My Search for Symmetrical Embeddings of Regular Maps

Various approaches are discussed for obtaining highly symmetrical and aesthetically pleasing space models of regular maps embedded in surfaces of genus 2 to 5. For many cases, geometrical intuition and preliminary visualization models made from paper strips or plastic pipes are quite competitive with exhaustive computer searches. A couple of particularly challenging problems are presented as detailed case studies. The symmetrical patterns discovered could be further modified to create Escher-like tilings on low-genus handle bodies.