Plane group packings

 

Densities of the densest plane group packings of regular convex n-gons obtained by the Entropic trust region packing algorithm. Each table entry contains a visualization of the packing configuration.

3-gon4-gon5-gon6-gon7-gon8-gon9-gon10-gon11-gon12-gon13-gon14-gon15-gon16-gon17-gon18-gon19-gon20-gon21-gon22-gon23-gon24-gon25-gon30-gon36-gon37-gon39-gon42-gon55-gon89-gonDisc
p20.99999990.99999990.92131060.99999990.89269060.90616360.90102990.91371990.90766480.92820310.90437050.90746550.90471500.90901710.90692160.91622180.90595920.90733860.90577860.90789620.90683030.91211520.90642630.91022920.90920880.90671540.90657320.90859480.90682580.90688310.9068996
p2gg0.99999990.99999990.92131060.99999990.89269060.90616360.89989110.91371990.90766480.92820320.90437050.90746550.90403060.90901710.90692160.91622180.90595920.90733860.90536630.90789620.90683030.91211520.90642630.91022920.90920880.90671540.90643310.90859480.90682580.90688310.9068996
pg0.99999990.99999990.92131060.99999990.89269060.90616360.89860880.91371990.90766480.92820320.90437050.90746550.90235330.90901710.90692160.91622180.90595920.90733860.90523760.90789620.90683030.91211520.90642630.91022920.90920880.90671540.90641170.90859480.90682580.90688310.9068996
p30.66666660.77376840.87048000.99999990.88085000.87665060.88773540.88775060.89896290.92820320.90005140.89722130.90017410.89950670.90348310.91622180.90379710.90218080.90349220.90300300.90500540.91211520.90513660.91022920.90920880.90610770.90591680.90859480.90654510.90676870.9068996
p10.66666660.99999990.81725590.99999990.86019090.90616360.88773540.91371990.88912790.92820320.89383630.90746550.90017410.90901710.89947940.91622180.90084360.90733860.90349220.90789620.90284560.91211520.90341670.91022920.90920880.90531570.90591680.90859480.90618450.90662810.9068996
p2mg0.99999990.99999990.85410190.85714280.84226100.86555340.83419840.83722130.83116090.86156120.83306330.83823750.83993880.84856350.84215190.84346700.84189910.84571070.84052280.84068060.83929540.84662830.83952300.84239120.84393060.84076690.84147470.84209520.84180530.84179610.8417872
cm0.99999990.99999990.85410190.85714280.84226100.86555340.83419840.83722130.82795300.86156120.83212870.83823750.83993880.84856350.84215190.84346700.84189910.84571070.84052280.84068060.83865010.84662830.83916180.84239120.84393060.84058660.84147470.84209520.84180530.84179610.8417872
p40.71281270.99999990.84211370.81776890.84219450.85031060.83030530.83004040.83780420.86156120.83691050.83527880.83765860.84613840.84190330.83945770.84177460.84367770.85031060.83933890.84072730.84662830.84061520.84095760.84393060.84127380.84118360.84136520.84179120.84179000.8417872
p4gm0.69615240.99999990.71119040.74613380.76477350.82842710.76593230.77205630.77628330.80384750.77662800.77869770.78028790.79564940.78043390.78137100.78213470.79192220.78220900.78271110.78313530.78991490.78317870.78395750.78739790.78440800.78458620.78466440.78498450.78523170.7853981
c2mm0.66666660.99999990.71714030.74999990.76253210.82842710.76697420.77254240.77570960.80384750.77697420.77882320.78006200.79564940.78058880.78141670.78202370.79192220.78229180.78273160.78307280.78991490.78322740.78396340.78739790.78442310.78457340.78466590.78497990.78523280.7853981
pm0.49999990.99999990.69098300.74999990.73825040.82842710.75712160.77254240.76655080.80384750.77193720.77882320.77530310.79564940.77754670.78141670.77911720.79192220.78025930.78273160.78111580.78991490.78177470.78396340.78739790.78374540.78391070.78466590.78465050.78511260.7853981
p2mm0.49999990.99999990.69098300.74999990.73825040.82842710.75712160.77254240.76655080.80384750.77193720.77882320.77530310.79564940.77754670.78141670.77911720.79192220.78025930.78273160.78111580.78991490.78177470.78396340.78739790.78374540.78391070.78466590.78465050.78511260.7853981
p60.99999990.72193580.75933020.85714280.75740120.76438820.78535280.76932140.77293480.79560270.77254250.77326660.77997690.77425980.77536100.78533300.77523950.77536410.77863840.77571940.77622020.78181300.77616830.78019660.77932180.77682460.77770290.77879550.77711350.77726140.7773425
p31m0.74999990.66323010.70166190.71999990.72084060.71565440.73410840.71653940.72182800.74613390.72205890.72380320.73087470.72583400.72735160.72825520.72787740.72698910.72996480.72662410.72722640.73320160.72725520.72874410.73086530.72822730.72928700.72887480.72886910.72895340.7290091
p3m10.99999990.49742260.53854070.66666660.57196550.57980830.63041490.58887750.59164260.61880210.59535790.59664150.61359790.59851850.59921670.61081450.60029510.60071650.60915080.60139270.60166640.60807680.60211810.60681950.60613920.60346840.60591090.60572980.60408800.60440440.6045997
p4mm0.57735020.49999990.55537630.52148720.51446180.56854240.52443260.53314420.54240650.53589830.54144120.53607620.53385700.54604740.53500760.53725020.54014840.53792030.53994210.53783730.53683210.54211180.53716820.53838270.53867850.53931150.53825600.53869160.53863220.53882750.5390120
p6mm0.49999990.46410160.49372790.47999990.49025210.48235320.48940560.48305400.47750210.49742260.47995120.48454480.48724980.48518380.48675890.48550350.48660960.48548640.48664320.48542530.48409000.48880100.48418860.48582940.48724350.48526760.48619130.48591650.48607840.48603370.4860060