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The words spiral and helix are both used for curves that "wind around". For example, both searches "DNA spiral" and "DNA helix" (with quotation marks) result in many thousands of Google hits.

Is there a difference between helix and spiral, or are these words synonyms?

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A helix is a curve in 3-d space with an axis, where the tangent line makes a constant angle with the axis. See here for some info on the helix. A circle would be the only helix which could lie in a plane, since the tangent line to a plane curve is always at 90 degrees from an axis normal to that plane. So a 2-d helix is not much of a helix, which more typically resemble corkscrews.

[Note: I take that back about plane curves --- the tangent line to any curve in the $xy$ plane (at any point on it) is at a 90 degree angle from the $z$ axis, so that any curve in the $xy$ plane would qualify as a helix under the constant angle definition. That makes such curves not good candidates for the term ""helix", i.e. the notion is trivially true for all plane curves.]

Spirals on the other hand do not need to have that property, and are typically curves in a 2-d plane. See the link here for some information about various types of spiral. Some of them are quite interesting.

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