Girl's Surface

Sue Goodman, Alex Mellnik and Carlo H. Séquin
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (2013)
Pages 383–388 Regular Papers


Boy's surface is the simplest and most symmetrical way of making a compact model of the projective plane in R3 without any singular points. This surface has 3-fold rotational symmetry and a single triple point from which three loops of intersection lines emerge. It turns out that there is a second, homeomorphically different way to model the projective plane with the same set of intersection lines, though it is less symmetrical. There seems to be only one such other structure beside Boy's surface, and it thus has been named Girl's surface. This alternative, finite, smooth model of the projective plane seems to be virtually unknown, and the purpose of this paper is to introduce it and make it understandable to a much wider audience. To do so, we will focus on the construction of the most symmetrical Möbius band with a circular boundary and with an internal surface patch with the intersection line structure specified above. This geometry defines a Girl's cap with C2 front-to-back symmetry.