  A tile Hamiltonian (TH) replaces the actual atomic interactions in a
quasicrystal with effective interactions between and within tiles. We studied
Al-Co-Cu decagonal quasicrystals described as decorated Hexagon-Boat-Star (HBS)
tiles using {\em ab-initio} methods. The dominant term in the TH counts the
number of H, B and S tiles. Phason flips that replace an HS pair with a BB pair
lower the energy. In Penrose tilings, quasiperiodicity is forced by arrow
matching rules on tile edges. The edge arrow orientation in our model of AlCoCu
is due to Co/Cu chemical ordering. Tile edges meet in vertices with 72$^\circ$
or 144$^\circ$ angles. We find strong interactions between edge orientations at
72$^\circ$ vertices that force a type of matching rule. Interactions at
144$^\circ$ vertices are somewhat weaker.
