The Banach-Tarski Paradox

[Magical Realist]

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

[Syne]

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

#2 & 3 mean this cannot be done in reality, only in theory.

[Magical Realist]

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.