Zio Chris' Lorenz Attractor by Chris Culy

is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

The Lorenz Attractor^{?} is another classic fractal,
discovered by Edward Lorenz, as he was trying to model the weather. He also coined the term "the butterfly
effect", to capture the idea that small changes in one place and time can have unpredictable consequences
elsewhere later. The term comes from the idea of a butterfly flapping its wings setting off a chain of
events that leads to a hurricane weeks later in another part of the world. Is it a coincidence that
the first version of the attractor that Lorenz created sort of looks like a butterfly? Hmm...

Like the De Jong attractor, and different from the
Mandelbrot set and
the Burning Ship fractal, here we have three parameters
that we can adjust to get different images.
Unlike the other fractals here, the Lorenz attractor is 3 dimensional. However, we are only plotting
two dimensions at a time. We use the third dimension as the source of information about the color.
As with the De Jong attractor, there is no standard use of colors.

The examples here include the standard "butterfly" one, as well as some more unusual ones, not typically
seen on the web. As with the other fractal examples here, this pages owes a lot to Paul Bourke,
specifically his Lorenz Attractor page
and the
Wikipediage Lorenz Attractor page.

The Lorenz attractor, like the De Jong attractor is visualized as a series of points,
calculated by equations with parameters, and starting from a particular point. The equations are:

nextX = previousX + dt * r * (previousY - previousX);

nextY = previousY + dt * (previousX * (b - previousZ) - previousY);

nextZ = previousZ + dt * (previousX * previousY - p * previousZ);

*dt* is a small factor that incorporates the notion of change of time. The
Wikipediage Lorenz Attractor page
has some useful technical details about the parameters and their range of values, including the
idea that the parameters are typically positive.

p:
r:
b:
iteration power:
increment:

Initial point
x:
y:
z:

Dimension for color:
Color schemes:

Some examples:

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