In multicellular organisms, cells pack together to form tissues of intricate and well defined morphology. How such cell-packing geometries arise is an important open question in biology, because the functionality of many differentiated tissues depends on their reliable formation. We show that combining adhesive forces due to E- and N-cadherin with a quantitative description of cell membrane elasticity in an interfacial energy model explains not only the qualitative neighbor relations, but also the detailed geometry of a tissue. The characteristic cellular geometries in the eyes of both wild-type Drosophila and genetic mutants are accurately reproduced by using a fixed set of few, physically motivated parameters. The model predicts adhesion strengths in the eye epithelium, quantifies their role relative to membrane elasticity, and reveals how simple minimization of interfacial energy can give rise to complex geometric patterns of important biological functionality.