...and other mathematical objects

What is chaos?  Mathematically speaking, it is a sensitive dependence on initial conditions.  It's found in flowing water, the stock market, celestial orbits, coastlines, weather...almost anything that's difficult to predict.  What these systems have in common is that in order to know what will happen in the future, we need to know everything about the present with absolute precision.  Thanks to Heisenberg, we never will.

Chaos is everywhere in nature, but it was a surprising discovery that it also appears in simple systems of equations. 

At left is a quaternion Julia set.

A common type of chaos generator is the iterative system, in which equations are applied to their own results over and over again.  The resulting path never settles down in one place or starts repeating itself: instead of forming a simple shape, which is what you'd expect from simple equations, it keeps skipping around forever within a finite volume, slowly filling in a lacy fractal design.

At right, a Lorenz-84 attractor generated in Chaoscope. 

With the help of Nicolas Desprez's Chaoscope, I've etched a few such systems.  I'm not stocking any attractors at present, but if you have a favorite that you'd like to see in glass, feel free to write.  The most three-dimensional sets work the best.

If you'd like to start exploring the world of strange attractors, Chaoscope is an excellent tool: it's free, easy to use, and contains many systems. 

In other news, this etching of the Feigenbaum function was commissioned by Wolfram Research for Dr. Feigenbaum's 60th birthday, and I'm proud to say that it appears at the Mathworld site for that function.

Photo Amy Young, stylist Michael Trott.