Wonder why I didn’t discover Jason Rampe’s Softology blog much earlier, as it deals with most of the math related topics of my own blog (and many more). His expertise and visualizations, as well as his software application Visions of Chaos, live at the opposite end of the spectrum when compared to my modest tinkering. As a result, most of his visualizations are far beyond the specs of a microcontroller and a small display, but I discovered a few mathematical models that I simply had to try right away:
- Diffusion-limited Aggregation
“Diffusion-limited aggregation creates branched and coral like structures by the process of randomly moving particles that touch and stick to existing stationary particles.” – [Softology blog].
After coding a 2D version for my M5Stack Fire (ESP32), I was surprised to see how this simple principle generates these nice structures. The process starts with a single stationary (green) particle in the center. Then, one by one starting from a randomly chosen point at the border, every next particle takes a random walk inside the circle until it hits a stationary particle to which it sticks, becoming a stationary (green) particle itself.
- Reaction-Diffusion model
It was Alan Turing who came up with a model to explain pattern formation in the skin of animals like zebras and tigers (morphogenesis). It’s based on the reaction of two chemical substances within cells and the diffusion of those chemicals across neighbour cells. Certain diffusion rates and combinations of other parameters can produce stable Turing Patterns.
The above (Wikipedia) picture shows the skin of a Giant pufferfish. The pictures below show some first results of my attempts with different settings on a 200×200 pixel grid.
From left to right: ‘curved stripes’ settings (C), ‘dots’ settings (D), 50% C + 50% D, 67% C + 33% D
There’s still plenty to experiment for me here. The algorithm is strikingly simple and fully parameterized, but also quite heavy on the ESP32….
- Mitchell Gravity Set Fractals
A simulation of particles, following Newton’s law of gravity. This simple example has six equal masses placed at the corners of a regular hexagon. Particles (pixels) are colored according to the time they need to cross an escape range and disappear in ‘space’.
- Agent-based models
After coding my own variant of the Foraging Ant Colony example from the Softology blog, I’m currently working on a couple of obstacle escape mechanisms. Coming soon.