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 R^{3} 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 C_{2}
front-to-back symmetry.

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