211469037742 0.099386097475237854515035674439 23.5377524758343263668687648446 0.793867321991546728965486354273 135 6 28 [5]
71 0.042115911651114051392641615673 0.098377840214804831710441895724 23.7439951029422384399149411876 0.791280801977985254884915338238 141 8 28 [6]
72 0.041828699597051416701987863185 0.097595171966147097903996090666 23.9070305707158983982824668333 0.791518538295412737222133671760 144 6 28 [5]
73 0.041597294201379434344754464127 0.096965881252392168466377198381 24.0400251794944480696209143614 0.793657074936233804743975608094 143 4 30 [5]
74 0.041327719538519885419563026930 0.096234251809130105794547342246 24.1968347435173793744803369535 0.794135245261342517836824674706 161 5 29 [6]
75 0.041032850590849648929006267366 0.095435766821584991832020067678 24.3707172570408895440924970912 0.793422495196473113238894716813 154 6 29 [6]
76 0.040804704927251857119396324645 0.094819245705666116355484230773 24.5069778542164960268756194162 0.795085702654816843115558768458 154 7 29 [6]
77 0.040558779096203082107171923521 0.094155925556115242070254814202 24.6555745089875052972472358026 0.795866711997174785310565368528 164 4 29 [5]
78 0.040294467192844828258033240885 0.093444453868176181742235479971 24.8173029615731478107861952313 0.795729220063152884335604005778 151 8 31 [5]
79 0.040035822097006118702450300740 0.092749675936786006890440995718 24.9776312217847541832291941348 0.795617745623855299726388612671 163 4 31 [6]
80 0.039844578067413915145714662697 0.092236865513465111126045988623 25.0975176172797733478472908955 0.798009974696707760068174318856 159 7 29 [6]
81 0.039598896812237962565791616340 0.091579221019747353056037495260 25.2532287639627357427101845524 0.798051763023851700484615403959 174 2 30 [5]
82 0.039393416735785588980126316269 0.091030166755467165692653524617 25.3849521788645597488118081743 0.799541520116696457707666354192 169 6 29 [6]
83 0.039171058445401154434968508937 0.090437014849319306511540734148 25.5290523076841741299602364105 0.800181625300250892218634937018 178 5 29 [6]
84 0.038906886690802678971918807096 0.089733673317322938087103263967 25.7023906319493082912538574697 0.798936227900518380252281480907 170 3 31 [6]
85 0.038664292457572917885535785240 0.089089068135313097483564074994 25.8636570447349967988503235098 0.798397058785871868483596244526 184 5 29 [6]
86 0.038483258924123575320205308525 0.088608838911977289488578724160 25.9853252545911866367647030163 0.800243223515082061049314749104 181 7 32 [5]
87 0.038291635579757780019559443384 0.088101262027837264489221263762 26.1153639655087739908163920822 0.801506327351666768712585669947 179 7 31 [6]
88 0.038087925226711666783978530630 0.087562505947546614958533776458 26.2550399909597627639074475988 0.802115986789931065752615064215 185 8 32 [6]
89 0.037863497233666749164449824658 0.086969954024228718628267735971 26.4106612716914826266685538424 0.801698967981872706127622169563 192 6 33 [6]
90 0.037692475487242589573876187843 0.086519109048157772430150679652 26.5304941390347365892574086795 0.803399764204909570662687404503 175 7 32 [5]
91 0.037471662651535436206963530619 0.085937898533063812745460157828 26.6868329088947984138221058210 0.802836645773774975836803869788 184 8 34 [6]
92 0.037264690039514402311535477755 0.085394029110194849892065853754 26.8350548183717306612439503935 0.802717483367222385133206820440 193 8 33 [8]
93 0.037087301465658458898371786150 0.084928599119251415876899847013 26.9634068934879218988914840513 0.803735749612069643735958441786 183 5 33 [5]
94 0.036894699782078599751850736775 0.084423983523159607776307226761 27.1041641728101166046749509948 0.803962299147896513148584475800 196 6 34 [6]
95 0.036698643855145706404699424636 0.083911097075895669310105374215 27.2489633117542253530789977102 0.803902733343150651673077653467 203 6 33 D1 [6]
96 0.036528541360465405736724439656 0.083466740924861234781902767178 27.3758535861575152885676015460 0.804851508650084192038746076317 184 8 34 [6]
