The Densest Packing of 11 Equal Circles in a Circle: Two Solutions

The case n=11 is interesting because the densest packing comes in two different forms.  These have exactly the same density.  Here is a picture showing the two possibilities:
None of the 11 small disks overlap and all apparent tangencies are exact tangencies.  The curious thing is that the gray disk in the left picture can be picked up and put back down as shown in the right hand picture.  It fits exactly.  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, where R = 1+csc(Pi/9) exactly.

An approximate value is R = 3.9238044001630872522327544133662917.

R is an algebraic number with minimal polynomial 3*x^6-18*x^5+9*x^4+84*x^3-75*x^2-66*x-1. This factors over Q(sqrt(3)) and we find that R is a root of the polynomial x^3+(-3-2*3^(1/2))*x^2+(4*3^(1/2)+3)*x+2/3*3^(1/2)-1. This cubic has three real roots so exact solutions involve cube roots of complex numbers.

Let s = (-12*sqrt(3)+36*i)^(1/3) where i^2 = 1, then R = s/3 + 4/s + 1 + 2/sqrt(3) exactly.

A similar thing, i.e. two equally good optimal packings occurs for n = 13 , which, so far as I know has not been previously noted2.

    1  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.

    2  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.
        See Hugo's web site for lots of additional packing information.

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