  We introduce a new class of algebraic varieties which we call frieze
varieties. Each frieze variety is determined by an acyclic quiver. The frieze
variety is defined in an elementary recursive way by constructing a set of
points in affine space. From a more conceptual viewpoint, the coordinates of
these points are specializations of cluster variables in the cluster algebra
associated to the quiver.
  We give a new characterization of the finite--tame--wild trichotomy for
acyclic quivers in terms of their frieze varieties. We show that an acyclic
quiver is representation finite, tame, or wild, respectively, if and only if
the dimension of its frieze variety is $0,1$, or $\ge2$, respectively.
