Designing Geometry from Chaos: Discovering Attractor Line Forms

From Chaos to Line

In a previous post, I shared sketches of lines generated from dynamical systems like Lorentz, Rossler, and Rikitake. Beyond the scientific curiosities - they also offer a rich space for design exploration. This post reflects on how the geometry behind those lines was discovered and shaped.

🌐 Why Attractors?

Attractor systems are great starting points for generating “predictable randomness” because they balance structure and unpredictability. The equations are deterministic, but the output feels organic. Each point flows from the last, forming a continuous path in 3D space. That makes them perfect candidates for generating line-based geometries.

🔧 Development Process

1. Choosing Systems
I started with well-documented attractors: Lorentz, Rossler, and Rikitake. These are known to produce visually engaging trajectories. The goal was to convert them into something drawable—not just as dots in phase space, but as clean, uninterrupted lines.

2. Numerical Integration
Each attractor is defined by a system of ordinary differential equations (ODEs). To visualize them, I used Euler’s method for simplicity, stepping forward with small time increments. For example, in the Lorentz system:

dx = sigma * (y - x);
dy = x * (rho - z) - y;
dz = x * y - beta * z;

Small updates to x, y, and z move the point through space. Connecting these movements gives us a line.

3. Visualization
The visualization was built using Three.js. This allowed for smooth rendering of the continuous line geometry in real-time, with interactive camera controls to explore the forms from different angles. The points generated from the integration step were connected using THREE.Line geometry, creating a visual representation of the attractor’s path through space.

✨ Discovering Form

As the systems evolved, so did the goals. Beyond visualizing known equations, I began adjusting parameters, blending systems (like a Lorentz-Rossler hybrid), and experimenting with perturbations. Some observations:

  • Small parameter changes drastically alter form—tight spirals can open into loose waves.
  • Line continuity feels more expressive than point plots. The path tells a story through motion.
  • Hybrid systems (interpolating between equations) generate surprising transitional geometries.

🧪 Future Directions

Some areas I’m exploring next:

  • 3D geometry export: Converting these line forms into meshes for fabrication or 3D printing.
  • Live interaction: Real-time tweaking of parameters to “sculpt” the attractor shape.
  • Machine discovery: Using evolutionary algorithms to find new systems with interesting geometry.

Final Thought

Attractor functions give us a playground where math meets art. The geometry isn’t drawn from intuition—it’s discovered, step by step, through equations and iteration. What begins as chaos can become a line, and what begins as a line can become design.

If you’re curious to try your own, fork the JSFiddle sketches and start shaping your own geometries.