## The Densest Packing of 13 Equal Circles in a Circle: Two Solutions

The case *n*=13 has the curious property that the densest packing
not only comes in two varieties but one is totally rigid while the other
has three loose circles. These have exactly the same density.
Here is a picture showing the two possibilities:
None of the 13 small disks overlap and all apparent tangencies are exact
tangencies. The three red circles in the right hand picture (which are
now loose) can be pushed down until they are tightly against other circles and
then there will be just enough room to pick up the top circle and place is
lower, fitting exactly and transforming the right had picture into the left
hand picture. (Unfortunately, the circles are numbered differently in the
two picures.)
I discovered this in 1996. Very likely it was already known^{1}.
My proof is brute-force computation. A slick geometric proof would be
nice.
When the small circles are unit circles both circumcircles have radius, R,
exactly equal to 1+csc(Pi/10) = 2+sqrt(5).

A similar thing, i.e. two equally good optimal packings occurs for
*n* = 11 , which, so far as I know has not
been previously noted^{2}.

1 Hugo Pfoertner informs me that both of the *n* = 13
arrangements have been found independently before,

one attibuted to Kravits in 1967, and that they have both (in separate places) appeared on the web.

I still don't know for sure if it has been previously noted
that their circumradii are exactly equal.

2 Hugo Pfoertner informs me H. Milissen proved the optimality
of the eleven circle arrangement in:

Densest Packing of Eleven Congruent
Circles in a Circle, Geom. Dedicata 50 (1994), p. 15-25.

Presumably he mentions both arrangements, but I have not yet seen his paper to verify this.

See Hugo's web site
for lots of additional packing information.

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