97 0.036370285227438766405826136730 0.083053860023443209730331620826 27.4949727159569226119869683495 0.806204126100330452643869270795 182 11 32 [6]
98 0.036188454424976049430221322261 0.082580102676824111845235849880 27.6331226599673116333282119193 0.806391635762240157928952557906 221 5 33 [6]
99 0.036005719267126801921579945796 0.082104664730410793301982080510 27.7733654639972529087158770201 0.806413974212951661023788085345 198 8 35 [6]
100 0.035803503873716013690544482287 0.081579331143843909073394168376 27.9302272628718249835124183232 0.805435800316324240655906711002 207 8 33 [5]
101 0.035658964130103636934822646848 0.081204338795983619187108206105 28.0434394098338510512890830626 0.806935251341079709983770362312 207 9 35 [6]
102 0.035495740294867397500361418047 0.080781378970496905117919483506 28.1723945378481851983169915224 0.807481379903009845379145849137 205 7 35 [6]
103 0.035321486940073031617824515957 0.080330430112890309266255330321 28.3113788979669827624827226362 0.807411719600508113363656855723 194 8 35 [6]
104 0.035161727350453294736088134374 0.079917525361418865372985467739 28.4400134849207939038583622360 0.807892565633920838683393309335 212 11 36 [6]
105 0.035002899389078017195423633237 0.079507535550413631474933510446 28.5690619192543461992458105751 0.808308613563586414273192155503 227 7 34 [6]
106 0.034864593631030463076200801255 0.079150931754149919254186442436 28.6823936794711997443765900222 0.809571011348573274486374373642 209 7 37 [6]
107 0.034698078329801354932758582117 0.078722100110945588619449579496 28.8200398447173878044805897498 0.809421045642221947166082852160 231 7 34 [6]
108 0.034542778791871884710963290664 0.078322650940848352613021772907 28.9496107428191554088395964551 0.809688864802184175416732095047 236 7 34 [6]
109 0.034372259323998769332424170875 0.077884607044462099704159053767 29.0932286578496257388816562458 0.809137865913420110860707345792 231 8 38 [6]
110 0.034233576903981078701085691481 0.077528774546236856408912694099 29.2110871967839376704257429996 0.809985251705645576319263094586 196 10 34 [6]
111 0.034086416480706564837550946824 0.077151606146844006306329180207 29.3371994843170257013817904218 0.810336757486760989742062085648 240 8 36 [8]
112 0.033944935518406306598892913973 0.076789398252722952525582189528 29.4594756103678517719664353262 0.810863715546345010852481163947 222 6 35 [6]
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229 0.024084500459467290103007722168 0.052484807087231654444187586108 41.5204791846497932384155750610 0.834623553657383629277582698997 481 11 53 [6]
230 0.024036148148198053021410571416 0.052370017880378149349740043537 41.6040038459726422108724341998 0.834905743504081653653649381269 464 11 53 [6]
231 0.023986316043926012758399259040 0.052251758814860704677248786145 41.6904370879090114785975099452 0.835062442724222028998815854203 482 14 56 [6]
232 0.023935164074611907684427262278 0.052130413089493577365342229466 41.7795339477410420080159247921 0.835104206068493131115663036133 526 13 55 [6]
233 0.023893526215715307133559732738 0.052031671371384578173730353405 41.8523407123674199016618928127 0.835788295705684405839004978935 540 13 54 [6]
234 0.023843951158961837355723027786 0.051914146914887685656018579190 41.9393578410407998267151745531 0.835895857353228014279021843826 543 12 54 [6]
235 0.023789163431004572040761513113 0.051784315478143222458716583506 42.0359464468049965173824285560 0.835614698179789074241235307004 530 10 56 [6]
236 0.023743899908977693673052675853 0.051677093591257801167389567104 42.1160805020869688952250281180 0.835980172382326452891683315865 537 11 55 [6]
237 0.023695255577671141004767540353 0.051561903317023587991058420757 42.2025412100781302363466225207 0.836086111170115504965164509592 541 13 57 [6]
238 0.023646080813452248897244692365 0.051445499277879373509811488896 42.2903062832763505543233798240 0.836132615298075787511884163724 519 13 58 [6]
239 0.023588960994927703479526497723 0.051310341472024801206313247688 42.3927107351200588764884202191 0.835594155811079745076292029959 545 12 57 [6]
