[seqfan] Re: Number of touching points between unit circles in the optimal packing of circles in a circle
Hugo Pfoertner
yae9911 at gmail.com
Sun Mar 19 23:11:24 CET 2017
Density records for the circles in circle packing are given in
http://oeis.org/A084644
On Sun, Mar 19, 2017 at 7:00 PM, Benoît Jubin <benoit.jubin at gmail.com>
wrote:
> 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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