From: mk_thisisit
The Banach-Tarski paradox is a notable mathematical concept that, while counterintuitive, is considered “absolutely understandable today” and “well described” [00:01:42].
Core Principle
The Banach-Tarski paradox states that a single sphere can be disassembled into five parts [05:37:00]. These parts can then be moved and rotated without changing their shape [05:40:00]. From these five parts, it is possible to create two new spheres, each identical in volume to the original [05:41:00].
Abstraction vs. Physical Reality
This concept challenges intuition because it appears to contradict the principle of conservation of mass or volume [05:41:00]. However, the paradox exists solely at the level of mathematical abstraction [05:53:00]. The “spheres” in question are not physical objects with mass, but rather collections of geometric points [05:48:00]. If the object consisted of atoms, such a transformation would be impossible [05:58:00].
The paradox operates purely within the realm of mathematics, demonstrating how abstract concepts can behave differently from physical reality [08:08:00]. Despite its abstract nature, there have been attempts in physical works to explain certain behaviors of quarks using concepts related to the Banach-Tarski paradox [05:53:00].
Dimensionality
An interesting aspect of the Banach-Tarski paradox is its dependence on dimension:
- It does not exist in one dimension (on a line) or two dimensions (on a plane) [05:57:00].
- It is impossible to construct a “paradoxical circle” in two dimensions that could be disassembled and reassembled into two identical circles [06:02:00].
- The paradox requires at least three dimensions [06:06:00].