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Full-Text Articles in Number Theory
On The Spectrum Of Quaquaversal Operators, Josiah Sugarman
On The Spectrum Of Quaquaversal Operators, Josiah Sugarman
Dissertations, Theses, and Capstone Projects
In 1998 Charles Radin and John Conway introduced the Quaquaversal Tiling. A three dimensional hierarchical tiling with the property that the orientations of its tiles approach a uniform distribution faster than what is possible for hierarchical tilings in two dimensions. The distribution of orientations is controlled by the spectrum of a certain Hecke operator, which we refer to as the Quaquaversal Operator. For example, by showing that the largest eigenvalue has multiplicity equal to one, Charles Radin and John Conway showed that the orientations of this tiling approach a uniform distribution. In 2008, Bourgain and Gamburd showed that this operator …
A Visual Tour Of Dynamical Systems On Color Space, Jonathan Maltsman
A Visual Tour Of Dynamical Systems On Color Space, Jonathan Maltsman
HMC Senior Theses
We can think of a pixel as a particle in three dimensional space, where its x, y and z coordinates correspond to its level of red, green, and blue, respectively. Just as a particle’s motion is guided by physical rules like gravity, we can construct rules to guide a pixel’s motion through color space. We can develop striking visuals by applying these rules, called dynamical systems, onto images using animation engines. This project explores a number of these systems while exposing the underlying algebraic structure of color space. We also build and demonstrate a Visual DJ circuit board for …