Banach–Tarski Paradox: Duplicating a Solid Sphere
The video addresses mathematically literate viewers who have heard of the Banach–Tarski paradox—duplicating a ball from finitely many pieces—but want to understand how such an extreme result can be made precise.
Interval-to-Circle Mapping and Infinite Points from a Punctured Circle
This construction targets learners building intuition for how infinite sets can still “cover” spaces even when missing points, a stepping-stone toward accepting paradoxical decompositions.
Free Group of Rotations and Paradoxical Decompositions
The free-group construction is aimed at viewers comfortable with basic algebra who want to understand the algebraic “engine” behind Banach–Tarski rather than treating it as a black box.
Orbits of Rotation Groups and Duplication of Sphere Sections
This note targets learners trying to visualize how free-group rotations carve the sphere into structured orbits that can be duplicated.
Axiom of Choice and Paradoxical Sphere Decomposition
Abide by Reason speaks here to viewers interested in the foundational assumptions that make Banach–Tarski possible, particularly the role of the Axiom of Choice.