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.
.
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..