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In this thesis, we use an area of mathematics called dynamical systems to study two types of objects: number expansions and translation surfaces.  The most familiar way of representing numbers is via decimal expansions, e.g., 51 or 3.14159....  However, there are several other expansions of numbers, including binary expansions, Lüroth series expansions, and continued fractions.

Our first chapter builds a broad, unifying theory for a large class of new continued fraction algorithms. Within this theory, we find several well-studied algorithms; a new subfamily of superoptimal continued fractions with arbitrarily good convergence and approximation properties and a unifying framework to prove several old and new results on certain rational approximations of irrational numbers.

Our second chapter introduces a new family of functions called skewed symmetric golden maps, each of which generates an expansion of each number in its domain. Using tools from ergodic theory, we study the relative frequencies of digits typically occurring in these number expansions.  The central tool for our analysis is a mysterious phenomenon of our functions called matching, which has been recently observed and exploited to understand several other families of functions generating number expansions.

Our final chapter deals with symmetries of translation surfaces, which are objects constructed by glueing together certain edges of polygons. We develop a novel algorithm to construct translation surfaces with prescribed, large symmetry groups.  Our ideas are also used to obtain obstructions for the realisability of certain symmetry groups of translation surfaces.