  We show that a single prototile can fill space uniformly but not admit a
periodic tiling. A two-dimensional, hexagonal prototile with markings that
enforce local matching rules is proven to be aperiodic by two independent
methods. The space--filling tiling that can be built from copies of the
prototile has the structure of a union of honeycombs with lattice constants of
$2^n a$, where $a$ sets the scale of the most dense lattice and $n$ takes all
positive integer values. There are two local isomorphism classes consistent
with the matching rules and there is a nontrivial relation between these
tilings and a previous construction by Penrose. Alternative forms of the
prototile enforce the local matching rules by shape alone, one using a
prototile that is not a connected region and the other using a
three--dimensional prototile.
