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We propose a 3D object construction methodology built on face-loop modeling operations. Our Face Extrusion Quad (FEQ) meshes, have a well designed face-loop structure similar to artist crafted 3D models. Furthermore, we define a construction graph which encodes a sequence of primitive extrude/collapse and bridge/separate operations that operate on admissible face-loops. We show that FEQs are imbued with a meaningful face-loop induced shape skeleton, part segmentation, plausible construction history, and possess the many advantages of extrusion-based 3D modeling. Our evaluation is threefold: we show a gallery of challenging 3D models transformed to FEQs with compelling face-loop structure; we showcase the potential of an inherent construction graph, using FEQ-based cut-paste and inverse modeling applications; and we demonstrate the impact of various algorithmic and parameter related …