Cross-Caps—Boy Caps—Boy Cups

Carlo H. Séquin
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (2013)
Pages 207–216


We present a constructive, pictorial introduction to low-genus, non-orientable surfaces such as Möbius bands, cross-caps, and Boy's surface. We explore the use of these compact surface elements as general building blocks to make topologically equivalent models of arbitrary complex surfaces immersed in 3D Euclidean space, such as Klein bottles and single-sided surfaces of higher genus. As a by-product we generate some models for geometrical sculptures and for other whimsical artifacts, such as self-intersecting hats and furniture with unusual shapes.