Fractals: Using mathematics as a generator of art and beauty

I've always had an interest in patterns and numbers, and in how simple rules and computations can lead to complex and interesting outcomes, leading often to fascinating results from otherwise trivial starting positions. Patterns are visible all around us in nature, both in physical manifestations of various objects, as well as in the mathematics that attempts to describe the physical world around us and to put forth a basic foundation for what constitutes an absolute truth in the universe.

This is a story about some of my early explorations into the world of patterns and numbers and their visual representations in the form of fractals. Fractals are complex visual representations of numbers, generated by nothing more than some basic mathematical rules and a computer to help with the tedious task of crunching the numbers and assigning the colours to each numerical output value. While often ordinary, with the right rules and settings some fractal output can be considered quite beautiful, and the feeling of discovery that goes along with it is quite exhilarating.

The above is an example of one of my earlier discoveries. Here we see a fractal that shows self-symmetry and numerous curious patterns that one might find everywhere in nature, including spirals, branches, contrasting features, and yet it maintains an amazing uniqueness of its own at the same time.

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Below is another example showing some similar patterns in a different colour scheme. A hub of activity in the centre with similar components encircling it in the periphery.

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How about some more fractal examples that might more closely resemble what you might commonly see around you in the physical universe. Here's one that might bring to mind some sort of a flower, or perhaps a plant seed even if you're very imaginative.

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And this one certainly reminds me of a flower of some kind, tipped on its side!

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Continuing with the plant theme, doesn't this one look a bit like some leaves on the end of branches?

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This one reminds me of a spider, but all the other patterns embedded within it give it some extra mystery and enjoyment to view.

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Here's another curious one that I think resembles an eye of some sort.

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This one reminds me of a ball of fire. Maybe a representation of our sun? Or some of you might see another eye looking your way?

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How about some alien species dancing around...Or are those brain cells connecting to one another?!

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Although these are all two-dimensional, sometimes the patterns and colour schemes really take it to another level and appear to give it a 3D effect, thanks to the appearance of what can be interpreted as shadows, like in this one.

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And here are a few more that I think are worth sharing for their fascinating colours, mysterious patterns and other interesting attributes.

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There's nothing quite like the feeling of discovering something perhaps never before having been seen by another sentient being until it is generated into a visual manifestation from its underlying mathematics.

Although exploring fractals has always been a solitary activity for me, I really hope you enjoyed viewing them as much as I enjoyed the process of their discovery. I have many more from these collections that I'd be happy to share if there's interest!

All images contained in this posting were generated by me, and are free to be used by anyone for any purpose (CC0).