A surface which can be interpreted as a self-intersecting Rectangle in 3-D. It is given by the parametric
equations

(1) | |||

(2) | |||

(3) |

for . The center of the ``plus'' shape which is the end of the line of self-intersection is a Pinch Point. The coefficients of the first Fundamental Form are

(4) | |||

(5) | |||

(6) |

and the coefficients of the second Fundamental Form are

(7) | |||

(8) | |||

(9) |

giving Gaussian Curvature and Mean Curvature

(10) | |||

(11) |

**References**

Francis, G. K. *A Topological Picturebook.* New York: Springer-Verlag, pp. 8-9, 1987.

Geometry Center. ``Whitney's Umbrella.'' http://www.geom.umn.edu/zoo/features/whitney/.

Gray, A. *Modern Differential Geometry of Curves and Surfaces.* Boca Raton, FL: CRC Press, pp. 225 and 309-310, 1993.

© 1996-9

1999-05-26