The "Lorenz Attractor" is a "simple" set of three deterministic equations developed by Edward Lorenz while studying the non- repeatability of weather patterns. The weather forecaster's basic problem is that even very tiny changes in initial patterns ("the beating of a butterfly's wings" - the official term is "sensitive dependence on initial conditions") eventually reduces the best weather forecast to rubble.

The lorenz attractor is the plot of the orbit of a dynamic system consisting of three first order non-linear differential equations. The solution to the differential equation is vector-valued function of one variable. If you think of the variable as time, the solution traces an orbit. The orbit is made up of two spirals at an angle to each other in three dimensions. We change the orbit color as time goes on to add a little dazzle to the image. The equations are:

dx/dt = -a*x + a*y

dy/dt = b*x - y -z*x

dz/dt = -c*z + x*y

We solve these differential equations approximately using a method known as the first order taylor series. Calculus teachers everywhere will kill us for saying this, but you treat the notation for the derivative dx/dt as though it really is a fraction, with "dx" the small change in x that happens when the time changes "dt". So multiply through the above equations by dt, and you will have the change in the orbit for a small time step. We add these changes to the old vector to get the new vector after one step. This gives us:

xnew = x + (-a*x*dt) + (a*y*dt)

ynew = y + (b*x*dt) - (y*dt) - (z*x*dt)

znew = z + (-c*z*dt) + (x*y*dt)

(default values: dt = .02, a = 5, b = 15, c = 1)

We connect the successive points with a line, project the resulting 3D orbit onto the screen, and voila! The Lorenz Attractor!

We have added two versions of the Lorenz Attractor. "Type=lorenz" is the Lorenz attractor as seen in everyday 2D. "Type=lorenz3d" is the same set of equations with the added twist that the results are run through our perspective 3D routines, so that you get to view it from different angles (you can modify your perspective "on the fly" by using the command.) If you set the "stereo" option to "2", and have red/blue funny glasses on, you will see the attractor orbit with depth perception.