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+--- Thread: The Banach-Tarski Paradox (/thread-20095.html)
The Banach-Tarski Paradox - Magical Realist - Apr 1, 2026
This is a stellar example of how something mathematically proven can yet seem to defy all logic and common sensibility. Take apart a sphere into several pieces and you can reassemble them into two spheres of the exact same size! Any explanations anybody? Warning: do not try this at home!
https://www.youtube.com/watch?v=GWNifHPh8fk
RE: The Banach-Tarski Paradox - Syne - Apr 1, 2026
The video's graphic is actually duplicating the pieces as it comes apart. Not a true depiction of the paradox.
Key Steps of the Paradox
The process follows a specific mathematical procedure often broken down into these phases:
1. Decomposition into "Pieces": A solid ball in 3D space is partitioned into a finite number of disjoint subsets—as few as five pieces.
2. Nature of the Pieces: These are not "solid" chunks like slices of an orange; they are infinitely complex, scattered "clouds" of points.
3. Non-Measurability: Because these pieces are so complex, they are considered non-measurable, meaning they do not have a well-defined volume in the traditional sense.
4. Rigid Motion Reassembly: Using only rotations and translations (rigid motions), these point clouds are moved and regrouped.
5. Final Result: The reassembled pieces form two complete, solid balls identical to the first
- Google AI
RE: The Banach-Tarski Paradox - Magical Realist - Apr 1, 2026
Ah yes..the imaginary infinitely point-composed sphere and line segment. Kind of a variation on Zeno's infinite distance between two bodies moving towards each other.