Banach–Tarski in plain words: why 'doubling a ball' needs infinity

A 5-beat faceless video outline for “Banach–Tarski in plain words: why 'doubling a ball' needs infinity” in the Math paradoxes explained niche, derived from caption timing of 2 reference videos.

Angles

• Cold-open on the concrete hook from the primary reference — Vsauce leads with a scene, not a definition dump
• Make the mechanism explicit with one visual beat: Up, down, left and right around two perpendicular axes. We are going to need every single …
• Contrast or scale-up using the second reference (TED-Ed): and then go back in time and choose again, we'd know how to find the same marble. If we ha…
• Close on what the viewer can check themselves; original narration only — do not copy reference wording

Beat-by-beat structure

1. Hook (0:00–0:30): open on the puzzle of 'Banach–Tarski in plain words' in one image.
2. Setup (0:30–2:00): define the key terms in plain words; borrow structure from Vsauce, not lines.
3. Mechanism (2:00–5:00): walk the cause-and-effect with one diagram; cite the evidence beat (Up, down, left and right around two perpendicular axes. We are going t…).
4. Contrast (5:00–7:00): show the common myth or the second case from TED-Ed.
5. Recap card: three takeaways + one 'check this yourself' action. Original narration only.

Reference videos

• The Banach–Tarski Paradox s86-Z-CbaHA
• Does math have a major flaw? - Jacqueline Doan and Alex Kazachek seugK4PrW48

Method

yt-dlp VTT captions (manual/uploader only) → timed beat heuristics → editorial topic synthesis (no ASR). Structure only; write original narration.