  Bipartite, periodic, planar graphs known as brane tilings can be associated
to a large class of quivers. This paper will explore new algebraic properties
of the well-studied del Pezzo 3 quiver and geometric properties of its
corresponding brane tiling. In particular, a factorization formula for the
cluster variables arising from a large class of mutation sequences (called
$\tau-$mutation sequences) is proven; this factorization also gives a recursion
on the cluster variables produced by such sequences. We can realize these
sequences as walks in a triangular lattice using a correspondence between the
generators of the affine symmetric group $\tilde{A_2}$ and the mutations which
generate $\tau-$mutation sequences. Using this bijection, we obtain explicit
formulae for the cluster that corresponds to a specific alcove in the lattice.
With this lattice visualization in mind, we then express each cluster variable
produced in a $\tau$-mutation sequence as the sum of weighted perfect matchings
of a new family of subgraphs of the dP3 brane tiling, which we call Aztec
castles. Our main result generalizes previous work on a certain mutation
sequence on the dP3 quiver in [Zha12], and forms part of the emerging story in
combinatorics and theoretical high energy physics relating cluster variables to
subgraphs of the associated brane tiling.
