---
title: "The Aizawa Attractor · generative art"
description: "A sphere pierced by a tube. Generative art plate #16 by Theodore Vorillas, drawn live in your browser from its equations."
url: "https://vorillaz.com/lab/aizawa-attractor"
---

![](https://vorillaz.com/_astro/aizawa-attractor.B398ufUo_Zfu3W3.webp)

fig. #16 · rendered live in your browser from the equations below

# #16 the aizawa attractor

A sphere pierced by a tube.

- date

  2026.09.07

- tags

  generative-art · javascript

The Aizawa attractor, also known as the Langford attractor, is a system of three ordinary differential equations that produces one of the most sculptural shapes in the chaotic zoo. Trajectories wrap around the surface of a sphere-like body, then dive through a twisted tube along its axis and emerge on the other side, again and again, never repeating the same path.

Unlike the butterfly-like Lorenz family, the Aizawa attractor needs six parameters to keep its form. Small changes to `e` or `f` are enough to tear the sphere open or collapse the inner tube.

## The algorithm

```javascript title="aizawa-attractor.js"
const a = 0.95;
const b = 0.7;
const c = 0.6;
const d = 3.5;
const e = 0.25;
const f = 0.1;

const next = (x,y,z) => {
  const dx = (z - b) * x - d * y;
  const dy = d * x + (z - b) * y;
  const dz = c + a * z - Math.pow(z, 3) / 3 - (x * x + y * y) * (1 + e * z) + f * z * Math.pow(x, 3);
  return {dx,dy,dz};
};
```

The system is quite stiff around the tube, so a smaller time step (`dt = 0.005`) or a Runge–Kutta integrator keeps the surface smooth.

## References

- W\.F. Langford, _Numerical studies of torus bifurcations_, International Series of Numerical Mathematics, 70, 1984.
- [Elegant Chaos: Algebraically Simple Chaotic Flows](https://www.researchgate.net/publication/268721148_Elegant_chaos_Algebraically_simple_chaotic_flows_Dedicated_to_the_memory_of_Edward_Norton_Lorenz)