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What is PascGalois?
 The PascGalois project has its origin in a simple exercise with Pascal's triangle. Take each entry in the triangle and replace it with its congruent value mod n, where n is a positive integer larger than 1. By assigning each of the values 0, 1, ..., n with a distinct color, patterns reminiscent of fractals appear. These structures can be treated as types of 1-dimensional cellular automata. Our interest in this construction lies in the fact that arithmetic mod n is the multiplication for the cyclic group Zn and the patterns seen in the triangles are related to the structure of these groups. We generalize this construction using other groups. If G is a group with a; b 2 G, then a PascGalois triangle is formed by placing a down the left side of the triangle and b down the right. An entry in the interior of the triangle is determined by multiplying the two entries above it using the group multiplication.
Like Pascal's triangle mod n, PascGalois triangles can have self-similar properties. Furthermore, many of these properties can be described using subgroups, quotients, and automorphisms of the group G. A related structure is a 2-dimensional cellular automaton. 2-D automata consist of rectangular grids of cells which take on various state values that change discretely over time according to some local rule. We use groups (and sometimes other algebraic structures) as our alphabets and group multiplication for the various local rules. The long term behavior of these systems can often be understood in terms of the subgroup lattice of the underlying group. The focus of this research will be on creating PascGalois triangles and other 1-D and 2-D cellular automata generated using group and ring multiplication rules, as well as rules over alphabets with other algebraic structure. Undergraduate students from a variety of backgrounds, including mathematics, statistics, computer science, and secondary education have already completed successful PascGalois research projects. Current areas of research interests include, but are not limited to, the following:
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Periodicity Properties of finite cellular automata |
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Fractal Dimensions for Infnite cellular automata |
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Cellular Automata as algebraic Systems |
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Combinatorial and p-adic approaches |
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Virtual Reality Rendering for Higher Dimensional Systems |
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Information Entropy of Group Generated Cellular Automata |
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Group Actions on Group Generated Systems |
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