Brad, nice exploration.
I am not sure if you are being playful when you say you don't understand the arguments to the CellularAutomaton function.
I wouldn't use the offsets in the radius, but instead use weights, e.g., CellularAutomaton[{113724623751, {4, {{1,0,1},{1,1,1},{0,1,0}}},{1,1}}]
or
ArrayPlot /@
CellularAutomaton[{113724623751, {4, {{1, 0, 1}, {1, 1, 1}, {0, 1,
0}}}, {1, 1}}, {{{1}}, 0}, {{25, 30}}]
and it seems that you have the neighborhood upside down.
