Catastrophe Theory for Dummies Part 2

Perhaps you were wondering, “Does the robot manage to keep the broomstick balanced forever?”

This is actually a complicated problem, but there are some simple facts. I forgot to stipulate that the robot’s hand can move only horizontally. But, since there is a floor under the robot, this detail does not change the eventual conclusion.

  • If the broomstick falls 90 degrees from vertical,  all the way over to a horizontal position, no horizontal movement of the robot’s hand can rebalance the stick to vertical. Donald Trump would say to our hapless robot, “You’re fired.”
  • If the stick is very close to horizontal, the force-through-a-distance required of the robot’s hand to rebalance the stick is very large. Force-through-a-distance is another word for “energy.”
  • The energy required to knock the stick  further over does not increase with angle. In fact, once started, the stick proceeds to fall by itself. A shove just speeds it up. But for the robot, the energy required to raise the stick from very-near-to-horizontal, in the words of math, “tends towards infinity.” So a vandal seeking to wreak havoc on the robot has a terrific advantage.
  • If the stick is very close to horizontal, and the robot happens to have a direct connection to a Japanese nuke plant that actually works, so that it could exert huge force to right the stick, the stick would disintegrate, busting up the whole scenario.
  • If the stick is almost vertical, what math calls “infinitesimally close” the problem becomes part of what is called “linear control theory”, making it “easy.” But this is a deception, because if the stick is not perfectly vertical, faulty intuition might maintain this assumption,  now false.
  • This is a “nonlinear problem.” All such problems are complicated.

But this problem is just simple enough that there may be a paper that addresses whether the stick can be stabilized at any angle above the horizontal. The answer depends upon how the movements of the crowd, who are constantly jostling the robot, are modeled.  For this we owe much to the German mathematician, Carl Friedrich Gauss. If the crowd is modeled with Gauss’s distribution, the stick will always fall, eventually.

So the little Japanese robot, wandering the crowded streets of Ginza, is a disaster waiting to happen.

This is a nice addition to the three examples of the Ted G. Lewis paper, “Cause-and-Effect or Fooled by Randomness?” , to be discussed next.