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The regular polyhedra have the highest order of 3D symmetries and areexceptionally at- tractive templates for (self)-assembly using minimal types ofbuilding blocks, from nano-cages and virus capsids to large scale constructionslike glass domes. However, they only represent a small number of possiblespherical layouts which can serve as templates for symmetric assembly. In thispaper, we formalize the necessary and sufficient conditions for symmetricassembly using exactly one type of building block. All such assembliescorrespond to spherical polyhedra which are edge-transitive andface-transitive, but not necessarily vertex-transitive. This describes a newclass of polyhedra outside of the well-studied Platonic, Archimedean, Catalanand and Johnson solids. We show that this new family, dubbed almost-regularpolyhedra, can be pa- rameterized using only two variables and provide anefficient algorithm to generate an infinite series of such polyhedra.Additionally, considering the almost-regular polyhedra as templates for theassembly of 3D spherical shell structures, we developed an efficient polynomialtime shell assembly approximation algorithm for an otherwise NP-hard geometricoptimization problem.

Characterization and Construction of a Family of Highly Symmetric Spherical Polyhedra with Application in Modeling Self-Assembling Structures