  We prove a complexity dichotomy theorem for counting planar graph
homomorphisms of domain size 3. Given any 3 by 3 real valued symmetric matrix
$H$ defining a graph homomorphism from all planar graphs $G \mapsto Z_H(G)$, we
completely classify the computational complexity of this problem according to
the matrix $H$. We show that for every $H$, the problem is either polynomial
time computable or \#P-hard. The P-time computable cases consist of precisely
those that are P-time computable for general graphs (a complete classification
is known) or computable by Valiant's holographic algorithm via matchgates. We
also prove several results about planar graph homomorphisms for general domain
size $q$. The proof uses mainly analytic arguments.
