Abstract: Interactive 3D computer graphics extends beyond the creation and manipulation of fancy, virtual, on-screen models. It also plays an important role for making repetitive or symmetrical patterns, constructing 3D CAD files for rapid prototyping, or drawing 2D nets that can be folded up into 3D paper or cardboard models. To make effective mathematical visualization models one or more of these modes may have to be engaged. This will be demonstrated in the context of realizing highly symmetrical embeddings of regular maps on surfaces of higher genus.