The theorem 1

In 1988 it was conjectured that any set of disjoint convex objects in 3-d would have a subset that could be translated to infinity without disturbing the others. Mathematicians had shown sets where no single convex object could move without disturbing other objects, but half of the objects could be moved together as a unit.

This "twisted tetrahedron" is the simplest configuration of objects that cannot be taken apart by translation, giving a counterexample to this conjecture. One can prove it cannot be taken apart by checking, for each moving subset, that every motion vector has positive projection on the normal of a moving tetrahedron in contact with a stationary tetrahedron. This configuration can be taken apart if we also allow rotation.

However, we can nest five copies together to form a configuration that cannot be taken apart by arbitrary rigid motions. That is, no group of sticks can be removed without disturbing the rest. Again, this is proved by testing, for each possible subset (all 2 to the 30th or about 1 billion of them), that every motion pushes a moving object into a stationary object. (An infinitesimal motion can be represented as a vector in a six dimensional force/torque space. Many subsets can be ruled out by symmetries or simple geometric arguments, leaving about 120 thousand subsets to check.)

(Here are gzipped polygon descriptions of the six-stick example (1K) and the 30-stick example (20K) in a Mathematica format (ASCII) if you'd like to experiment with them. I reserve the copyright.)

Next, Prev, Start, Jack's Home