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The Banach-Tarski Paradox and Equidecomposability and Their Applicability to the Infinite

CSEF · 2018 Mathematical Sciences

Leonardo S. Park · Chino Hills High School, Chino Hills, San Bernardino County

Overview

Objectives/Goals The objective of this project was to formulate two mathematical proofs(one through a graphical demonstration in the context of the mechanisms of the Hyperwebster, an infinite dictionary, and one through compiling) to affirm the validity of the Banach-Tarski Paradox. Methods/Materials Personal computer, Painting Software(Paint), and the Pyth Compiler/Executor. This project was carried out through two experiments to show how it is possible to create two duplicates from one starting sphere. The first experiment was through the computer and the software, Paint, to draw 8 graphs and sketch lines to represent a hypothetical three-dimensional sphere rotating in a graphical plane and making its mechanisms imitate that of the Hyperwebster, an infinitely countable dictionary. The second experiment was conducted through the software Pyth Compiler, to basically code a three-dimensional sphere and also "rotate" it to highlight how its movement is representing the workings of the Banach-Tarski Paradox. Results I was able to prove that the Banach-Tarski Paradox was valid by converting the mechanisms of the Hyperwebster, an infinitely countable dictionary, to a three-dimensional graphical sense for a solid sphere and by utilizing the Pyth Compiler/Executor to compile a rotating sphere to imitate the workings of the Banach-Tarski Paradox. Conclusions/Discussion I was able to confirm my hypothesis of the legitimacy of the Banach-Tarski Paradox in a pure mathematical sense. I have realized that currently, it is only able to be proven in the abstract mathematical world because in this context, this paradox is not constrained by the limitations of the physical boundaries that are present in the real world. However, it is quite true that what is able to be proven mathematically is often proved through physics as well, and it may be possible for an application in the physical world to be created if our technology ever reaches that level.

Summary statement

I devised two mathematical proofs through a graphical context and through a compiler to test the validity of the Banach-Tarski Paradox, which states that one 3-D ball, in set-theoretic geometry, can be cut out to yield two identical copies.

Help received

I did not receive any assistance and worked alone on my experiments.

Competition history

  • CSEF 2018 Mathematical Sciences · Entry S1513

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