  Many remarkably robust, rapid and spontaneous self-assembly phenomena in
nature can be modeled geometrically starting from a collection of rigid bunches
of spheres. This paper highlights the role of symmetry in sphere-based assembly
processes. Since spheres within bunches could be identical and bunches could be
identical as well, the underlying symmetry groups could be of large order that
grows with the number of participating spheres and bunches. Thus, understanding
symmetries and associated isomorphism classes of microstates correspond to
various types of macrostates can significantly reduce the complexity of
computing entropy and free energy, as well as paths and kinetics, in high
dimensional configuration spaces. In addition, a precise understanding of
symmetries is crucial for giving provable guarantees of algorithmic accuracy
and efficiency in such computations. In particular, this may aid in predicting
crucial assembly-driving interactions.
  This is a primarily expository paper that develops a novel, original
framework for dealing with symmetries in configuration spaces of assembling
spheres with the following goals. (1) We give new, formal definitions of
various concepts relevant to sphere-based assembly that occur in previous work,
and in turn, formal definitions of their relevant symmetry groups leading to
the main theorem concerning their symmetries. These previously developed
concepts include, for example, (a) assembly configuration spaces, (b)
stratification of assembly configuration space into regions defined by active
constraint graphs, (c) paths through the configurational regions, and (d)
coarse assembly pathways. (2) We demonstrate the new symmetry concepts to
compute sizes and numbers of orbits in two example settings appearing in
previous work. (3) We give formal statements of a variety of open problems and
challenges using the new conceptual definitions.
