  A tile Hamiltonian (TH) replaces the actual atomic interactions in a
quasicrystal with effective interactions between and within tiles. We study
Al-Co-Ni and Al-Co-Cu decagonal quasicrystals described as decorated
Hexagon-Boat-Star (HBS) tiles using {\em ab-initio} methods. A dominant term in
the TH counts the number of H, B and S tiles, favoring tilings of H and B only.
In our model for Al-Co-Cu, chemical ordering of Cu and Co along tile edges
defines tile edge arrowing. Unlike the edge arrowing of Penrose matching rules,
however, the energetics for Al-Co-Cu do not force quasiperiodicity.
Energetically favored structures resemble crystalline approximants to which the
actual quasicrystalline compounds transform at low temperature.

Condensed Matter