Infinity shapeshifter vs. Banach-Tarski paradox

Take on solid ball, cut it into a couple of pieces and rearrange those pieces back together into two solid balls of exactly the same size as the original ball. Impossible? Not in mathematics!
In this prequel to the Vsauce video the Mathologer takes you on a whirlwind tour of mathematical infinities off the beaten track. At the end of it you'll be able to shapeshift any solid into any other solid. At the same time you'll be able to appreciate like a mathematician what's really amazing about the Banach-Tarski paradox.

Enjoy :)

[Pokemon]

You guys know what an anagram of Banach-Tarski is? Banach-Tarski Banach-Tarski

[salvatorederosa]

That's why when you dismantle a computer and re build it you always have extra parts. Murphy's law corollary states that if you iterate it to infiniity you get enough parts for two computers. Pure genius

[Mathologer]

Vsauce's video on the Banach-Tarski paradox is one of the best things I saw on YouTube last year. Having said that I am pretty sure that among the 5M+ people who sat through the ingeniously complicated construction about 5M came away from it being impressed by the Banach Tarski paradox for the wrong reason, namely that "you can double volume using no extra material". In reality this part is pretty easy if you know a little bit about infinity.
To put things in perspective, I thought it would be a good idea to make this video in which I give an off the beaten track introduction to mathematical infinities with Banach Tarski as the guiding principle :)

[prateekgurjar]

i love his innocent nerdy kid laugh...I want his collab with numberphile

[lilayhagos]

Please, please never give this up. The majority of people don't understand math's capacity to explain everything in our universe so beautifully, and are made to hate it from the beginning. I've always been good at math, but never enjoyed it and have had teachers tell me that I'll never use this but I have to learn it for the test - such a cynical way of looking at it.

You make it interesting, explain the narrative behind the formula and work so hard on all your animations. The short term result will be more subscribers but the long term result could help change the world's view in mathematical education which has been needed for centuries now.

[constantly-confused]

Since you mentioned it in the video: How about a video about the axiom of choice (and its equivalents)?

[tobuslieven]

Take a finite thingy and divide it into an infinite number of pieces (hard work is good for you). Now divide your infinity pieces into 2 groups (this may take some time). Both groups are still considered infinite (no problem). Infinity is enough to make whatever (obviously). It is certainly enough to create a thingy Create 1 thingy. You now have 1 thingy and one spare infinity. Now use your remaining infinity to create another thingy. You have 2 thingies. Or my arithmetic isn't good. What would a good arithmetic look like in this context? Maybe it would have to disallow infinity? What kind of arithmetic would exclude banach-tarsky silliness?

[simongreve]

The V Sauce video from "just a few weeks ago" was uploaded in July :P

[MichaelClark-uwex]

The old dividing infinity trick.

No matter how you divide it, you always get infinity.

[timothemalahieude]

There is a little mistake in the description... It says "Take 2 solid balls, cut them [...] into 2 solid balls of exactly the same size". Well, I think it won't be difficult ^^

[cathodius]

This paradox has two really seriously solid balls

[richardfarrer]

My favourite analogy. Consider W, the set of all possible 'words' made up of any finite, non-zero combination of letters, such as "A", "ABCD", "XYZ" etc. Now split these up into one set containing all 26 single letter words, and another 26 sets containing each word of two or more letters beginning with a particular letter. For example, the 'A' set contains "ABCD" and all other words beginning with A. But the 'A' set is just the original set, W, with the letter A put in front of each string. Thus the set W can be split into 26 copies of itself, together with 26 single-letteer strings. Banach-Tarski does something similar, except it folds the extra set containing the individual letters into the copies. The advantages of three dimensions instead of one.

[rudolphjansen]

get some better audio for the guys asking the questions...that would be great

[chasemarangu]

At 2:30 well, the reason this scaling thing works is like "the dart board paradox" which is where you try to figure out: on a dart board, what is the probability that you will hit a specific point? You can't answer it because there are infinitely many points, each infinitesimally small! The infinitesimal is 1/∞, so you know. Scaling something up like this is not increasing its measure, rather the scale of how you look at it. You are increasing the size of the unit measurement too, because the infinitesimal is the only thing that stays the same size.
*Do not use the infinitesimal as a unit of measurement!* unless you want to encounter paradoxes.

[TheAbbeyClinic]

Love this guy. Thanks for so many interesting vids, and your jolly and thoughtful tutorials.

[Achrononmaster]

@7:43 wait a minute! No one knows the answer to the Continuum Hypothesis, so if it turns out quantum mechanics is based on discrete spacetime _topology, _ not discretized spacetime itself, then the smooth continuum spacetime of general relativity has some plausibility. Then in effect with gravity we could be "seeing" the effects of higher infinities above ℵ₁ if the CH is false (in some platonic realist sense) in fact, there is no reason to believe a physical continua, like smooth space, has much at all to do with the reals _c_ = 2^ℵ₀ since we are free to regard the physical continuum as containing many more sets of points than the reals.

[PaulOMalleyDublin]

As always enjoyed your video.
There might be a rumour that David Hilbert went to a hotel, it was full, but that was okay because everyone moved to room n plus one to make room for him.

[pyjamadeus]

Nice vid! One thing that makes Banach-Tarski so counter intuitive is that the transformations you perform on the pieces are all isometries of 3d space, simple rotations and translations, whereas mapping squares to lines can be bijective but not distance preserving.

The real assault to our intuition thanks to Banach-Tarksi, I think, is the idea that we can't define a concept of volume that simultaneously applies to all subsets of R^3 while also matching our intuitive notions of what volume should be, namely that it should be preserved by isometries.

[GhostlyGorgon]

Great video! I only have one observation: I don't think those maps where you interleave the decimal expansions are bijective as many rational numbers have more than one decimal expansion, i.e. 1/2 can be written as 0.500... or .49999... so you either choose one to go with, in which case your function isn't surjective or you map the point (1/2, x) to both points on the line in which case your map is not a function. I may be wrong here; if I am I would love an explanation.

[greg]

Please do a video on the axiom of choice.