Theorem

The unit ball $\mathbb D^3 \subset \R^3$ is equidecomposable to the union of two unit balls.

Proof

Let $\mathbb D^3$ be centered at the origin, and $D^3$ be some other unit ball in $\R^3$ such that $\mathbb D^3 \cap D^3 = \varnothing$.

Let $\mathbb S^2 = \partial \mathbb D^3$.

By the Hausdorff Paradox, there exists a decomposition of $\mathbb S^2$ into four sets $A, B, C, D$ such that $A, B, C,$ and $B \cup C$ are congruent, and $D$ is countable.

For $r \in \R_{>0}$, define a function $r^{*} : \R^3 \to \R^3$ as $r^{*}(\mathbf x ) = r \mathbf x$, and define the sets:

$\displaystyle W = \bigcup_{0 \mathop < r \mathop \le 1} r^{*}(A)$
$\displaystyle X = \bigcup_{0 \mathop < r \mathop \le 1} r^{*}(B)$
$\displaystyle Y = \bigcup_{0 \mathop < r \mathop \le 1} r^{*}(C)$
$\displaystyle Z = \bigcup_{0 \mathop < r \mathop \le 1} r^{*}(D)$

Let $T = W \cup Z \cup \left\{{ \mathbf 0 }\right\}$.

$W$ and $X \cup Y$ are clearly congruent by the congruency of $A$ with $B \cup C$, hence $W$ and $X \cup Y$ are equidecomposable.

Since $X$ and $Y$ are congruent, and $W$ and $X$ are congruent, $X \cup Y$ and $W \cup X$ are equidecomposable.

$W$ and $X \cup Y$ as well as $X$ and $W$ are congruent, so $W \cup X$ and $W \cup X \cup Y$ are equidecomposable.

Hence $W$ and $W \cup X \cup Y$ are equidecomposable, by Equidecomposability is Equivalence Relation.

So $T$ and $\mathbb D^3$ are equidecomposable, from Equidecomposability Unaffected by Union.

Similarly we find $X$, $Y$, and $W \cup X \cup Y$ are equidecomposable.

Since $D$ is only countable, but $\mathbb{SO}(3)$ is not, we have:

$\exists \phi \in \mathbb{SO}(3): \phi \left({D}\right) \subset A \cup B \cup C$

so that $I = \phi \left({D}\right) \subset W \cup X \cup Y$.

Since $X$ and $W \cup X \cup Y$ are equidecomposable, by a theorem on equidecomposability and subsets, $\exists H \subseteq X$ such that $H$ and $I$ are equidecomposable.

Finally, let $p \in X - H$ be a point and define $S = Y \cup H \cup \left\{{p}\right\}$.

Since:

• $Y$ and $W \cup X \cup Y$
• $H$ and $Z$
• $\left\{{0}\right\}$ and $\left\{{p}\right\}$

are all equidecomposable in pairs, $S$ and $\mathbb B^3$ are equidecomposable by Equidecomposability Unaffected by Union.

Since $D^3$ and $\mathbb D^3$ are congruent, $D^3$ and $S$ are equidecomposable, from Equidecomposability is Equivalence Relation.

By Equidecomposability Unaffected by Union, $T \cup S$ and $\mathbb D^3 \cup D^3$ are equidecomposable.

Hence $T \cup S \subseteq \mathbb D^3 \subset \mathbb D^3 \cup D^3$ are equidecomposable and so, by the chain property of equidecomposability, $\mathbb D^3$ and $\mathbb D^3 \cup D^3$ are equidecomposable.

$\blacksquare$

Axiom of Choice

This theorem depends on the Axiom of Choice, by way of Hausdorff Paradox.

Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.

Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.

However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.

Conclusion

Whether this is a veridical paradox or an antinomy is being hotly discussed to the present day. There are even Facebook groups anti and pro.

Pick your personal choice of philosophy and start ranting.

If you feel really adventurous, check out Rudy Rucker's novel White Light.

Source of Name

This entry was named for Stefan Banach and Alfred Tarski.

They raised this question in a collaborative paper in 1924.