From Möbius Bands to Klein-Knottles

Carlo H. Séquin
Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture (2012)
Pages 93–102


A construction of various immersed Klein bottles that belong to different regular homotopy classes, and which thus cannot be smoothly transformed into one another, is introduced. It is shown how these shapes can be partitioned into two Möbius bands and how the twistedness of these bands defines the homotopy type. Some wild and artistic variants of Klein bottles are presented for their aesthetic appeal and to serve as study objects for analysis.