Catastrophe Theory
Catastrophe Theory is a mathematical theory of sudden change. It was created by the French mathematician Rene Thom in the early 1960's.
It was mentioned as part of Ari Bosronov's lecture (pg 121?) and by Abu BinBob as being related to the caustic waves in the swimming pools (pg 130-131?). However, it is explained in more detail in Dr. Jennifer Beseret's introductory lecture on the topic of Catastrophe Theory in the section of the same name beginning on pg 137. The claim is that catastrophe theory is involved in the sudden change from one level of complexity to the next.
"So lets get started with some mathematics you may not be familiar with, Catastrophe Theory. The French mathematician, Ren Thom, created the theory in the 1960's. There are other people involved, Hassler Whitney, for example, but this is not a history of mathematics course so we won’t discuss who did what any further."
"Isn’t that the theory of dynamic systems. I thought that was called Chaos Theory?" one of them said, ...
Jen responded ..., "You are confused. Chaos and Catastrophe may sound alike to you, and both fields of mathematics are theories about dynamic systems, but Chaos deals with self-similarity and fractal geometry. Catastrophe is topological. You are familiar with Topology of course?"
"Thom worked out the relationship of topological singularities to the maxima and minima of calculus. He could see how the former would ‘unfold’ into arrangements of the latter, imposing a structure on them. To know the arrangement of the max and min of a dynamic process would be to know its qualitative behavior. How many topologically different structures were possible? For a very wide range of processes, only seven stable unfoldings, the seven ‘elementary catastrophes’ are possible. These unfoldings are called catastrophes because each of them has regions where a dynamic system can jump suddenly from one state to another, although the factors controlling the process change continuously." She paused for breath, ...
"Each of the seven catastrophes represents a pattern of behavior determined only by the number of control factors, not by their nature or by the interior mechanisms that connect them to the systems behavior."
..."These catastrophes model a wide variety of processes ...," she said, and began to display the graphs, "... caustics ... engineering ... phase transitions, as in explosions ..."
"Boom!" said a boy in the middle, ...
She continued her list of applications of catastrophe theory, "... the nervous system of an animal or a human being can convey an infinite variety of messages. The transmission of every nerve impulse depends on appropriate levels of just a few substances, especially sodium and potassium, inside and outside each nerve cell. Key features such as these can serve as the control factors in models based on catastrophe theory ..."
"So the models are multidimensional?" Sarah asked, bringing the crowd back from the brink.
"Yes, exactly," Jen responded showing relief, "Your recognition of that truth was a sudden change in your mental state, a good catastrophe." Jen was waiting for the bad catastrophe.
"Let’s examine this in more detail." Jen returned to the first graph. "We can use the cusp catastrophe graph to model examples of territorial behavior, aggression, and group formation to illustrate observed patterns of behavior and their changes from one mode into another."
..."Will this be on the exam?" "No, it won’t be on the exam, look at the list," Jen said sharply, ...
"This is the least interesting ..." " ... of the elementary catastrophes. Because it’s simple!" she snarled, thinking, "Simple, just like you!" But she still had control of herself, if not the class. So she tried to continue, "Obviously, only a few things can happen."
"The system can be balanced at the point of inflection," she said raising her voice over the increasing cross conversations. "Boom" boy started trying to balance his pen on the tip of his finger. Jen grabbed for it. "Oooh!" from the class. "The system can be essentially unstable, having no minimum," Jen said a bit louder. "Or the system can move to a state of minimal potential, if one can exist," she said more quietly, as if giving up.
"Consider a rubber band ..." The fat kid in the second row, woke up and took the wide rubber band from around his overstuffed notebook. Then the rubber bands started flying. The nasty comments started flying.
Finally, she had lost control of the class. Everything in the classroom was in motion. Desks were being pushed back as some students started to walk out. Others were throwing papers and shooting rubber bands like a junior high school class. Someone had unplugged his earphones from his laptop and the unfiltered music spilled out into the classroom. There was only one student not moving. It was the girl with the green ribbon and pony tail, Sarah. Suddenly, Jen stopped yelling at the class and looked directly at Sarah and smiled. Then she raised the projection screen from in front of the blackboard and pointed to what was written on it. "Homework #1: Describe what just happened. State the control variables, and draw the appropriate catastrophe graph. Due Thursday."
I don't have the tools to show you the pure mathematics of catastrophe theory on this page at this moment. It involves drawing three dimensional graphs. In the future I will update this page. However, the mathematics is embedded in the reality around you, and I can supply an example of one of the elementary catastrophes and its associated graph which should make you understand that this theory describes your universe.
Imagine standing in a large room. On the floor of the room is a large rug. Perhaps it is a hyperrug, woven by members of FoM from Hotan. The pattern is intricate, but the hyperthreads form a golden grid that shines through the material. You want to move the rug to the other side of the room, so you grab hold of it in the middle of one of the long sides, and try to pull it across the floor. But the rug is heavier than you thought and all you manage to do before stopping for breath is to fold your edge onto itself a bit. If you look at your edge of the rug, the fold forms a backwards "S", with your hands holding the top in the air. The other three edges are still flat on the floor, although they may have moved some.
As you stand there trying to catch your breath, look at the pattern on the rug. In the center is a large red circle enclosing a green leaf pattern. Outside the circle, the pattern is a brown and yellow image of rocks and pebbles. When the rug was flat on the floor, you placed your pet mouse down on the rug so that she could get her daily exercise. Round and round, lap after lap she ran, following the red circle. But now, as she makes one more circuit (counter-clockwise) she falls off the edge of the fold you have made. She stops and looks at you reproachfully for such a catastrophe.
Now to get more mathematical. Assume that the far left corner of the rug is the origin of a three-dimensional coordinate system. Along the far edge of the rug you can measure from the origin and let it be the positive x-axis. The left edge of the rug is the y-axis, with positive values increasing as the approach your edge. The third axis, the z-axis, measures the height of the rug above the floor. The red circular path on the surface of the rug could now be described by a formula, but we won't get that detailed here. Most of the rug still has z-value of 0. The path of your pet mouse changes gradually, or smoothly, step by step, even as it climbs slightly up hill onto the folded part of the rug. Then, suddenly, it changes "discontinuously" from a positive value of z at the edge of the fold, to a 0 value of z back on the floor. This event is the "catastrophe" or discontinuous change that the theory is interested in.
Catastrophe theory is especially interesting when used as applied mathematics. Instead of describing your pet mouse running around the red circle on the hyperrug, the same model can be used to describe the physical events of boiling and condensation. The x-axis and the y-axis in general represent control factors, while the z-axis represents behavior. Let the x-axis represent pressure, and the y-axis represent temperature. The z-axis represents the behavior, the density of water. On the right side the water is liquid. On the left side, the water is gas. Choose a point, A, in the liquid region, and a point, D, in the gaseous region. You can draw a path from A to D that avoids the fold and gradually increases in z-value. You can also follow a path from D to A that drops off the fold before continuing along the bottom surface to A.
Woodcock, Alexander, and Monte Davis. Catastrophe Theory: The Revolutionary New Way of Understanding How Things Change. New York: E. P. Dutton, 1978. ISBN: 0-525-07812-6