FOAM2D

Packing 2^10 disks of radius r= 2^N[ µ=0 , s= 2^x ] by central accretion

Example: x= -1 -> r= 2^N[0, ½]

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the density relative to the hexagonal packing for r=1 = N[1,0] seems to be worst around x=-2.

For n from 2^4 to 2^12 disks, and x from -6 to +1 :

{ x , average relative density, st. dev } of 9 trials each:

{{{1, 1.02617}, {0, 0.991686}, {-1, 0.968466}, {-2, 0.964752}, {-3, 0.967135}, {-4, 0.975938}, {-5, 0.988774}, {-6, 0.995003}},

{{1, 1.0161}, {0, 0.972108}, {-1, 0.953107}, {-2, 0.949295}, {-3, 0.946075}, {-4, 0.961704}, {-5, 0.97937}, {-6, 0.987076}},

{{1, 1.00405}, {0, 0.953407}, {-1, 0.936764}, {-2, 0.938327}, {-3, 0.935906}, {-4, 0.951348}, {-5, 0.968828}, {-6, 0.982599}},

{{1, 0.993151}, {0, 0.959808}, {-1, 0.930754}, {-2, 0.927054}, {-3, 0.928213}, {-4, 0.939786}, {-5, 0.962663}, {-6, 0.977621}},

{{1, 0.997315}, {0, 0.944644}, {-1, 0.922496}, {-2, 0.920333}, {-3, 0.920602}, {-4, 0.929189}, {-5, 0.949664}, {-6, 0.970883}},

{{1, 0.988552}, {0, 0.937059}, {-1, 0.919512}, {-2, 0.914515}, {-3, 0.913327}, {-4, 0.919621}, {-5, 0.937403}, {-6, 0.956857}},

{{1, 0.969229}, {0, 0.938508}, {-1, 0.912795}, {-2, 0.910102}, {-3, 0.911495}, {-4, 0.916161}, {-5, 0.92741}, {-6, 0.947013}},

{{1, 0.96712}, {0, 0.931581}, {-1, 0.910554}, {-2, 0.907608}, {-3, 0.908906}, {-4, 0.913843}, {-5, 0.922806}, {-6, 0.93496}},

{{1, 0.969043}, {0, 0.929894}, {-1, 0.908601}, {-2, 0.905374}, {-3, 0.906803}, {-4, 0.912763}, {-5, 0.919392}, {-6, 0.927296}}}

It is notable that lager values of n result in worse packing. This is most evident at very narrow distributions (x=-6).

Probably an effect of accumulation of small deviations from the ideal N[1,0] packing

zooming in on the interval x= -3 .. –1 for 1024 disks, averaged over 16 trials seems to confirm this

{{-3.4, 0.912441, 0.00203675}, {-3.2, 0.911916, 0.00139081}, {-3., 0.91177,

0.000993791}, {-2.8, 0.911028, 0.00107375}, {-2.6, 0.910904,

0.00133503}, {-2.4, 0.910417, 0.000908881}, {-2.2, 0.909868,

0.00130885}, {-2., 0.910446, 0.00177819}, {-1.8, 0.910387,

0.0015151}, {-1.6, 0.910886, 0.00169118}, {-1.4, 0.911392,

0.00127439}, {-1.2, 0.913151, 0.00165397}, {-1., 0.913942, 0.00176514}}