The Minimal Surface having a Helix as its boundary. It is the only Ruled Minimal
Surface other than the Plane (Catalan 1842, do Carmo 1986). For many years, the helicoid remained the only known
example of a complete embedded Minimal Surface of finite topology with infinite Curvature. However, in 1992 a
second example, known as Hoffman's Minimal Surface and consisting of a helicoid with a Hole, was discovered (*Sci. News* 1992).

The equation of a helicoid in Cylindrical Coordinates is

(1) |

(2) |

(3) | |||

(4) | |||

(5) |

which has an obvious generalization to the Elliptic Helicoid. The differentials are

(6) | |||

(7) | |||

(8) |

so the Line Element on the surface is

(9) |

and the Metric components are

(10) | |||

(11) | |||

(12) |

From Gauss's Theorema Egregium, the Gaussian Curvature is then

(13) |

(14) |

(15) |

The helicoid can be continuously deformed into a Catenoid by the transformation

(16) | |||

(17) | |||

(18) |

where corresponds to a helicoid and to a Catenoid.

If a twisted curve (i.e., one with Torsion ) rotates about a fixed axis and, at the same time, is displaced parallel to such that the speed of displacement is always proportional to the angular velocity of rotation, then generates a Generalized Helicoid.

**References**

Catalan E. ``Sur les surfaces réglées dont l'aire est un minimum.'' *J. Math. Pure Appl.* **7**, 203-211, 1842.

do Carmo, M. P. ``The Helicoid.'' §3.5B in
*Mathematical Models from the Collections of Universities and Museums* (Ed. G. Fischer).
Braunschweig, Germany: Vieweg, pp. 44-45, 1986.

Fischer, G. (Ed.). Plate 91 in
*Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume.*
Braunschweig, Germany: Vieweg, p. 87, 1986.

Geometry Center. ``The Helicoid.'' http://www.geom.umn.edu/zoo/diffgeom/surfspace/helicoid/.

Gray, A. *Modern Differential Geometry of Curves and Surfaces.*
Boca Raton, FL: CRC Press, p. 264, 1993.

Kreyszig, E. *Differential Geometry.* New York: Dover, p. 88, 1991.

Meusnier, J. B. ``Mémoire sur la courbure des surfaces.'' *Mém. des savans étrangers* **10** (lu 1776), 477-510, 1785.

Peterson, I. ``Three Bites in a Doughnut.'' *Sci. News* **127**, 168, Mar. 16, 1985.

``Putting a Handle on a Minimal Helicoid.'' *Sci. News* **142**, 276, Oct. 24, 1992.

Wolfram, S. *The Mathematica Book, 3rd ed.* Champaign, IL: Wolfram Media, p. 164, 1996.

© 1996-9

1999-05-25