Two lines of formula and four numbers: how do they draw pictures like these?

These are the Clifford attractor equations (attributed to Clifford Pickover):

Since $|\sin|$ and $|\cos|$ never exceed 1, the point always stays inside the box $|x|\le1+|c|$, $|y|\le1+|d|$ and cannot escape. The stage does something simple: iterate three million times, count how often each pixel is hit, and map the counts to brightness on a log scale.

Take two starting points extremely close together and iterate both. If their distance is multiplied on average by $e^{\lambda}$ each step, $\lambda$ is the (largest) Lyapunov exponent. The stage computes it by carrying a tangent vector along with the Jacobian matrix.

• $\lambda>0$: tiny differences grow exponentially; that is chaos, and the points spread into a picture with fractal detail.
• $\lambda<0$: differences shrink, and every point is drawn into the same cycle. At $a=-0.95$, $b=1.6$, $c=1$, $d=0.7$ it is a 3-point cycle.
• $\lambda\approx0$: the point circles along a closed curve, neither chaotic nor settling.

All six pictures in the gallery have $\lambda$ above 0.

Chaos does not mean the points are sprinkled at random over the box. The map stretches and folds the plane again and again: stretching pulls neighbouring points apart (the chaos), folding brings them back into a bounded region. What survives endless stretching and folding is a set with zero area but infinite layers of detail, and the points cover it with a fixed density, which is the light and shade in the picture.

The Lorenz attractor (wonder No. 3 on this site) is the same kind of object in continuous time; here time moves in discrete hops, so millions of points take only a fraction of a second.