  Fermions moving in a two-dimensional honeycomb lattice (graphene) have, at
low energies, chiral symmetry. Generalizing this construction to four
dimensions potentially provides fermions with chiral symmetry and only the
minimal fermion doubling demanded by the Nielsen-Ninomiya no-go theorem. The
practical usefulness of such fermions hinges on whether the action has a
necessary set of discrete symmetries of the lattice. If this is the case, one
avoids the generation of dimension three and four operators which require fine
tuning. We construct hyperdiamond lattice actions with enough symmetries to
exclude fine tuning; however, they produce multiple doublings. The limit where
the actions exhibit minimal doubling does not possess the requisite symmetry.
