Tarski-Banach Decompositions

Two theorems I am going to state are mind boggling results associated with the names of F. Hausdorff, A. Tarski, S. Banach, J. von Neumann and R. M. Robinson. References are given in both books by Gelbaum and Olmsted. (The second of which actually proves the first theorem below.)

Both theorems use the notion of a rigid motion. A rigid motion of a space is a transformation that does not change the (euclidean) distance between two points. In the theorems below, B_r = {xR³: dist₂(x,0)r}.

Tarski-Banach Theorem 1

There exists a decomposition of B₁ into 5 pairwise disjoint sets A₁,...,A₅ of which the last is a single point such that there exist rigid motions R₁,...,R₅ with

B₁ = R₁(A₁)R₂(A₂) and B₁ = R₃(A₃)R₄(A₄)R₅(A₅),

where all unions are disjoint.

This means breaking a ball into five pieces such that it's possible to combine these pieces into two balls equal in size to the original one. No seams are visible after the operation. No cavities are created under the surface. 1 = 2. I just wonder what prevents me from taking up this occupation professionally. Could have taken a few bowling balls' manufacturers out of business. The problem is one can't be sure that dropping a ball will break it into five pieces, let alone into the right pieces of which one is a point.

Tarski-Banach Theorem 2

For any two positive numbers e and M, B_M can be split into a disjoint union of sets A₁,A₂,...,A_n such that there exist R₁,R₂,...,R_n rigid motions with the property that

B_e = R₁(A₁)R₂(A₂)...R_n(A_n)

where all the unions are disjoint.

What this theorem claims is that, for example, a ball of a pea size can be broken into several pieces which after rearrangement can be combined into a ball of, say, Sun's size. Weird and abstract as the theorem appears, it found a real life application to lion hunting. In order to catch a lion, apply to the lion Tarski-Banach decomposition. Put pieces back together to get a feline of a customary size of a domesticated cat. Now you may expect only a minor harm from the lion. Go after it fearlessly. After caging the beast, rearrange the pieces into their original configuration.

Both results may appear more acceptable if you recollect the strange behavior of infinite sets. Substituting for a moment the notion of 1-1 correspondence for that of congruence, we may remember that a union of two countable sets is again countable. From this we learn to anticipate counterintuitive results dealing with infinite sets.

Both theorems are proven with what is known as Axiom of Choice whose usage (a clear intuitive appeal notwithstanding) was questioned by the stream of Intuitionistic Mathematics in the early decades of the century. The axiom states that given a set X whose elements are other sets X_a indexed by a from a set A, it is then possible to form a set Y by selecting and putting into it one element x_a from each X_aX. The objection intuitionist had is that in general it's impossible to provide a constructive rule for selection of a single element from each X_a. They also reject any proof of pure existence (see, e.g., 3 Glass Puzzle, Best Distance). Most of mathematicians, however, prefer adjusting their intuition to rejecting the Axiom of Choice. All the more so because some results (Tarski-Banach Decompositions is one example) are only obtainable with the help of this axiom or equivalent statements.

The down side is that, for example, the sets whose existence is claimed by Banach and Tarski could not be visualized, let alone built. Such that neither theorem may be of immediate pragmatic value in any getting rich schemes.

Reference

1. B.R.Gelbaum and J.M.H.Olmsted, Counterexamples in Analysis, Holden-Day, 1964
2. B.R.Gelbaum and J.M.H.Olmsted, Theorems and Counterexamples in Mathematics, Springer-Verlag
3. Lion Hunting & Other Mathematical Pursuits. A Collection of Mathematics, Verse and Stories by Ralph P. Boas, Jr., G.L.Alexanderson, D.H.Mugler, eds, MAA, 1995
4. W. Sierpinski, On the Congruence of Sets and Their Equivalence by Finite Decomposition, in Congruence of Sets and Other Monographs, Chelsea Publ. Co., 1960