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
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(1)
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(2)
|
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(3)
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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
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(4)
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and the locus of the centers of curvature of a helix is another helix. The arc length is given by
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(5)
|
The torsion of a helix is given by
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(6)
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so
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(7)
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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
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(8)
|
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(9)
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The minimal surface of a helix is a helicoid.
