Helix in a helix I am trying to work out a "helix in a helix" mathematically.  Intuitively I think of this as a steel cable, which is made up of a number of smaller steel cables all bound together in spiral.  If I wanted to find the length of one of the individual cables, it would be bound in a spiral in the smaller cable and then those cables bound in a larger spiral cable.  I know that if I wanted to do a helix whose ends meet, I would use the parametrization
$$((a+b\cos(\omega{t}))\cos{t},(a+b\cos(\omega{t})\sin{t},b\sin(\omega{t})),t=0..2\pi$$
I've been trying to map out in my head how to, instead of curling the helix, making the helix travel in the path of a helix.  I've achieved it partially with
$$((a+b\cos(\omega{t}))\cos{t},(a+b\cos(\omega{t})\sin{t},t),t=0..\infty$$
But this doesn't keep the smaller helix in tact, and turns it into a sine wave helix.  I've also tried 
$$((a+b\cos(\omega{t}))\cos{t},(a+b\cos(\omega{t})\sin{t},tb\sin(\omega{t})),t=0..2\pi$$
But this gives me sort of a nautilus shape where the helix curls into itself and increases in size and curls around into itself.  What am I missing?
EDIT:  Also, what if we wanted to do this again, like a 'helix in a helix in a ... in a helix'?
 A: As in an earlier answer I use the local frame along the helix to help parametrize the desired curve.
From that answer I reuse: a parametrization for a helix along the $x$-axis
$$
\vec{r}(t)=(ht,R\cos t, R\sin t).
$$
Its tangent vector
$$
\vec{t}=\frac{d\vec{r}(t)}{dt}=(h,-R\sin t,R\cos t).
$$
Its normal vector
$$
\vec{n}(t)=
\frac{\frac{d\vec{t}}{dt}}{\left\Vert\frac{d\vec{t}}{dt}\right\Vert}=(0,-\cos t,-\sin t).
$$
And it binormal vector
$$
\vec{b}(t)=\frac1{\Vert\vec{t}\Vert}\vec{t}\times\vec{n}=\frac{1}{\sqrt{R^2+h^2}}(R,h\sin t,-h\cos t).
$$
This is, of course, orthogonal to both $\vec{t}$ and $\vec{n}$.
The tube around the helix (with radius $a$) then has a parametrization
$$
S(t,u)=\vec{r}(t)+a\vec{n}(t)\cos u+ a\vec{b}(t)\sin u
$$
with $t$ ranging over as many loops as you wish, and $u$ ranging over the interval $[0,2\pi]$.
To get a curve looping around the helix along that surface we simply set $u=kt$,
where $k$ indicates the number of rotations around the tube per single rotation of the tube around the $x$-axis.
Here's the image of the resulting curve with $R=3$, $h=1$, $a=1$ and $k=12$. For clarity I included both the tube as well as the curve.

The parametrization of that thin "cable" on the tube surface is (with the above values for the constants)
$$
\left\{\begin{array}{ccl}
x&=&t+\frac{3\sin 12 t}{\sqrt{10}},\\
y&=&3\cos t-\cos t\cos 12t+\frac{\sin t\sin 12t}{\sqrt{10}},\\
z&=&3\sin t-\sin t\cos 12t-\frac{\cos t\sin 12t}{\sqrt{10}}.
\end{array}\right.
$$
