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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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    Standard terms are 'super-helix' or 'super-coil'. See DNA. – 2017-01-18
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    But DNA is a double helix, which is a different thing from this. – 2017-01-19
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    @aml the term supercoil includes other forms of secondary coiling, such as plectonemes, which are different from the example OP gave. So maybe it is a subset of supercoils. – 2017-01-19
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    @user50229: True, but DNA also super-coils. https://www.youtube.com/watch?v=N5zFOScowqo – 2017-01-19
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    Informally, people might understand the analogy with a telephone wire – 2017-01-20
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    @Jyrki I don't see anywhere in the thread where the name of the curve is given. – 2017-01-20
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    Ah! You wanted a name rather than a parametrization. Ok, sorry. – 2017-01-20
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    I do not know the name, but this sounds like an intriguing take on the tube plot 3D model which is the helix or path itself with a tube around it. In this case, it seems like we have a path along the surface of a tube plot surface with a tube around it. Very intriguing. I wonder if helix shapes like that can be made around arbitrary paths. – 2017-01-25
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    I believe this is the shape of a tungsten light bulb filament. – 2017-01-25

5 Answers 5

47

According to this journal article, we may call it a doubly-twisted helix or generally a multiply-twisted helix.

enter image description here

37

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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    fascinating.... – 2017-01-18
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    @rschwieb Even more fascinating: Tungsten is brittle, and still they manage to coil it like that. See https://www.youtube.com/watch?v=DIGqBb3iZPo – 2017-01-19
17

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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    Very funny. Improved barbedwire! – 2017-01-18
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    Since "slinky" = "superhelix", the last one can be called a **superslinky**? – 2017-01-19
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    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} – 2017-01-19
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    @Ng, yeah, I'd have made an arclength parametrization of the helix first so that the normal and binormal vectors would come out nice... – 2017-01-19
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    P.S. The plots were of course generated with *Mathematica*. – 2017-01-19
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    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 :-) – 2017-01-20
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    "I don't want to do that with paper & pencil" - I'm sure you don't, @Jyrki. *Mathematica* actually gags if I try to iterate beyond the fifth, as the required Frenet frame expression is incredibly complex. – 2017-01-20
8

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.

4

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.