A Spiral’s Parametric equation is

x = Cos[a t];    y = Sin[a t];    z = t;

where a is a constant and t is a common variable for the x, y and z coordinates.

The Mathematica Command for plotting this is, ParametricPlot3D[{Sin[10 t], Cos[10 t], t}, {t, 0, 2 \[Pi]}]. This graph is shown below.

When its ends are joined, we get a Toroid.

When spiral is coiled but its ends are not allowed to join, we get a spiral on a spiral. Let us call it Spiral level 2

Spiral level 2 can be given a similar treatment as earlier and thus we get Toroid Level 2.

The Spiral Level 3

Toroid Level 3:

The Spiral Level 4:

Spiral Level 1 was straightforward.

Spiral Level 2 took 2 hours to figure out.

Spiral Level 3 took a week of thinking and after which Spiral Level 4 followed immediately.

The insanity apparently did not end there. After very slow results given by the slow computer, I have done Toroid Level 4 and Spiral Level 5 which are shown below.

Toroid Level 4

Spiral Level 5

In the final two images, the pattern is so intricate that the smallest curls cannot be seen at the current resolution. Also I am limited by what  I can present in a webpage.

The concepts can be extended to any general levels since the trick has been found. I have no intention of doing it because the equations are getting scarier for each advanced level. Also I do not have enough computing power at my disposal. If I get hold of a super-computer I can write the program for it and see it live.

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