  We investigate the self-adjointness of the two dimensional Dirac operator
with infinite mass boundary conditions on an unbounded domain with an infinite
number of corners. We prove that if the domain has no concave corners, then the
operator is self-adjoint. On the other hand, when concave corners are present,
the operator is no longer self-adjoint and self-adjoint extensions can be
constructed. Among these, we characterize the distinguished extension as the
unique one whose domain is included in the Sobolev space $H^{s}$, where $s>1/2$
depends on the amplitude of the corners. Lastly, we study the spectrum of this
distinguished self-adjoint extension.
