The Banach-Tarski Paradox
by aoum, Mar 8, 2025, 12:19 AM
The Banach-Tarski Paradox: A Mathematical Duplicating Machine
Imagine taking a solid sphere, cutting it into a finite number of pieces, and reassembling them—without stretching or adding anything—into two identical spheres of the same size. This seems impossible, but the Banach-Tarski Paradox states that it is mathematically possible! This paradox is one of the most counterintuitive results in set theory and relies heavily on the Axiom of Choice. In this blog, we’ll explore the history, proof, and implications of this famous theorem.
1. What Does the Banach-Tarski Paradox State?
Formally, the Banach-Tarski Paradox states:
A three-dimensional solid sphere can be decomposed into a finite number of disjoint subsets, which can be reassembled (using only rigid motions such as rotations and translations) into two spheres identical to the original.
That is, there exists a partition of a solid ball
into finitely many disjoint sets:
![\[
B = A_1 \cup A_2 \cup \dots \cup A_n
\]](//latex.artofproblemsolving.com/a/2/e/a2eb3f98dd3a76c178c7f258522b789fe7854df3.png)
such that there exist rotations and translations
satisfying:
![\[
T_1(A_1) \cup T_2(A_2) \cup \dots \cup T_n(A_n) = B \cup B.
\]](//latex.artofproblemsolving.com/1/2/f/12f487cdbd41aeb2a158f0dbade241b83ba87f21.png)
In other words, we "duplicate" the ball using only rearrangements!
This paradoxical result was proven in 1924 by Stefan Banach and Alfred Tarski, following earlier work by Felix Hausdorff.
2. The Role of the Axiom of Choice
The proof crucially depends on the Axiom of Choice (AC), which states that:
Given any collection of nonempty sets, it is possible to choose one element from each set, even if there is no explicit selection rule.
The Axiom of Choice allows the construction of certain "non-measurable" sets, which do not have a well-defined volume. This is key to understanding why the paradox works.
3. The Proof: How Does It Work?
The proof relies on group theory, measure theory, and properties of rotations in three-dimensional space. Below is an outline of the main ideas.
Step 1: The Free Group Structure of SO(3)
The paradox exploits the fact that the group of rotations in three-dimensional space,
, contains a subgroup that is non-abelian and behaves like a free group.
A free group
on two generators 
and 
consists of all possible words made from 
and their inverses 
, with no algebraic simplifications beyond cancellation.
The crucial result is that there exist two rotations
that generate a free group, meaning that any word formed from 
and their inverses corresponds to a unique rotation.
Step 2: Constructing the Paradoxical Decomposition
We now construct a decomposition of the sphere using the free group structure.
1. Consider the unit sphere
in 
and remove a single point 
(which makes the sphere equivalent to an open set).
2. Using the rotations
and 
, define orbits of points on .
3. Partition the sphere into four sets:
![\[
A, B, R_1(A), R_2(B)
\]](//latex.artofproblemsolving.com/f/3/4/f34df0abc214bd2f6d1e954fc8fbe4b192ccd7e5.png)
where
and 
are defined recursively.
4. Using the fact that the free group acts on without preserving volume, we obtain a paradoxical duplication.
By applying rotations, we can rearrange these sets to form two identical copies of the original sphere.
Step 3: Extending to a Solid Ball
To extend this result from to a three-dimensional ball 
:
1. Use the fact that can be decomposed into countably many nested spheres.
2. Apply the Banach-Tarski decomposition to each sphere.
3. The union of all these decompositions results in two copies of the original ball.
Thus, the entire solid sphere is duplicated using only rigid motions!
4. Why Doesn’t This Work in Reality?
Although the Banach-Tarski Paradox is mathematically valid, it does not apply to the real world for several reasons:
Thus, while the paradox is a fascinating result in pure mathematics, it does not imply that we can duplicate objects in the real world!
5. Implications and Related Results
The Banach-Tarski Paradox has deep implications in mathematics and related paradoxes:
6. Conclusion: A Strange but Beautiful Theorem
The Banach-Tarski Paradox is one of the most striking results in modern mathematics. It shows that our intuitive notions of volume and shape do not always hold in higher mathematics. While it cannot be realized physically, it remains a fascinating example of the power of abstract mathematics and the strange consequences of the Axiom of Choice.
References
Imagine taking a solid sphere, cutting it into a finite number of pieces, and reassembling them—without stretching or adding anything—into two identical spheres of the same size. This seems impossible, but the Banach-Tarski Paradox states that it is mathematically possible! This paradox is one of the most counterintuitive results in set theory and relies heavily on the Axiom of Choice. In this blog, we’ll explore the history, proof, and implications of this famous theorem.
