1 research outputs found
Characterization and Construction of a Family of Highly Symmetric Spherical Polyhedra with Application in Modeling Self-Assembling Structures
The regular polyhedra have the highest order of 3D symmetries and are
exceptionally at- tractive templates for (self)-assembly using minimal types of
building blocks, from nano-cages and virus capsids to large scale constructions
like glass domes. However, they only represent a small number of possible
spherical layouts which can serve as templates for symmetric assembly. In this
paper, we formalize the necessary and sufficient conditions for symmetric
assembly using exactly one type of building block. All such assemblies
correspond to spherical polyhedra which are edge-transitive and
face-transitive, but not necessarily vertex-transitive. This describes a new
class of polyhedra outside of the well-studied Platonic, Archimedean, Catalan
and and Johnson solids. We show that this new family, dubbed almost-regular
polyhedra, can be pa- rameterized using only two variables and provide an
efficient algorithm to generate an infinite series of such polyhedra.
Additionally, considering the almost-regular polyhedra as templates for the
assembly of 3D spherical shell structures, we developed an efficient polynomial
time shell assembly approximation algorithm for an otherwise NP-hard geometric
optimization problem.Comment: 25 pages, 12 figure