Spiraling Spirals

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.

Spiral Level 1

When its ends are joined, we get a Toroid.

Toroid Level 1

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

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

Toroid Level 2

The Spiral Level 3

Spiral Level 3

Toroid Level 3:

Toroid Level 3

The Spiral Level 4:

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

toroid highest

Spiral Level 5

Spiral Highest

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.

.

One thought on “Spiraling Spirals

  1. 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..

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s