[seqfan] Re: Number of touching points between unit circles in the optimal packing of circles in a circle
Benoît Jubin
benoit.jubin at gmail.com
Sun Mar 19 19:00:46 CET 2017
The page http://hydra.nat.uni-magdeburg.de/packing/cci/ (linked from
wikipedia) seems to count the minimal number of contacts (and includes
contacts with the boundary, but those appear also separately in
another column). I would say that to get the maximum number of
contacts, one should add to the minimum number of contacts twice the
number of loose circles (also given on the page), but again, this is
for the configuration given on the page, and there might exist other
optimal configurations.
Therefore, these numbers are probably conjectural for n > 9 (as Neil
noted, there is a configuration that has been proven optimal for each
n <= 13, but to obtain the sequence of mutual contacts, one needs
more, namely, a description of all optimal configurations).
A sequence that would be interesting would be the n's which give
record densities (and also record lows). There are similar sequences
for circles in a square and many others
(http://hydra.nat.uni-magdeburg.de/packing/).
Benoit
On Sun, Mar 19, 2017 at 3:58 PM, Neil Sloane <njasloane at gmail.com> wrote:
> The Wikipedia page only say that the circumradius has been
> proved to be optimal (for n <= 13 circles)
> It does NOT say that the total number of contacts
> is optimal.
>
> It would seem to me to be equally interesting to maximize
> the number of pairwise contacts (B.J. suggested
> minimizing them). We should probably have both
> sequences - but only as far as they have been proved to be optimal.
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
>
> On Sun, Mar 19, 2017 at 10:30 AM, Benoît Jubin <benoit.jubin at gmail.com> wrote:
>> For a given n (number of disjoint unit disks), there can be many
>> configurations acheiving the minimum r (smallest radius of an
>> enclosing circle) with different t's (tangency numbers). You could
>> define a(n) to be the minimal t among the optimal configurations.
>>
>> This reminds me of the "kissing numbers", which should be in the OEIS.
>>
>> Regards,
>> Benoit
>>
>> On Sun, Mar 19, 2017 at 10:48 AM, Felix Fröhlich <felix.froe at gmail.com> wrote:
>>> Dear Sequence Fans,
>>>
>>> Let A be an arrangement of n unit circles in the plane, let r the radius of
>>> the smallest circle that can enclose A and let t be the number of points
>>> where two unit circles in the arrangement touch each other.
>>>
>>> A sequence arising from the above is the following:
>>>
>>> a(n) = the value of t for the specific A such that r is minimal over all
>>> possible A.
>>>
>>> https://en.wikipedia.org/wiki/Circle_packing_in_a_circle seems to suggest
>>> that the sequence with offset 2 starts
>>>
>>> 1, 3, 4, 5, 6, 12, 7, 8, 12, 14
>>>
>>> This is not in the OEIS. Are the terms correct and should this be in the
>>> OEIS?
>>>
>>> Best regards
>>> Felix Fröhlich
>>>
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>>
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