Abstract
We study the design and optimization of polygonal meshes with concave planar faces. The motivating applications of this work are architecture, product design, and art. To discretize freeform surfaces into polyhedral meshes, we propose a novel class of regularizers for mesh aesthetics based on symmetries. They are useful to generate concave polygons on negative Gaussian curvature region, and provide the necessary flexibility to create smooth transformation of planar faces across the region where Gaussian curvature alternated between positive and negative.