  We associate an square to any two dimensional evolution algebra. This
geometric object is uniquely determined, does not depend on the basis and
describes the structure and the behaviour of the algebra. We determine the
identities of degrees at most four, as well as derivations and automorphisms.
We look at the group of automorphisms as an algebraic group, getting in this
form a new algebraic invariant. The study of associative representations of
evolution algebras is also started and we get faithful representations for most
two-dimensional evolution algebras. In some cases we prove that faithful
commutative and associative representations do not exist, giving raise to the
class of what could be termed as "exceptional" evolution algebras (in the sense
of not admitting a monomorphism to an associative algebra with deformed
product).
