Naturally fractal

On my wall paper is a photograph of a beautiful Japanese cherry tree in bloom. The shape is highly irregular, yet there is a sort of aesthetic beauty to the whole like a hint of a deeper symmetry at play. How can such a shape arise?
cherrytree

A Japanese Cherry Tree

Well for starters, we know that all information about the shape of the tree must lie in its DNA. Also the climate and sunshine play a major role. But how can such an ornate shape be stored efficiently?

Actually, the information needed to get the shape of the tree can be stored quite efficiently in a very few sets of rules. Let us consider what a growing tree needs. It needs sunlight, so we can say that if the leaves on a particular branch receive a more sunlight, then it will grow stronger. Conversely if the leaves on some branch don’t get much sunlight, that branch will not develop. This means that over time, the tree will tend to grow in the direction it gets more sunlight. If sunlight can come from any direction, the tree canopy will likely grow wide like this image of the African Savannah.

savanna tree

Israeli Babool Tree

Another key idea is that of fractals. Each small piece of the tree looks very similar to the whole tree. The tree is essentially composed of a basic pattern repeated over and over in different scales. When the tree sprouts out of the ground, the trunk splits into branches. These branches thicken as the tree grows and when they are thick enough, they in turn split into smaller branches. These in turn will grow and split in turn into smaller branches till at last we reach the extremities of twigs.

All the information needed to represent this is the knowledge of when and how to split. Thus any branch which reaches a prescribed thickness will split and create smaller branches fiving rise to the overall shape of the tree. Thus any part of the tree down to the tiny twigs exhibit the same underlying shape as the tree as a whole.

fractalbranches

Tree branches showing fractal nature

This is the essence of what mathematicians call fractals. One pattern keeps repeating itself at different scales. But it is not just tree trunks that show a fractal nature. See if you can spot the repeating structure in these close up shots of leaves.

Fractal structure of leaves

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