The Building Game: From Enumerative Combinatorics to Conformational Diffusion

We study a discrete attachment model for the self-assembly of polyhedra called the building game.We investigate two distinct aspects of the model: (i) enumerative combinatorics of the intermediate states and (ii) a notion of Brownian motion for the polyhedral linkage defined by each intermediate that we term conformational diffusion. The combinatorial configuration space of the model is computed for the Platonic, Archimedean, and Catalan solids of up to 30 faces, and several novel enumerative results are generated. These represent the most exhaustive computations of this nature to date. We further extend the building game to include geometric information. The combinatorial structure of each intermediate yields a systems of constraints specifying a polyhedral linkage and itsmoduli space.Weuse a randomwalk to simulate a reflected Brownian motion in each moduli space. Empirical statistics of the random walk may be used to define the rates of transition for a Markov process modeling the process of self-assembly.