A Quartic Nonorientable Surface, also known as the Steiner Surface. The Roman surface is one of the three possible surfaces obtained by sewing a Möbius Strip to the edge of a Disk. The other two are the Boy Surface and Cross-Cap, all of which are homeomorphic to the Real Projective Plane (Pinkall 1986).

The center point of the Roman surface is an ordinary Triple Point with , and the six endpoints of the three lines of self-intersection are singular Pinch Points, also known as Whitney Singularities. The Roman surface is essentially six Cross-Caps stuck together and contains a double Infinity of Conics.

The Roman surface can given by the equation

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) | |||

(8) | |||

(9) |

in the former gives

(10) | |||

(11) | |||

(12) |

for and . Flipping and and multiplying by 2 gives the form shown by Wang.

A Homotopy (smooth deformation) between the Roman surface and Boy Surface is given by the equations

(13) | |||

(14) | |||

(15) |

for and as varies from 0 to 1. corresponds to the Roman surface and to the Boy Surface (Wang).

**References**

Fischer, G. (Ed.). *Mathematical Models from the Collections of Universities and Museums.*
Braunschweig, Germany: Vieweg, p. 19, 1986.

Fischer, G. (Ed.). Plates 42-44 and 108-114 in
*Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume.*
Braunschweig, Germany: Vieweg, pp. 42-44 and 108-109, 1986.

Geometry Center. ``The Roman Surface.'' http://www.geom.umn.edu/zoo/toptype/pplane/roman/.

Gray, A. *Modern Differential Geometry of Curves and Surfaces.*
Boca Raton, FL: CRC Press, pp. 242-243, 1993.

Nordstrand, T. ``Steiner's Roman Surface.'' http://www.uib.no/people/nfytn/steintxt.htm.

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

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

© 1996-9

1999-05-25