Chapter 1: The Banach-Tarski Paradox

The Banach-Tarski Paradox is a captivating illustration of the strange and counterintuitive aspects of mathematics. Introduced by mathematicians Stefan Banach and Alfred Tarski in 1924, this theorem posits that it is possible to take a solid sphere, divide it into a finite number of pieces, and then rearrange those pieces to create two identical spheres of the same size as the original. This concept challenges our basic understanding of volume and space.

To clarify this perplexing idea, I will begin with a simplified analogy before diving into more complex theories. While I won't be providing a formal proof—since it requires advanced knowledge of group theory, measure theory, and geometric set theory—I aim to illustrate the intriguing challenges posed by infinity, which may shed light on the plausibility of the Banach-Tarski theorem.

Understanding the Infinite Dictionary: The HyperWebster

This thought experiment, proposed by Dr. Ian Stewart, involves a hypothetical dictionary containing every possible combination of letters. Starting with the ‘empty word’—which we will denote as ‘ ’—we can generate sequences of letters such as a, aa, aaa, and continue infinitely. We then move on to combinations starting with ab, aba, abaa, and so forth, creating a comprehensive collection that includes the entirety of classic literature, variations of historical events, and every conceivable combination of letters.

The scale of this dictionary is astronomical—essentially, it is infinite.

Volumes of the HyperWebster

To manage this vast collection, we categorize the HyperWebster into volumes by alphabetical order. Each volume corresponds to a letter, containing all words that begin with that specific letter. However, one unique entry—the empty word—is reserved in its own special volume.

Cost-Effective Publishing

In a bid to reduce printing costs, the publishers of the HyperWebster decide to eliminate the first letter of every word. Readers would then refer to the volume title to know which letter to prepend. For example, the word "aadfff" found in the K volume would actually represent "kaadfff."

Thus, by removing the leading letter from words in the A volume, we retrieve every possible word combination beginning with 'a'. Remarkably, this also means we recover the empty word—an essential element for our understanding of the sphere. By applying this approach across all volumes, we effectively reconstruct the entire HyperWebster 26 times over.

Connecting Back to Banach-Tarski

This analogy illustrates that if we can label points on a sphere with sequences of letters and manipulate these labels carefully, it is conceivable to recreate two spheres from a single original. While the proof of this theorem involves more intricate steps, this foundational concept holds true.

The Formal Statement of Banach-Tarski

The formal statement of the Banach-Tarski theorem asserts that for any two bounded subsets A and B in three-dimensional space, if both have non-empty interiors, they can be equidecomposed through isometries. The key components in this proof hinge on the axiom of choice, which allows for the selection of elements from an infinite collection of sets. This axiom is crucial because it ensures that the proof remains valid in the absence of constructive methods.

Interestingly, the theorem does not require the decomposed pieces to be Lebesgue measurable. If they were, both A and B would share the same measure, imposing a significant limitation compared to the original theorem's conditions. Recent advancements, such as those from Grabowski, Mathé, and Pikhurko in 2019, have highlighted that if A and B are measurable and of equal measure, they can still be equidecomposed under the conditions stated in the Banach-Tarski theorem.

This fascinating result transcends traditional geometry, applying equally to hyperbolic and spherical geometries in dimensions three and beyond.

A Mathematical Joke to Conclude

To wrap up this exploration of the Banach-Tarski Paradox, here's a light-hearted mathematical joke:

17. What is an anagram of Banach-Tarski?

1. Banach-Tarski Banach-Tarski.