Equivalence of deterministic walks on regular lattices on the plane

We consider deterministic walks on square, triangular and hexagonal two dimensional lattices. In each case, there is a scatterer at every lattice site that can be in one of two states that forces the walker to turn either to his/her immediate right or left. After the walker is scattered, the scatterer changes state. A lattice with an arrangement of scatterers is an environment. We show that there are only two environments for which the scattering rules are injective, mirrors or rotators, on the three lattices. On hexagonal lattices Webb and Cohen (2014), proved that if a walker with a given initial position and velocity moves through an environment of mirrors (rotators) then there is an environment of rotators (mirrors) through which the walker would move with the same trajectory. We refer to these trajectories on mirror and rotator environments as equivalent walks. We prove the equivalence of walks on square and triangular lattices and include a proof of the equivalence of walks on hexagonal lattices. The proofs are based both on the geometry of the lattice and the structure of the scattering rule.