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We introduce a fully automatic approach to simultaneously segment and detect similarities within an existing 3D architectural model. Our framework approaches the segmentation problem as a weighted minimum set cover over an input triangle soup, and maximizes the repetition of similar segments to find a best set of unique component types and instances. The solution for this set-cover formulation starts with a search space reduction to eliminate unlikely combinations of triangles, and continues with a combinatorial optimization within each disjoint subspace that outputs the components and their types. We show the discovered components of a variety of architectural models obtained from public databases. We demonstrate experiments testing the robustness of our algorithm, in terms of threshold sensitivity, vertex displacement, and triangulation variations of the original model. In addition, we compare our components with those of competing approaches and evaluate our results against user-based segmentations. 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