240 0.023540144212784773869791745828 0.051194875870916130930251785207 42.4806233539085472795228250725 0.835621004980658392943896675977 540 15 56 [6]
241 0.023489510803114194681394161845 0.051075157638242061251264027808 42.5721935370157602989221303111 0.835496923935939446828839892943 559 10 55 [6]
242 0.023449691072916169631910709134 0.050981039003774944453644391512 42.6444850335353031060947623172 0.836121682288207395397915675939 546 15 58 [6]
243 0.023402793568819185372229773591 0.050870226883763610820089182391 42.7299414943502464932563492382 0.836221916158600927732212025546 552 17 54 [6]
244 0.023361701426167729351469291911 0.050773163702686788991995413066 42.8051014674764950953826530604 0.836717077290580847403037613294 574 12 56 [6]
245 0.023309894997663702713759402136 0.050650834624581914879280774593 42.9002361486496487990688491838 0.836424195129477122011092804902 561 13 57 [6]
246 0.023260295185694073822574375711 0.050533760015853059129108835102 42.9917157979592757896928309478 0.836267885352791821884689240165 480 11 57 [6]
247 0.023220619522626652354396589463 0.050440141213531054074720385258 43.0651731331103934989852861464 0.836805308420142692123812633870 584 13 56 [6]
248 0.023168220061789816647317582681 0.050316541510664810618468102496 43.1625734447010819487013360057 0.836405516411483543029252420367 545 11 60 [6]
249 0.023128949875853259864392333018 0.050223942582111717805012388641 43.2358583233389698973586308468 0.836933680348609072899201405156 578 17 57 [6]
250 0.023079967224267566546971350123 0.050108479480595501008513922289 43.3276178550437519850973479728 0.836739463953839159868712619753 570 12 58 [6]
251 0.023035567155365513321671194263 0.050003854772122120397421315558 43.4111299824053606498941459931 0.836857299616542061834936558081 542 15 56 [6]
252 0.022990008318987469857239762320 0.049896535315458044523891801121 43.4971569442234191644868009499 0.836871284072452125417831976742 579 12 53 [6]
253 0.022944466813648199355759296254 0.049789292887805113718769745019 43.5834926181489562497586290097 0.836866781673469892244608348573 579 11 60 [6]
254 0.022895468700979210923055041016 0.049673951144251837697842304514 43.6767647371740083611240495524 0.836589986593448285206078023226 575 9 61 [6]
255 0.022858649079053652195678937719 0.049587305159309110427864086144 43.7471171870931923110559361592 0.837184482177806744180303994252 601 13 58 [6]
256 0.022823354557619637628769097697 0.049504270303698063622033252134 43.8147686605581535786201111194 0.837874142044614021854753689838 584 16 59 [6]
257 0.022778605996431135977367687870 0.049399024812460694693245152440 43.9008427537960899919539108276 0.837851934929548376858189475355 596 8 57 [6]
258 0.022734369655630379315260803513 0.049295018296754301998028361795 43.9862646357710052206865190462 0.837848330867287409541046123034 595 12 59 [6]
259 0.022695320701969673119656578088 0.049203236434214036436245880847 44.0619462104896525630256684194 0.838208924495503054835370988236 572 15 58 [6]
260 0.022647631758477380235674196963 0.049091182849863258403468610659 44.1547271107356992891701157061 0.837912764246184713597193954609 423 23 48 [6]
261 0.022610418449000517419990391520 0.049003771082118959311931296014 44.2273990751464626226208546096 0.838373563927015784775816788818 586 14 59 [6]
262 0.022570701609367717300745092083 0.048910505226279307298698618186 44.3052244146881629385030658501 0.838631707224962828833393271446 558 13 58 [6]
263 0.022529136858314441215856762693 0.048812929319623387491877630939 44.3869645911865989733318797974 0.838734917059021884018270376137 548 16 59 [6]
264 0.022495082146221514494580806220 0.048733006066445376266846469445 44.4541608472396444401012522420 0.839380666957305068562072617112 592 15 58 [6]
265 0.022444160230695194823588056264 0.048613534600429261433058554787 44.5550196452605516342702882951 0.838749865061492569972426379524 614 11 60 [6]
266 0.022398697814338573794248496356 0.048506910053774867773555473809 44.6454525298273314943984194038 0.838507682968463287221372251085 587 14 59 [6]
