Tree Branch Self-Similarity

Trees are beautiful when they are lush and green in summer or when they flare with red and gold in autumn. What about winter, when the branches are bare? That’s when it’s easiest to see the self-similar structure in the tree’s branching habit. At each juncture where a bud begins a new branch, the angle of the branch is determined by the tree’s genetics and specific influences in the environment. Each tree species has a standard pattern, which is repeated with every bud, so that the shape of the junctures is the same, broadly speaking, in everything from the first heavy branches on the trunk to the newest fine twig. I’ve talked about this before, but I just can’t get enough of the beauty of these elegant patterns.

There are important biological factors behind a tree’s branching habit, but from a mathematical point of view it reveals a fractal pattern, which has the property of being self-similar. A self-similar structure looks roughly the same at every scale from the largest zoomed-out view to the narrowest closeup. There are practical limits in the real world (if you zoom in far enough you get to individual cells and eventually molecules, which aren’t particularly similar to the tree trunk), but within the range that we can call the tree structure the idea holds. If you zoom in on the picture above each branching region looks like the overall structure of the tree.

Fractals are important in nature and mathematics (as I’ve commented before). They are also very beautiful, and reveal the beauty in the structure of everything around us.

Posted in response to the WordPress photo challenge: Structure

Branches of Chaos

Chaos means one thing in everyday language: unpredictable, out-of-control anarchy. In math and science it means something else. This kind of chaos is unpredictable in detail but predictable in its overall form. I recommend James Gleick’s great book on this topic: Chaos: Making a New Science, and I’ve talked about it before so I won’t go into it a lot here, but the one thing we often see here is self-similarity, where zooming in closer and closer you see the same overall pattern. Last winter I took this shot of bare tree branches against the sky. Big branches sprout smaller ones, which sprout twigs, which sprout twiglets (is that a word?), and so on down and down. At each level the type of branching that happens is the same as at the levels above and below. This is just really cool.

Posted in response to the WordPress Photo Challenge: Chaos

The Ornate World: Fractals and Complexity #photochallenge

Romanesco cauliflower

Sunflower

Fern

Julia set

Mandelbrot set

This week’s WordPress photo challenge is to celebrate the ornate. Where do we find complexity and complication, and find it beautiful? What make complexity beautiful, at least for me, is when there is a pattern behind the complexity. Something that makes each shape inevitable, even as the whole is intricate and surprising. This is one of the hallmarks of fractals, and perhaps of good writing as well.