I discovered this in 1996. Very likely it was already known1. 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 noted2.
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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