267 0.022363264304949120895764261604 0.048423831492912564321078059620 44.7161910874833315058548318980 0.838999154129281163037782252058 480 11 57 [6]
268 0.022317605020243469805457169785 0.048316809330659906307366732234 44.8076753349177556151237750085 0.838706167435457434594633827196 599 16 60 [6]
269 0.022290287491545429323127887859 0.048252796226434964782784578777 44.8625887117559128392123863574 0.839776056778986375570318017512 626 17 60 D1 [6]
270 0.022245690608903079690293744197 0.048148320218505447797511831979 44.9525266525006903104519880599 0.839528449976898881230787855877 635 15 61 [6]
271 0.022195409688557981616261999271 0.048030569642073897448415004712 45.0543609706611014481013250985 0.838832967756145319872325957429 561 16 61 [6]
272 0.022160697725320837229754005722 0.047949304810767444732066636943 45.1249329960129782648675187560 0.839296925650241616394721951588 643 9 63 [6]
273 0.022117229076365001134750496389 0.047847568919983564741017976678 45.2136204109141260876018671458 0.839081116904870774082036045281 635 13 57 [6]
274 0.022086131505356453817858168329 0.047774806871146916299398752054 45.2772817981942347418253096229 0.839788143166421763584539505442 620 16 60 [6]
275 0.022055886261172340114160973353 0.047704055123724002833214786100 45.3393705498210543072339960468 0.840546200210349052191058536760 563 18 60 [6]
276 0.022010358605427197713609911864 0.047597583503298766352321338331 45.4331534495501265243590021808 0.840123604845244509167144250509 650 11 60 [6]
277 0.021971840901308172088323012152 0.047507533399066964917883501895 45.5127999739184997199161178714 0.840219059177334062133934498929 615 18 62 [6]
278 0.021924249634895798156805791058 0.047396305673679019134608934567 45.6115952268827928027589880062 0.839603308165187843799940688910 627 12 61 [6]
279 0.021885676951730618952621482638 0.047306184353898722283608587629 45.6919839493895383754309633697 0.839661113416673094556304410901 642 11 59 [6]
280 0.021853005235358596381471286167 0.047229870133549212240993787212 45.7602965463981193817440936078 0.840156592003833585987130263705 651 9 62 [6]
281 0.021819887352002277079957747134 0.047152532535896084232434743476 45.8297508079589668631835863889 0.840603505113541033263075151808 651 17 61 [6]
282 0.021786052404929988593179584923 0.047073539948429115887915592477 45.9009269514888779810696890065 0.840980768726725053518308529013 651 9 61 [6]
283 0.021745284972951021241503348584 0.046978388656945590412415168221 45.9869806831182421230653523034 0.840807372057279626983056821831 602 12 62 [6]
284 0.021695998052797688780586047086 0.046863391057112465455800114083 46.0914495644071312346338693497 0.839957814384691199405636687040 669 10 62 [6]
285 0.021661292323165651734278444524 0.046782439768707085352861600309 46.1652972999469103946307424549 0.840220851193001615276155651148 663 14 62 [6]
286 0.021631326997472829715029431219 0.046712562159628250137895485246 46.2292489090858449941397414426 0.840837798652647837567144618249 642 13 61 [6]
287 0.021589188419877711311659524984 0.046614323264530837623338446168 46.3194808693815804197853398896 0.840493575969427208571607532259 685 8 59 [6]
288 0.021555939988186400630352833205 0.046536831761737473956468526817 46.3909252182017487947682649547 0.840826300756040728385829407681 624 16 59 [6]
289 0.021531503839036265433328712811 0.046479890989074171970032390535 46.4435743771420323619634676372 0.841833953975872739950658882158 681 14 61 [6]
290 0.021489806016526542749545618876 0.046382750983066842541848921615 46.5336913339729060781161203284 0.841478175671708764684510043053 638 16 63 [6]
291 0.021449966547123511348330417476 0.046289968109162666830194003267 46.6201193276034383459368638783 0.841251972932149065779801481944 671 10 61 [6]
292 0.021424778601857855307697129991 0.046231321491165439004601295523 46.6749280626538067078283322161 0.842161542284540566144179484364 693 18 63 [6]
