by Dirk Brockmann

This explorable illustrates how fractal patterns observed in natural systems, particularly structural properties of some plants, can approximately be modeled by simple iterative models. Sometimes these models are refered to as Lindenmayer systems.

The pattern that you see is generated by a basic structure to start with and applying a simple replacement rule.

Initially we have a root branch (stem) of base length $L_0$. To this root we attach 3 leaf branches of lengths $l_1$, $l_2$, $l_3$ and at angles $\theta_1$, $\theta_2$, and $\theta_3$, respectively. The lengths $l_i$ are a bit shorter than the base length $L_0$. The lengths and angles are parameters of the structure and can be varied using the sliders.

The stem plus its three leaves are the starting structure that you see in the little cartoon in the control panel.

Now we iterate the process, replacing the open leaf branches again by a smaller version of the original structure (stem + 3 branches), effectively treating the leaf branches as new root branches. Repeating the process ad infinitum (in our case 8 times) we obtain the self-similar structure.

In addition to the basic parameters of 3 lengths and 3 angles, you can add a little randomness to the structure by increasing the angle noise and the length noise. This way lengths and angles at each step of the generation process are modified by a little random perturbation.

To get more tree like structures, you can vary the width of the root branch with the bottom slider.

You can also choose from the list of preset structures by clicking on the buttons on the bottom right.

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Diffusion Limited Aggregation

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The distribution of primes along number spirals

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