We present a systematic method for the elucidation of crystalline ground states and phase behavior of multicomponent colloidal materials. In two-dimensional space there are exactly seventeen âwallpaper groupsâ which represent distinct combinations of the four isometries of a Euclidean plane: translations, rotations, reflections, and glide reflections. These groups define all sets of operations that produce a unique, regular arrangement of points that tile a two-dimensional plane. Using properties of these groups, we develop an algorithm to cover a plane with a fixed number of arbitrary components in all ways that satisfy a desired stoichiometric ratio. These combined symmetry-stoichiometry rules dramatically reduce the number of possible configurations, which generally suffer from a so-called âcombinatorial explosionâ otherwise making extensive, random structure searching computationally infeasible . These candidates represent a complete, systematic coverage of all wallpaper groups, which helps to ensure that the correct ground state is discovered. With subsequent continuum relaxation, this approach is able to predict crystal structures ab initio
for multicomponent colloidal mixtures. We use this approach to investigate the ground state phase behavior of multicomponent systems inspired by multifunctional DNA-coated colloidal mixtures,  with a particular focus on stable, low-density âopenâ crystals. We demonstrate the approach for binary and ternary mixtures at zero ambient pressure in order to explore how complexity can be achieved through the combination of many components with simple interactions rather than a smaller number of components with more complicated potentials.
 A. M. Oganov, A. O. Lyakhov, M. Valle, âHow evolutionary crystal structure prediction works - and whyâ, Accounts of Chemical Research, 2011, 44, 227-237.
 N. A. Mahynski, H. Zerze, H. W. Hatch, V. K. Shen, J. Mittal, âAssembly of multi-flavored two-dimensional colloidal crystalsâ, Soft Matter, 2017, 13, 5397-5408.