Here we presented a framework to explore quad mesh topologies. The core of our work is a systematic enumeration algorithm that can generate all possible quadrangular meshes inside a defined boundary with an upper limit of v3-v5 pairs. The algorithm is orders of magnitude more efficient than previous work. The combination of topological enumeration and shape-space exploration demonstrates that mesh topology has a powerful influence on geometry. The Fig. 18. A gallery of different quadrilateral meshes for a Shuriken. The quadrilaterals of the model were colored in a postprocess. Topological variations have distinctive, interesting patterns of mesh lines. © 2014 ACM 0730-0301/2014/01-ART3 15.00.