## Roman Surface

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)

Solving for gives the pair of equations
 (2)

If the surface is rotated by 45° about the z-Axis via the Rotation Matrix
 (3)

to give
 (4)

then the simple equation
 (5)

results. The Roman surface can also be generated using the general method for Nonorientable Surfaces using the polynomial function
 (6)

(Pinkall 1986). Setting
 (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).

See also Boy Surface, Cross-Cap, Heptahedron, Möbius Strip, Nonorientable Surface, Quartic Surface, Steiner Surface

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/.