1. What Does the Banach-Tarski Paradox State?
Formally, the Banach-Tarski Paradox states:
A three-dimensional solid sphere can be decomposed into a finite number of disjoint subsets, which can be reassembled (using only rigid motions such as rotations and translations) into two spheres identical to the original.
That is, there exists a partition of a solid ball
into finitely many disjoint sets:![\[
B = A_1 \cup A_2 \cup \dots \cup A_n
\]](http://latex.artofproblemsolving.com/a/2/e/a2eb3f98dd3a76c178c7f258522b789fe7854df3.png)
such that there exist rotations and translations
satisfying:![\[
T_1(A_1) \cup T_2(A_2) \cup \dots \cup T_n(A_n) = B \cup B.
\]](http://latex.artofproblemsolving.com/1/2/f/12f487cdbd41aeb2a158f0dbade241b83ba87f21.png)
In other words, we "duplicate" the ball using only rearrangements!
This paradoxical result was proven in 1924 by Stefan Banach and Alfred Tarski, following earlier work by Felix Hausdorff.
2. The Role of the Axiom of Choice
The proof crucially depends on the Axiom of Choice (AC), which states that:
Given any collection of nonempty sets, it is possible to choose one element from each set, even if there is no explicit selection rule.
The Axiom of Choice allows the construction of certain "non-measurable" sets, which do not have a well-defined volume. This is key to understanding why the paradox works.
3. The Proof: How Does It Work?
The proof relies on group theory, measure theory, and properties of rotations in three-dimensional space. Below is an outline of the main ideas.
Step 1: The Free Group Structure of SO(3)
The paradox exploits the fact that the group of rotations in three-dimensional space,
, contains a subgroup that is non-abelian and behaves like a free group.A free group
on two generators 
and 
consists of all possible words made from 
and their inverses 
, with no algebraic simplifications beyond cancellation.The crucial result is that there exist two rotations
that generate a free group, meaning that any word formed from 
and their inverses corresponds to a unique rotation.Step 2: Constructing the Paradoxical Decomposition
We now construct a decomposition of the sphere using the free group structure.
1. Consider the unit sphere
in 
and remove a single point 
(which makes the sphere equivalent to an open set).2. Using the rotations
and 
, define orbits of points on .3. Partition the sphere into four sets:
![\[
A, B, R_1(A), R_2(B)
\]](http://latex.artofproblemsolving.com/f/3/4/f34df0abc214bd2f6d1e954fc8fbe4b192ccd7e5.png)
where
and 
are defined recursively.4. Using the fact that the free group
acts on without preserving volume, we obtain a paradoxical duplication.By applying rotations, we can rearrange these sets to form two identical copies of the original sphere.
Step 3: Extending to a Solid Ball
To extend this result from
to a three-dimensional ball 
:1. Use the fact that
can be decomposed into countably many nested spheres.2. Apply the Banach-Tarski decomposition to each sphere.
3. The union of all these decompositions results in two copies of the original ball.
Thus, the entire solid sphere is duplicated using only rigid motions!
4. Why Doesn’t This Work in Reality?
Although the Banach-Tarski Paradox is mathematically valid, it does not apply to the real world for several reasons:
- Physical matter is not infinitely divisible: The proof relies on sets that are infinitely complex and non-measurable, which do not exist in physical reality.
- Volume is preserved in real-world physics: The paradox contradicts the conservation of mass and volume, which always hold in physics.
- Rotations in the real world are continuous: The proof relies on abstract mathematical rotations that do not have physical analogs.
Thus, while the paradox is a fascinating result in pure mathematics, it does not imply that we can duplicate objects in the real world!
5. Implications and Related Results
The Banach-Tarski Paradox has deep implications in mathematics and related paradoxes:
- Hausdorff Paradox: A similar result in two dimensions showing that a sphere minus a countable set can be paradoxically decomposed.
- Non-Measurable Sets: It highlights the existence of sets that do not have a well-defined volume.
- Measure Theory and Set Theory: It demonstrates that volume is not purely an algebraic concept but depends on underlying axioms.
6. Conclusion: A Strange but Beautiful Theorem
The Banach-Tarski Paradox is one of the most striking results in modern mathematics. It shows that our intuitive notions of volume and shape do not always hold in higher mathematics. While it cannot be realized physically, it remains a fascinating example of the power of abstract mathematics and the strange consequences of the Axiom of Choice.
References
- AoPS: Banach-Tarski Paradox
- Wikipedia: Banach-Tarski Paradox
- Wagon, S. The Banach-Tarski Paradox (1993)
- Tarski, A. Logic, Semantics, Metamathematics (1956).
- Hausdorff, F. Set Theory (1914).
- Jelonek, Z. The Role of the Axiom of Choice in Paradoxical Decompositions (1995).
March 2025