  A systematic, decoration-based technique to discover the atomic structure of
a decagonal quasicrystal, given pair potentials and experimentally measured
lattice constants, is applied to the ``basic'' cobalt-rich decagonal Al-Co-Ni
quasicrystal. First lattice-gas Monte Carlo simulations are performed, assuming
the atomic sites are vertices of a rhombus tiling with edge 2.45 A. This phase
is found to be dominated by 13 A diameter decagon-shaped clusters, each with a
pentagon of Co atoms at the center. These, and another smaller cluster,
decorate vertices of a ``binary tiling'' with rhombus edge 10.4 A. Further
simulations with a restricted site list show that Al arrangements on the
borders of the 13 A decagon cluster form Hexagon, Boat, and Star tiles with
edge 2.45 A; they indicate specific sites for Co versus Ni atoms, and how the
structure adapts to small composition changes. In the second half of the paper,
relaxation (augmented by molecular dynamics annealing) is used to obtain
realistic structures. The dominant new feature is a set of linear ``channels''
attractive to Al atoms and running transverse to the layers. Each is typically
occupied by three atoms in four layers, implying puckering and a spontaneous
period doubling to c $\approx$ 8 A. Puckering favors pentagonal long range
order of the cluster orientations. Our simulation captures most features of the
related $W$-AlCoNi crystal, except for its pentagonal bipyramid motif.
