Suppose I have a grid of black/white squares (not necessarily all white). How can I prove that Langton's ant is unbounded when it runs on this initial condition?
It seems as though the way to go about this is noting that if it were bounded, it's periodic and achieving a contradiction by looking at the neighbours of the initial square. However, this approach hasn't worked so far for me.
Using dimensionality reduction Turn the lattice into a single dimensional array, iterate the "Langtons ant wave equation" and observe that a) Langtons ant is created and b) $\Psi_\vartriangle$ is indeed a 1D array and look at the variable $k$, a plot of this can be seen in my group on linkedin 104 unbounded highway since this highway is a recurrence relation you can show by telescoping the index $t$ into the index $k$ into the array $\Psi_\vartriangle$ is unbounded in the limit as long as the size of the lattice $E^2$ is taken to the limit of infinity (Calculus).
Graham