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Does anyone know if there is a name for the curve which is a helix, which itself has a helical axis? I tried to draw what I mean:

enter image description here

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    $\begingroup$ Standard terms are 'super-helix' or 'super-coil'. See DNA. $\endgroup$ – amI Jan 18 '17 at 21:59
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    $\begingroup$ But DNA is a double helix, which is a different thing from this. $\endgroup$ – user50229 Jan 19 '17 at 19:40
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    $\begingroup$ @user50229: True, but DNA also super-coils. youtube.com/watch?v=N5zFOScowqo $\endgroup$ – LarsH Jan 19 '17 at 20:27
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    $\begingroup$ Informally, people might understand the analogy with a telephone wire $\endgroup$ – Ovi Jan 20 '17 at 0:12
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    $\begingroup$ Ah! You wanted a name rather than a parametrization. Ok, sorry. $\endgroup$ – Jyrki Lahtonen Jan 20 '17 at 10:09
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According to this journal article, we may call it a doubly-twisted helix or generally a multiply-twisted helix.

enter image description here

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For a light bulb the wire is called a "coiled coil filament" in this Wikipedia article.

coiled coil filament

The German word is "Doppelwendel" (roughly translated: double screw).

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    $\begingroup$ fascinating.... $\endgroup$ – rschwieb Jan 18 '17 at 17:28
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    $\begingroup$ @rschwieb Even more fascinating: Tungsten is brittle, and still they manage to coil it like that. See youtube.com/watch?v=DIGqBb3iZPo $\endgroup$ – Arthur Jan 19 '17 at 23:34
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I don't believe there is a standard term. Nevertheless, MathWorld calls the resulting curve a "slinky".

slinky

Of course, one can have a "slinky" of a "slinky":

superslinky

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  • $\begingroup$ Very funny. Improved barbedwire! $\endgroup$ – Jean Marie Jan 18 '17 at 22:20
  • $\begingroup$ Since "slinky" = "superhelix", the last one can be called a superslinky? $\endgroup$ – user21820 Jan 19 '17 at 11:28
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    $\begingroup$ Personally, I prefer the parametrization like this: \begin{bmatrix} a\cos t+r\left(\frac{b}{\sqrt{a^2+b^2}} \sin t \sin nt-\cos t \cos nt \right) \\ a\sin t+r\left(\sin t \cos nt-\frac{b}{\sqrt{a^2+b^2}} \cos t \sin nt \right) \\ bt+\frac{a}{\sqrt{a^2+b^2}} r\sin nt \end{bmatrix} with the parent/primary helix as: \begin{bmatrix} a\cos t \\ a\sin t \\ bt \end{bmatrix} $\endgroup$ – Ng Chung Tak Jan 19 '17 at 11:41
  • $\begingroup$ @Ng, yeah, I'd have made an arclength parametrization of the helix first so that the normal and binormal vectors would come out nice... $\endgroup$ – J. M. isn't a mathematician Jan 19 '17 at 12:49
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    $\begingroup$ J.M. Ng's formula is ok for the first iteration because the natural parameter is a constant multiple of $t$ (as I'm sure you know), so simple normalization of the normal/binormal will do just fine. You are, of course, right about the need to use arclength parametrization in all the succeeding iterations - I don't want to do that with paper&pencil :-) $\endgroup$ – Jyrki Lahtonen Jan 20 '17 at 7:44
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A corresponding construction on the torus is called an "iterated torus knot", so perhaps "iterated helix" would be a good name. I don't know of any standard name.

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Coiled coil as others also stated. More length is packed in a small volume. Used for electric bulb filaments, toroidal transformer primaries etc. Polar orbits of sun synchronous satellites around earth/sun is another example if the inner coil has no torsion.

EDIT1

A constant vector component of binormal is added to central coil vector if helicoids are to be defined as a surface... as a set of connected coils.

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