Abstract
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 its moduli space. We use a random walk 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.
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Notes
These data are publicly available at the Brown Digital Repository (Johnson and Menon 2016).
More precisely, we study a discretized reflected random walk on each algebraic variety. It has not been proven in general that these random walks converge to a reflected Brownian motion on each variety.
By ‘theoretical’ we mean results that are obtained without explicit enumeration on a computer. Closest in spirit to our work are enumerative results on polyominoes, configurations of n attached squares on a planar lattice.
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Acknowledgments
The authors are grateful to the referees of this article and the related paper Russell and Menon (2016) for their detailed, constructive criticism.
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Communicated by Robert V. Kohn.
Partially supported by NSF DMS Grants 1148284 and 1411278.
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Johnson-Chyzhykov, D., Menon, G. The Building Game: From Enumerative Combinatorics to Conformational Diffusion. J Nonlinear Sci 26, 815–845 (2016). https://doi.org/10.1007/s00332-016-9291-z
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DOI: https://doi.org/10.1007/s00332-016-9291-z