The Banach–Tarski Paradox: Turning One Ball into Two

Imagine this: You have a solid ball in three-dimensional space. Now, according to a mathematical theorem known as the Banach–Tarski Paradox, it’s possible to divide this ball into a finite number of disjoint subsets and reassemble them into two identical copies of the original ball. Yes, you read that correctly—two balls from one!

What’s Going On?

At first glance, this seems utterly nonsensical. How can you duplicate a physical object without adding any material? The key lies in the nature of the pieces involved. The paradoxical decomposition relies on sets that are non-measurable, meaning they don’t have a well-defined volume in the traditional sense. These sets are constructed using the Axiom of Choice, a principle in set theory that allows for the selection of elements from an infinite collection of sets.

How Does It Work?

The process involves:

• Dividing the original ball into a finite number of non-overlapping subsets.
• Reassembling these subsets through rotations and translations (without stretching or bending) to form two identical copies of the original ball.

This procedure defies our intuitive understanding of volume and space, leading to its classification as a veridical paradox—it contradicts our expectations but is mathematically sound.

Why Does It Matter?

While the Banach–Tarski Paradox doesn’t have practical applications in the physical world (since it involves non-measurable sets that can’t be physically constructed), it has profound implications in the realm of mathematics:

• It challenges our understanding of measure theory and the concept of volume.
• It highlights the importance of the Axiom of Choice in set theory.
• It has inspired further research into the properties of amenable groups and their relationship to paradoxical decompositions.

Mind-Blowing Mathematical Calculator

Based on the Banach-Tarski Paradox, this playful calculator shows the impossible result of dividing a ball into pieces and reassembling them. What will your result be?

Enter the number of pieces to divide the ball into: Calculate