  We consider deterministic walks on square, triangular and hexagonal two
dimensional lattices. In each case, there is a scatterer at every site that can
be in one of two states that force 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,
B. Z. Webb and E. G. D. Cohen, proved that given an initial position and
velocity of the walker and an environment of one type of scatterers, mirrrors
or rotators, there is an environment of the other type such that the walks on
both environments are equivalent, meaning they visit the same sites at the same
time steps. 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.
