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In geometry, Boy's surface is an immersion of the real projective plane in 3dimensional space found by Werner Boy in 1901 (he discovered it on assignment from David Hilbert to prove that the projective plane could not be immersed in 3space). Unlike the Roman surface and the crosscap, it has no singularities (i.e. pinchpoints), but it does selfintersect.
To make a Boy's surface:
Boy's surface is discussed (and illustrated) in JeanPierre Petit's Le Topologicon^{[1]}.
Boy's surface was first parametrized explicitly by Bernard Morin in 1978. See below for another parametrization, discovered by Rob Kusner and Robert Bryant.
Contents
Symmetry of the Boy's surface
Boy's surface has 3fold symmetry. This means that it has an axis of discrete rotational symmetry: any 120° turn about this axis will leave the surface looking exactly the same. The Boy's surface can be cut into three mutually congruent pieces. (proof)
Model at Oberwolfach
The Mathematical Research Institute of Oberwolfach has a large model of a Boy's surface outside the entrance, constructed and donated by MercedesBenz in January 1991. This model has 3fold rotational symmetry and minimizes the Willmore energy of the surface. It consists of steel strips which represent the image of a polar coordinate grid under a parameterization given by Robert Bryant and Rob Kusner. The meridians (rays) become ordinary Möbius strips, i.e. twisted by 180 degrees. All but one of the strips corresponding to circles of latitude (radial circles around the origin) are untwisted, while the one corresponding to the boundary of the unit circle is a Möbius strip twisted by three times 180 degrees — as is the emblem of the institute (Mathematisches Forschungsinstitut Oberwolfach 2008).
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