Multiplicity and Symmetry Breaking in (Conjectured) Densest Packings of Congruent Circles on a Sphere

The densest known packing of 15 congruent circles on a sphere occurs in two equally dense varieties. Previously, this was the only known example of a ‘best’ packing of n circles on a sphere with multiplicity greater than one. We present three new examples: n=62, 76 and 117. We discuss these and related examples as manifestations of symmetry breaking involving structures we call “rotational toggle hexagons”. Additionally, we present exact data for n=15, and observe that both the hexagonal packing of the plane and the icosahedral packing of 12 circles on a sphere arise as limiting cases from toggle hexagons