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The Banach-Tarski Paradox: summary

The Banach-Tarski Paradox

Vsauce · Aug 1, 2015 · about 24 min

Uploaded Aug 1, 2015 · 49.1M views (as of Oct 30, 2026)

Good value given

The chocolate bar trick that seems to make an extra piece is an illusion. The new bar is slightly smaller.
Infinity is a size more than a number, and some infinities are bigger than others.
There are as many even numbers as there are whole numbers, because each can be matched one to one.
In Hilbert's hotel, a fully booked infinite hotel can still fit a new guest by moving everyone down a room.
Some papers suggest a link between the paradox and particle collisions that produce more particles than they started with.

Can you really turn one ball into two identical balls?

Mathematically, yes. The Banach-Tarski paradox proves a sphere can be cut into five pieces and rearranged, with no stretching, into two spheres exactly like the original. The catch is that the pieces have to be infinitely complex, so you can't do it with a real knife. Vsauce walks there through ideas about infinity like Hilbert's hotel and Cantor's diagonal argument.

Worth watching?Good value given. One of the best explanations of a famous math result on YouTube. It builds up slowly through infinity puzzles, so it's worth watching in full if you're curious about math.

ActionableLessons

Actionable

Try explaining Hilbert's hotel to a friend.
Read Leonard Wapner's The Pea and the Sun, which Vsauce recommends for the full proof.
Rewatch the sphere section slowly, pausing after each rotation step.

Lessons

Intuition built on finite things breaks down with infinity.
Math that seems purely abstract has often turned out to describe the real world later.
Common sense is a tool with limits.

Quick answer

Vsauce posted this video on Aug 1, 2015. Here's our summary in our own words: the answer to the title, a verdict, 5 takeaways, and Actionable and Lessons tabs.

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Watch on YouTube: The Banach-Tarski Paradox

Channel: Vsauce. Published Aug 1, 2015. About 24 minutes.

No advice here. The suggestion is to be open to results that break common sense, since common sense only covers the small part of reality we can see.

Quotes from the video

“Infinity is not the biggest number, instead, it is how many numbers there are.”
“One plus one equals 1.”
“history continues to show us that the universe isn't strange. We are.”

A note on this summary

This page reports what the video says, in our own words. Quotes are copied word for word from the video's transcript. Views are as of Oct 30, 2026.

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FAQ: questions people ask

What is the Banach-Tarski paradox?

A proof that a solid ball can be split into five pieces and reassembled into two balls identical to the first.

Can it happen in real life?

Not with physical objects, because the pieces would need to be infinitely detailed. It's valid within math.

Is the video worth watching?

Good value given. It's long, but the explanation is excellent.