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The traditional approach to the quantitative study of segregation is to employ indices that are selected by ``desirable properties''. Here, we detail how information theory underpins entropy-based indices and demonstrate how desirable properties can be used to systematically construct models of segregation. The resulting models capture all indices which satisfy the selected properties and provide new insights, such as how the entropy index presumes a particular form of intergroup interactions and how the dissimilarity index depends on the regional composition. Additionally, our approach reveals that functions, rather than indices, tend to be necessary mathematical tools for a comprehensive quantification of segregation. We then proceed with exploratory considerations of two-group residential segregation, finding striking similarities in major U.S. cities, subtle segregation patterns that correlate with minority group diversity, and substantive reductions in segregation that may be overlooked with traditional approaches. Finally, we explore the promise of our approach for segregation forecasting.