We answer the question of whether optimal packings of circles on a sphere are equilibrium solutions to the logarithmic particle interaction problem for values of
N
=3–12 and 24, the only values of
N
for which the optimal packing problem (also known as the Tammes problem) has rigorously known solutions. We also address the cases
N
=13–23 where optimal packing solutions have been computed, but not proven analytically. As in Jamaloodeen & Newton (Jamaloodeen & Newton 2006
Proc. R. Soc. Lond. Ser. A
462
, 3277–3299. (
doi:10.1098/rspa.2006.1731
)), a logarithmic, or point vortex equilibrium is determined by formulating the problem as the one in linear algebra,
, where
A
is a
N
(
N
−1)/2×
N
non-normal configuration matrix obtained by requiring that all interparticle distances remain constant. If
A
has a kernel, the strength vector
is then determined as a right-singular vector associated with the zero singular value, or a vector that lies in the nullspace of
A
where the kernel is multi-dimensional. First we determine if the known optimal packing solution for a given value of
N
has a configuration matrix
A
with a non-empty nullspace. The answer is yes for
N
=3–9, 11–14, 16 and no for
N
=10, 15, 17–24. We then determine a basis set for the nullspace of
A
associated with the optimally packed state, answer the question of whether
N
-equal strength particles,
, form an equilibrium for this configuration, and describe what is special about the icosahedral configuration from this point of view. We also find new equilibria by implementing two versions of a random walk algorithm. First, we cluster sub-groups of particles into patterns during the packing process, and ‘grow’ a packed state using a version of the ‘yin-yang’ algorithm of Longuet-Higgins (Longuet-Higgins 2008
Proc. R. Soc. A
(doi:10.1098/rspa.2008.0219)). We also implement a version of our ‘Brownian ratchet’ algorithm to find new equilibria near the optimally packed state for
N
=10, 15, 17–24.
Optimal Random Packing of Spheres and Extremal Effective Conductivity
