Some of you may remember the Weeble, the roly-poly toy popular in the 1970s. Kids could knock them down, but they would always pop back to their original position (thanks to a weight in the bottom of the toy). The ad slogan said “Weebles wobble, but they don’t fall down.”
What made them so much fun was that no other item in the kids’ home would pop back up like that. A kid could be more curious than George (in the classic children’s book) and still never find an object that he or she could knock over and have pop back up again without any outside assistance. For a long time mathematicians believed that it was impossible to create a uniformly dense convex object that would pop back up like a Weeble.
Here’s the thinking behind that mathematical challenge.
If you have before you, resting quietly on a table, any object, you will usually be able to turn it over and around and allow it to rest on another part of itself. Such a position is called a “stable equilibrium point,” and there are often several such positions.
Now if the object is made uniformly and no little piece is heavier than any other little piece (that is if the object is uniformly dense), and if the object is convex (that is a line joining any two points in the object lies wholly within the object), then there will usually (but not invariably, as we shall soon see) be more than one way of placing the object motionless on the table.
In addition to the ways to place the object at rest, there will be ways to place the object so that it would not move if it were balanced absolutely perfectly, and there were no outside forces (like vibrations or breezes) acting on it. (An example of this would be the tip of a sharpened pencil. Under perfect conditions one might think of a pencil balanced on its sharpened tip.) Such a position is called an “unstable equilibrium point.”
Until recently no convex uniformly dense object was known which had exactly one stable and one unstable equilibrium point. A Russian mathematician named Vladimir Arnold conjectured it in 1995, and two Hungarian scientists, Gabor Domokos and Peter Varkonyi, proved it in 2006. The object was described mathematically, called a “Gomboc” (the Hungarian word for dumpling), and then constructed. The Gomboc has to be machined to tolerances below 10 microns (1/10th the thickness of a human hair), which is why the cost for one is in the $300 range (as listed by Hammacher Schlemmer).
Pick up the Gomboc, turn it over, place it down in any random position and it begins to rock back and forth, slowly and then faster, and then finally ends up in its original position. (You can see it in action on YouTube.)
I bought a Gomboc several months ago and have enjoyed looking at it and thinking about it. The story of the Gomboc illustrates the fact that there is more to mathematics than formulae. There are “things,” physical realizations of mathematical notions. The construction here required a lot of hard work since it is one thing to guess at a solution for this type of problem, and quite another to prove that your solution is correct. In addition, the solution is not so easy to describe, even mathematically.
#b#About the writer:#/b#
Lee Neuwirth, Princeton University Class of 1955, earned his Ph.D. in mathematics from Princeton in 1959. He served as director of the Princeton-based Institute for Defense Analyses (IDA) from 1977 until 1985. He retired from the research staff in 1999.
Neuwirth is the author of a memoir about the anti-war protests that were focused on the IDA during the Vietnam War: “Nothing Personal — The Vietnam War in Princeton 1965-1975.”
Neuwirth’s wife, Sydney, is an accomplished artist. Their daughter, Bebe, is a dancer, singer, and actress currently appearing in the CBS television drama, “Madame Secretary.” Their son, Peter, is an actuary with Towers Watson, a global professional services company that does risk management and other consulting. He recently published a book titled “What’s Your Future Worth? — Using Present Value to Make Better Decisions.”