This paper's objective is pedagogical and interpretive. Namely, it gives a simple geometric analysis of classical (by which I mean population-share-weighted or income-share-weighted) inequality decomposability in the simplest nontrivial setting of three individuals. Income distributions in this case can be represented as points on the two-dimensional income-share simplex. In this representation, classical decomposability translates into concrete geometric restrictions of within- and between-group components. The geometric framework makes it possible to localize and compare violations of decomposability across inequality measures. The analysis is applied to the Mean Log Deviation, the Gini coefficient, the coefficient of variation, and the Theil index.