Helix

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

			(1)
			(2)
			(3)

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

(4)

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

(5)

The torsion of a helix is given by

(6)

so

(7)

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

(8)

(9)

The minimal surface of a helix is a helicoid.

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