Sunday, August 2, 2009

Self-Similarity in Fractals

Fractals is a branch of mathematics that artists intuitively grasp because it appeals strongly to the pattern recognition part of the right brain. One of the principles of fractals is the concept of self-similarity. Many forms in nature repeat a certain structure or geometry at various levels of scale.

For example, this ginger root has bulbous branches that leave the main stem, and they tend to do the same thing as the branches get smaller, until the pattern degenerates at the smallest scale.

Romanesco broccoli demonstrates the principle of self-similarity even better. Each spiraling cone is composed of smaller spiraling cones. (Click to enlarge).

In many ferns, the shape of the leaflet resembles the frond. Since this comes down to math, computers can easily invent self-similarity, and this component has given realism to CGI renderings of rocks, plants, clouds, and water, where the phenomenon appears everywhere. Check out the digital fern below, which "grows" as you scroll down.

Fractal fern, link.
Wikipedia on fractals, link.
Fern glossary and more photos at: link.


Peter Underhill said...

I remember the first time I saw the Romansco Broccoli in a store. I thought it was too beautiful to eat. Isn't it a perfect example of the Fibonacci sequence in action?

Saskia said...

How very weird, I've been thinking about fractals yesterday and I even had the same examples in mind! lol

Stephen James. said...

Very interesting.

Daniel.Z said...

Great post.
The beauty of nature really is endless.

Roberto said...

Wow! This is fantastic stuff! I’ve been following your links for a very weird trip. The math stuff gives me a head-ache, but the concepts are psychedelic. Geometric shapes with finite volume and infinite circumference, what a concept! I especially like the ‘In creative works: Fractal art’ section on the wiki link. (Computer analysis of Jackson Pollock’s paintings, Max Ernst and African art fractals. What a gas!)
What really gets me is that the universe actually expresses itself in such beautiful and complexly simple mathematics.
And it’s not all just pretty pictures, check this out.
Our pulmonary bronchial tree bifurcates and branches like your ginger root example, so that the airways get smaller and smaller, down to the capillary bed where the O2 and CO2 exchange occurs. But as the airways get smaller and smaller, the air resistance actually decreases, because the overall volume of the airway increases!
Thanks for the journey -RQ

masta shake said...

Nature is reality. Fibonacci and other formulas are concepts based upon and then imposed on nature(reality) in order to understand it and control it via various fields of engineering. The variable of reality is why the computer generated images still look fake. Just like a drum machine is lifeless compared to a real drummer. Without the constant, perhaps only subconsciously perceptible, sway-swing the beat is dead. Lifeless as a mathematical equation.

Cthogua said...

Great post. If you're interested in fractals, and their relationship to man, art, and nature. Check out this TED talk by Ron Eglash about the use of fractal concepts by indigenous Africans in both material application, such as villiage structure, as well as cosmological organization. The universe is recursive and we are amongst those recursions.

tanaudel said...

That Romanesco broccoli is amazing! Crazy-beautiful (like passion flowers, some of the more bizarrely exuberant flowers I've seen). I've never even heard of it before.

treplovski said...

I see I'm not the first to point out the Fibonacci sequence in the broccoli, but I did want to point out that each spiral has a mirror-image counterpart; spirals clockwise and counter-clockwise bisecting each other. And Ernest Thomopson Seton pointed out that the "eyes" in a peacock's tail follow the same pattern. Gorgeous!

Roberto said...

And here’s a fractal video by PBS Nova: