Tori Story

Carlo H. Séquin
Proceedings of Bridges 2011: Mathematics, Music, Art, Architecture, Culture (2011)
Pages 121–130 Regular Papers


All possible immersions of a torus in 3D Euclidean space can be grouped into four regular homotopy classes. All possible immersions within one such class can be transfigured into one another through continuous, smooth, homotopy-preserving transformations that will put no tears, creases, or other regions of infinite curvature into the surface. This paper introduces four simple, easy-to-understand representatives for these four homotopy classes and describes several transformations that convert a more complex immersion of some torus into one of these representatives. Among them are operations that turn a torus inside out and others that will rotate its surface parameterization by 90 degrees. Some new, aesthetically interesting torus models are also presented.