We report on a group theory activity in which learners explored dihedral symmetries through a tangible geometric model. This approach has historical roots in the work of Felix Klein's Erlangen Program and his "Elementary Mathematics from an Advanced Standpoint." We situate our study with respect to this history as well as current educational research in spatial visual reasoning, abstract algebra, and teacher knowledge. Our findings highlight opportunities that tangible geometric models can provide for fostering structural and interconnected understanding characteristic of teachers' "knowledge at the mathematical horizon."