Moore and more and symmetry

In any spatially discrete model of pedestrian motion which uses a regular lattice as basis, there is the question of how the symmetry between the different directions of motion can be restored as far as possible but with limited computational effort. This question is equivalent to the question ''How important is the orientation of the axis of discretization for the result of the simulation?'' An optimization in terms of symmetry can be combined with the implementation of higher and heterogeniously distributed walking speeds by representing different walking speeds via different amounts of cells an agent may move during one round. Therefore all different possible neighborhoods for speeds up to v = 10 (cells per round) will be examined for the amount of deviation from radial symmetry. Simple criteria will be stated which will allow find an optimal neighborhood for each speed. It will be shown that following these criteria even the best mixture of steps in Moore and von Neumann neighborhoods is unable to reproduce the optimal neighborhood for a speed as low as 4.