293 0.021384735887631629126041174492 0.046138110014892047216471604839 46.7623264207987621723007887524 0.841889843879218790738728499509 604 20 60 [6]
294 0.021346547960303288556739989055 0.046049241723387969773718090438 46.8459819292389301971059079279 0.841748799534153250978933050272 680 16 64 [6]
294 0.021346547960303288556739989055 0.046049241723387969773718090438 46.8459819292389301971059079279 0.841748799534153250978933050272 673 16 64 [6]
295 0.021316856358852165442437038758 0.045980162761565566399122912023 46.9112322739245747557250755185 0.842263928960478429408655584434 690 13 63 [6]
296 0.021275607176299788668150888244 0.045884219587362520939488994103 47.0021838490213191320918148856 0.841851530152114815297737840543 711 10 63 [6]
297 0.021237377022300491080809459535 0.045795324512097750734228755397 47.0867941436431329149440263383 0.841662682198278796566906154171 605 9 62 [6]
298 0.021208993940380620791808476959 0.045729342628525885996359566441 47.1498083695550248530441143788 0.842240785846069285596394509203 720 9 62 [6]
299 0.021163661683864621636684778115 0.045623987746830819485292626323 47.2508025755490874023370872901 0.841458452895984933700402636277 664 14 63 [6]





Updates

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

20-May-2009: First complete presentation from N=1 to N=240. Packing of circles in this container was initiated by David Alan Paterson, CSIRO Materials Science and Engineering, Highett & Clayton, Australia. He had adopted the code for this case, great work. Many thanks to him!
28-May-2010: Improvements for N=104, 212 and 226 by E. Specht [6].
17-Oct-2010: Claudia Orquídea López Soto, PhD student of John E. Beasley at Brunel University, UK, sent me some difficult to find improvements for N=30, 60, 65, 90 and 100. Many thanks and welcome to the list of contributors!
18-Oct-2010: Improvements for N=92, 93, 138, 139, 147, 148, 149, 164, 177, 182, 183, 191, 192, 200, 211, 224, 225, 226, 230, 231, 233, 235, 240 and extension from N=241 to N=250 by E. Specht [6].
21-Oct-2010: Further improvements for N=159, 181, 183, 184, 188, 189, 191, 192, 194, 197, 198, 199, 200, 202, 204, 211, 212, 214, 215, 217, 219, 221, 222, 223, 224, 225, 226, 227, 229, 230, 231, 233, 234, 235, 236, 239 and 240 by E. Specht [6].
26-Oct-2010: More improvements for N=38, 50, 53, 64, 70, 77, 81, 86, 90, 93 and 100 by David W. Cantrell [5].
26-Oct-2010: Even more improvements for today for N=43, 92, 131, 143, 149, 151, 153, 173, 175, 182, 186, 187, 188, 191, 192, 195 and 197 by E. Specht [6].
27-Oct-2010: Next improvements for N=200, 201, 202, 203, 205, 206, 207, 208, 209, 210, 211, 212, 218, 219, 220, 221, 223, 224, 225, 228, 230, 231, 233, 234, 236, 237, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250 and extension up to N=299 by E. Specht [6].
26-Feb-2011: A bundle of better packings for N=13, 27, 36, 44, 48, 56, 57, 61, 68, 92, 111, 116 and 130 by Claudia Orquídea López Soto [8].
09-Mar-2011: Some slightly improvements for N=59, 61, 70, 72, 73 and 78 by David W. Cantrell [5]. The only remaining packings with cyan circles among the first 100 packings are now N=88 and 97.
18-Mar-2011: Only removals of "hidden rattlers" (i.e. such circles that have previously only one contact) by push them slightly apart for N=88, 101, 147, 156, 158 and 166 by E. Specht [6].

References

[1]   Yinfeng Xu, On the minimum distance determined by n(≤7) points in an isoscele right triangle, Acta Mathematicae Applicatae Sinica 12 (1996) 2, 169–175.
[2]   Hans Melissen, 1997 (referenced by Erich Friedman).
[3]   Erich Friedman, April 2005.
[4]   Maurizio Morandi, June 2005 (referenced by Erich Friedman).
[5]   , June 2005 (referenced by Erich Friedman) and private communication October 2010.
[6]   , program crt, 2009–2010.
[7]   , private communication, September 2010.
[8]   , private communication, February 2011.
[9]   C.O. López, J.E. Beasley, A heuristic for the circle packing problem with a variety of containers, European Journal of Operational Research 214 (2011) 3, 512–525.


©  E. Specht     08-Nov-2011