The densest known arrangement for n=64 circles in a circle is a symmetric arrangement which you can find on E. Specht's page here. However, this symmetric arrangement of 64 circles is not really a packing because it has 58 non-loose circles and so needs at least 116=2*58 contacts as a necessary condition for a packing. But it has only 115 contacts, i.e. there is one degree of freedom. The circles are not rigid, collectively they can move a tiny bit until another contact is achieved. At this point all the small circles can be slightly larger (equivalently, if we maintain their size as unit circles the circumcircle can be made slightly smaller). Here is a color coded picture showing the symmetric arrangement and the new non-symmetric arrangement which is slightly denser:
But for the colors, these arrangements look the same to the naked eye. The red circles are loose and can rattle a bit between their nearest neighbors. The yellow circles each have 5 contacts. This is key, since we now see that the right hand picture is a little tighter at the bottom. Circle 22 now touches circle 14, whereas they did not touch before. It is really too close to tell in these pictures, so look instead at the contact diagrams, one for each arrangement. Two circles are in contact just in case a line joins them.
And now for the new non-symmetric packing:
There is just one new contact: between circle 22 and circle 14. This allows for a slightly denser packing. (The new contact could have been between circle 21 and circle 13. That would have given the same new packing just rotated by 180 degrees.)
The green circles have 4 contacts and the pale blue circles have 3 contacts.
How much better is the new packing? Just a tiny bit, but this still means it is better. Here is the numeric data. First assume that all small circles have unit radius. Then, the circumradius, R is
8.96197110850392353216121 for the symmetric arrangement 8.96197110848573830216130 for the improved non-symmetric packing.
Symmetric: Radius of circumcircle = R = 8.96197110850392353216121, centers of
unit circles here.
Improved packing, no symmetry: Radius of circumcircle = R = 8.96197110848573830216130, centers of unit circles here.
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