A Nonorientable Surface which is one of the three possible Surfaces obtained by sewing a Möbius Strip to the edge of a Disk. The other two are the Cross-Cap and Roman Surface. The Boy surface is a model of the Projective Plane without singularities and is a Sextic Surface.

The Boy surface can be described using the general method for Nonorientable Surfaces,
but this was not known until the analytic equations were found by Apéry (1986). Based on the fact that
it had been proven impossible to describe the surface using quadratic polynomials, Hopf had conjectured that
quartic polynomials were also insufficient (Pinkall 1986). Apéry's Immersion proved this conjecture wrong,
giving the equations explicitly in terms of the standard form for a Nonorientable Surface,

(1) | |||

(2) | |||

(3) |

Plugging in

(4) | |||

(5) | |||

(6) |

and letting and then gives the Boy surface, three views of which are shown above.

The parameterization can also be written as

(7) | |||

(8) | |||

(9) |

(Nordstrand) for and .

Three views of the surface obtained using this parameterization are shown above.

In fact, a Homotopy (smooth deformation) between the Roman Surface and Boy surface is given by the
equations

(10) | |||

(11) | |||

(12) |

as varies from 0 to 1, where corresponds to the Roman Surface and to the Boy surface (Wang), shown below.

In , the parametric representation is

(13) | |||

(14) | |||

(15) | |||

(16) |

and the algebraic equation is

(17) |

(18) | |||

(19) | |||

(20) | |||

(21) |

gives another version of the surface in .

**References**

Apéry, F. ``The Boy Surface.'' *Adv. Math.* **61**, 185-266, 1986.

Boy, W. ``Über die Curvatura integra und die Topologie geschlossener Flächen.'' *Math. Ann* **57**, 151-184, 1903.

Brehm, U. ``How to Build Minimal Polyhedral Models of the Boy Surface.'' *Math. Intell.* **12**,
51-56, 1990.

Carter, J. S. ``On Generalizing Boy Surface--Constructing a Generator of the 3rd Stable Stem.''
*Trans. Amer. Math. Soc.* **298**, 103-122, 1986.

Fischer, G. (Ed.). Plates 115-120 in
*Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume.* Braunschweig, Germany: Vieweg, pp. 110-115, 1986.

Geometry Center. ``Boy's Surface.'' http://www.geom.umn.edu/zoo/toptype/pplane/boy/.

Hilbert, D. and Cohn-Vossen, S. §46-47 in *Geometry and the Imagination.* New York: Chelsea, 1952.

Nordstrand, T. ``Boy's Surface.'' http://www.uib.no/people/nfytn/boytxt.htm.

Petit, J.-P. and Souriau, J. ``Une représentation analytique de la surface de Boy.''
*C. R. Acad. Sci. Paris Sér. 1 Math* **293**, 269-272, 1981.

Pinkall, U. *Mathematical Models from the Collections of Universities and Museums* (Ed. G. Fischer).
Braunschweig, Germany: Vieweg, pp. 64-65, 1986.

Stewart, I. *Game, Set and Math.* New York: Viking Penguin, 1991.

Wang, P. ``Renderings.'' http://www.ugcs.caltech.edu/~peterw/portfolio/renderings/.

© 1996-9

1999-05-26