  We describe a quasiperiodic optical lattice, created by a physical
realization of the abstract cut-and-project construction underlying all
quasicrystals. The resulting potential is a generalization of the Fibonacci
tiling. Calculation of the energies and wavefunctions of ultracold atoms loaded
into such a lattice demonstrate a multifractal energy spectrum, a singular
continuous momentum-space structure, and the existence of controllable edge
states. These results open the door to cold atom quantum simulation experiments
in tunable or dynamic quasicrystalline potentials, including topological
pumping of edge states and phasonic spectroscopy.
