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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Fabulous work!!! Ashwini. It is a result of creative thinking. This is how one do modeling of complex systems as well. This

work can result to a creative book for beginners.

Take a real system- like nanotube, neural connections. Try to build that as you did by taking a simple spiral. It will be a innovative idea.

SPIRAL level 4 and 5 mimics biological system. Hence one can try to see that system at different resolution or basic building blocks. It is an interesting that SPIRAL 5 can be engineered from SPIRAL 1. If you get me then drop a mail to my inbox I,ll send you a biological system and you give me SPIRAL 1 for that with proper mathematical formulation, of course you are free to play with parameters..