A set of elementary building blocks undergoes self-assembly if local interactions govern how this set forms intricate structures. Self-assembly has been widely observed in nature, ranging from the field of crystallography to the study of viruses and multicellular organisms. A natural question is whether a model of self-assembly can capture the hierarchical growth seen in nature or in other fields of mathematics. In this work, we consider hierarchical growth in substitution rules; informally, a substitution rule describes the iterated process by which the polygons of a given set are individually enlarged and dissected. We develop the Polygonal Two-Handed Assembly Model (p-2HAM) where building blocks, or tiles, are polygons and growth occurs when tiles bind to one another via matching, complementary bonds on adjacent sides; the resulting assemblies can then be used to construct new, larger structures. The p-2HAM is based on a handful of well-studied models, notably the Two-Handed Assembly Model and the polygonal free-body Tile Assembly Model.
The primary focus of our work is to provide conditions which are either necessary or sufficient for the ``bordered simulation'' substitution rules. By this, we mean that a border made up of tiles is allowed to form around an assembly which then coordinates how the assembly interacts with other assemblies. In our main result, we provide a construction which gives a sufficient condition for bordered simulation. This condition is presented in graph theoretic terms and considers the adjacency of the polygons in the tilings associated to a given substitution rule. Alongside our results, we consider a collection of over one hundred substitution rules from various sources. We show that only the substitution rules in this collection which satisfy our sufficient condition admit bordered simulation. We conclude by considering open questions related to simulating substitution rules and to hierarchical growth in general.
| Identifer | oai:union.ndltd.org:USF/oai:scholarcommons.usf.edu:etd-8967 |
| Date | 03 July 2019 |
| Creators | Cruz, Daniel Alejandro |
| Publisher | Scholar Commons |
| Source Sets | University of South Flordia |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Graduate Theses and Dissertations |
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