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A helix, sometimes also called a coil, is a curve for which the tangent makes a constant angle with a fixed line. The shortest path between two points on a cylinder (one not directly above the other) is a fractional turn of a helix, as can be seen by cutting the cylinder along one of its sides, flattening it out, and noting that a straight line connecting the points becomes helical upon re-wrapping (Steinhaus 1999, p. 229). It is for this reason that squirrels chasing one another up and around tree trunks follow helical paths.

Helices come in enantiomorphous left- (coils counterclockwise as it "goes away") and right-handed forms (coils clockwise). Standard screws, nuts, and bolts are all right-handed, as are both the helices in a double-stranded molecule of DNA (Gardner 1984, pp. 2-3). Large helical structures in animals (such as horns) usually appear in both mirror-image forms, although the teeth of a male narwhal, usually only one which grows into a tusk, are both left-handed (Bonner 1951; Gardner 1984, p. 3; Thompson 1992). Gardner (1984) contains a fascinating discussion of helices in plants and animals, including an allusion to Shakespeare's A Midsummer Night's Dream.

The helix is a space curve with parametric equations

for , where is the radius of the helix and is a constant giving the vertical separation of the helix's loops.

The curvature of the helix is given by

and the locus of the centers of curvature of a helix is another helix. The arc length is given by

The torsion of a helix is given by

so

which is a constant. In fact, Lancret's theorem states that a necessary and sufficient condition for a curve to be a helix is that the ratio of curvature to torsion be constant.

The osculating plane of the helix is given by

The minimal surface of a helix is a helicoid.

See also

Generalized Helix, Helicoid, Seashell, Slinky, Spherical Helix, Spiral, Tangent Developable

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References

Bonner, J. T. "The Horn of the Unicorn." Sci. Amer. 184, pp. 42-43, Mar. 1951.Gardner, M. "The Helix." Ch. 1 in The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 1-8, 1984.Gardner, M. "The Helix." Ch. 9 in The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. New York: W. W. Norton, pp. 117-127, 2001.Geometry Center. "The Helix." http://www.geom.uiuc.edu/zoo/diffgeom/surfspace/helicoid/helix.html.Gray, A. "The Helix and Its Generalizations." §8.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 198-200, 1997.Isenberg, C. Plate 4.11 in The Science of Soap Films and Soap Bubbles. New York: Dover, 1992.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, pp. 83 and 155-156, 2002.Numberphile. "Three Gears Are Possible." Featuring H. Segerman. Jun. 10, 2016. https://www.youtube.com/watch?v=5Mf0JpTI_gg.Pappas, T. "The Helix--Mathematics & Genetics." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 166-168, 1989.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 329, 1958.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 229, 1999.Thompson, D'A. W. On Growth and Form, 2nd ed., compl. rev. ed. New York: Cambridge University Press, 1992.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, p. 95, 1991.

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Helix

Cite this as:

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

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