A Lorenz Attractor is a solution of the Lorenz equations, which are ordinary differential equations in 3D.
Some exhibit chaotic behavior and create nice looking patterns.
The Lorenz differential equations are:
dx/dt = o * (y - x)
dy/dt = x * (p - z) - y
dz/dt = x * y - b * z

These equations describe how, at each point in 'time', x, y and z affect the direction of the attractor.
Once we move a bit in that direction, x, y and z change, and so does the direction.
In order to simulate this without solving the differential equation,
we iterate over small enough timesteps.

To pick a specific set of Lorenz equations, we'll set the euqations constants to:
o = 10
p = 28
b = 8/3

Showing the 3D attractor on a 2D image will not let us see its 3D complexity,
but an animation of it turning around the y axis can let us see just that:

See the code here