In order to strengthen departmental collegiality and improve teaching, our mathematics department instituted a Teaching Polygon. Building on the faculty development idea of Teaching Squares, each member of our department visited one class taught by every other department member in a round-robin fashion during the school year. The visits were followed by structured reflection and conversations about our observations. We found that being in each other's classrooms strengthened our sense of community as a department, re-energized our enthusiasm for teaching, and encouraged us to reflect on ways to improve our own teaching.