From Fractals to Attractors

A more formal definition is as follows.

‘In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve’ …Wikipedia.

An attractor is a set of states (points in the phase space), towards which neighboring states approach in the course of dynamic evolution.

An attractor is defined as the smallest unit which cannot be itself decomposed into two or more attractors.

This restriction is necessary since a dynamical system may have multiple attractors.

Really, my purpose here today is not to teach you about attractors but rather show you how amazing they are when turned into art.

Unique shapes with unique color schemes turn the canvas into something beautiful.

And, it’s all done with mathematics.

For those who find my math frustrating or ‘ugly’ to study, It really can be and is a beautiful subject to explore more in depth.

So, on to the beauty!I will include the code here that creates the following art.

You can take that code and try to create your own art as well.

Simply importing needed librariesDefinitionHere a, b, c, and d are variables.

There can be more but we’ll focus on four and show the results of those.

The numbers below each figure are the ones substituted into the above equations.

Numbers can be entered into a dataframe and used at one rather than entering each individually.

This would render multiple figures one right after the other.

The are so many attractors and so much beautiful art that can be rendered simply from mathematical equations, numbers, and code.

The possibilities are endless.

a = -1.

4, b = 1.

6, c = 1.

0, d = 0.

7a = 1.

6, b = -0.

6, c = -1.

2, d = 1.

6a = 1.

7, b = 1.

7, c = 0.

6, d = 1.

2a = 1.

5, b = -1.

8, c = 1.

6, d = 0.

9a = -1.

7, b = 1.

3, c = -0.

1, d = -1.

2a = -1.

7, b = 1.

8, c = -1.

9, d = -0.

4a = -1.

8, b = -2.

0, c = -0.

5, d = -0.

9Code for Clifford AttractorsCode Continued for Clifford AttractorsHere there are fifty five strange attractors but there are so many more that can be created.

Below you will find the code for one specific Lazaro Alonso example.

See if it intrigues you.

Play with the code and try to create your own artwork.

From all this code, what do you get?.It’s amazing!!You can use this code to change the color, the background, the gradients, etc.

See what you can do as you change the code to make your own creation.

While I would have love to have an interactive attractor generator embedded here, accomplishing that came as more of a challenge than I anticipated.

Here is a link to The Lorenz Attractor.

There are worlds more out there.

I just can’t stop myself from coming back to the awe of knowing what the combinations of numbers and computers can produce that creates such beauty!.This is an amazing part of mathematics and data science that should be further explored.