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The (circular) helicoid is the minimal surface having a (circular) 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 helicoid is the only non-rotary surface which can glide along itself (Steinhaus 1999, p. 231).

The helicoid is a singly periodic minimal surface under axial translations.

The equation of a helicoid in cylindrical coordinates is

In Cartesian coordinates, it is

It can be given in parametric form by

which has an obvious generalization to the elliptic helicoid. Writing instead of gives a cone instead of a helicoid.

The first fundamental form coefficients of the helicoid are given by

and the second fundamental form coefficients are

giving area element

Integrating over and then gives

The Gaussian curvature is given by

and the mean curvature is

making the helicoid a minimal surface. The Gaussian curvature can be given implicitly by

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

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.

Helicoids occur in architectural forms such as helical staircases and ramps, as well as in screw-type machine-building products (Bock Hyeng et al. 2025).

See also

Calculus of Variations, Catenoid, Cone, Conoid, Elliptic Helicoid, Generalized Helicoid, Helix, Hoffman's Minimal Surface, Hyperbolic Helicoid, Minimal Surface, Seashell

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References

Bock Hyeng, C. A.; Krivoshapko, S. N.; Kouamou Nguessi, A.; Yamb Bell, E.; and Bahel, B. "Application of Curvilinear Analytical Surfaces in Forms of Architectural Objects and Machine Building Products." Int. J. Archit. Arts Appl. 11, 19-35, 2025. https://doi.org/10.11648/j.ijaaa.20251101.13.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 aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 87, 1986.Geometry Center. "The Helicoid." http://www.geom.uiuc.edu/zoo/diffgeom/surfspace/helicoid/.GRAPE. "Catenoid-Helicoid Deformation." https://archive.ins.uni-bonn.de/numod.ins.uni-bonn.de/grape/EXAMPLES/AMANDUS/cathel.html.GRAPE. "Helicoid." https://archive.ins.uni-bonn.de/numod.ins.uni-bonn.de/grape/EXAMPLES/AMANDUS/helicoid.html.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 449 and 644, 1997.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.Ogawa, A. "Helicatenoid." Mathematica J. 2, 21, 1992.Osserman, R. A Survey of Minimal Surfaces. New York: Dover, pp. 17-18, 1986.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.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 231-232, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, p. 94, 1991.

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Helicoid

Cite this as:

Weisstein, Eric W. "Helicoid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Helicoid.html

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