The Banach–Tarski Paradox

Learn set theory! Specifically ordinals (just one formal defintion of a number) This vsauce video is actually a really accurate (and entertaining!) introduction to set theory: ▶ this video

I don't know if this will explain your questions, but years ago I watched this video from Vsauce about this very topic, and IIRC it was explained quite nicely: ▶ this video

▶ this video nicely visualizes it. It's an infinite number of infinitely thin filaments from the center of the sphere going outward in every direction.

Note that the OP is about cuts that are nice, cutting polygons into smaller polygons, that could be done using a scissor. The article you linked is about using magical cuts that be non measurable, that look like magical interlocking 2D sponges. They are somewhat like the figures in the gif, but they actually more weird. It's somewhat similar to the Banach–Tarski Paradox. If you like that, I recommend to watch the video by Vsause ▶ this video that explains the Banach–Tarski Paradox with some technical details, but not too many.

I agree that most videos in YouTube simplify the problem too much or are just wrong. My recommendations anyway: * Mathologer: " Ramanujan: Making sense of 1+2+3+... = -1/12 and Co. " link * Vsauce: " The Banach–Tarski Paradox " ▶ this video Each one is like half an hour long, but you may need to pause them and rewatch them a few times to understand the details.

I have done differential topology and geometry for a time in University. There is a lot of visualisation going on in every lecture every day, even if it is not computer visualization. Everything in difftop is drawn at some point, just so you can get an understanding of what's happening. In fact, I would say that topology and subfields are formalisations of these beautiful and silly visual ideas. One problem, however is that visualizations are by necessity simplifications, even in pure 2d cases like this one. That's not a sphere, that's a tessellation of one. Everything on a screen is differential and "smooth" (or actually, everything is discrete and thus not differentiable), but some things are symbols for "not differentiable here". Even these break down in pathological cases, singularities and such. This doesn't make the visualisation any less useful, but IMHO it can in most cases not be "the" reasoning, only a guide to reasoning. Besides,as the article points out, there are many non-constructive proofs that don't have any visuals. Have a look at how vsauce explains Banach-Tarsky [1] or any of 3blue1brown's excellent videos to see the edges of what's possible with visualisations. [1] ▶ this video [2] link

Other good STEM video channels/series: Dave Ackley's series "Hyperspace Academy", "Robust First Computing", and "Artificial Life" are well animated: link Vsause's video on "The Banach–Tarski Paradox": ▶ this video (mindblown) XylyXylyX channel's series on "What is a Tensor?" and "What is a Manifold?": link Socratica's series on "Abstract Algebra": link Mathologer's channel on all-things math: link Ben Garside's series "Vector Spaces": link PatrickJMT's channel on calculus, game theory...(too many to list): link Numenta's ( link ) series "HTM School" on the Hierarchical temporal memory ( link ) model for ANNs: link Dan Shiffman's series "The Nature of Code: Simulating Natural Systems with Processing": link

Vsauce: The Banach–Tarski Paradox (24m13s) [2015-08-01]: ▶ this video (explainer video) Matematikrevyen: Matematikrevyen 2011: Banach-Tarski (3m42s) [2011-12-30]: link (music video)

I'm glad to have found this post. I discovered the Banach-Tarski theorem via Vsauce[0]. It was interesting but I couldn't get the significance of it. It either didn't seem like an unexpected result or too esoteric to appreciate. There's phrasing in the post that could be misunderstood (later clarified) but can leave unclarity from assumed understanding of the earlier description. > In R3, given a solid ball B of radius R, it is possible to partition B into finitely many pieces such that those pieces can be reassembled to form two solid balls B1 and B2 each of radius R. "finitely many pieces" could be confusing because though we may be talking about 6 'pieces' those pieces have an uncountable infinity of radial line slices. It's also missing the rigid motion part which is key. The part that didn't seem surprising (and is considered trivial) was in handling uncountable infinities of things. The 'number' of points on a line [0, 1) is the same as the number on a line [0, 2) so I wouldn't be surprised to map points from [0, 1) to [0, 2) filling the latter without 'gaps'. Similarly for areas. But what the theorem is saying is that this kind of mapping doesn't work in R1 nor R2 but does work in R3 (with rigid motions). The part that makes B-T surprising is that the extra volume/ball can be constructed with rigid motions of those uncountably infinite sets. This is where it seems beyond me to appreciate: that one or two balls have the same uncountably number of radial line slices is c

> The medium of YouTube is not conducive to lengthy expositions I've actually found a couple channels on YT specialising in lengthy reviews/essays, which I find very good. Off the top of my head: 1. Whitelight[1] - game reviews/critiques/analyses. Particularly worthy of note are the Assassin's Creed Unity[2] (1.5h), Batman Arkham City[3] (3h 10min) and Watch Dogs[4] (1h 15min) reviews. 2. MauLer[5] - critiques of mostly Star Wars, but sometimes also other mainstream films. The ongoing series of TFA critiques has 4 parts so far (there is going to be at least 6 total), each part taking anywhere between 2 and 4 hours. I also feel that I'm forgetting some, which I don't watch regularly, but periodically am reminded about their existence and then after a long break spend several days watching, to then forget about them again. But the channels I listed aren't fringe. Quite the contrary, they're quite popular. I think it's also interesting that one of, if not the most popular Vsauce video is the one about the Banach-Tarski paradox[6], which is almost 30 minutes long. His other videos also show this trend, where the long-form ones seem to get more views in general. And those are just essays/reviews/etc. There is a whole genre of podcasts on youtube dedicated to 3-4h in-depth interviews. Everybody knows a whole bunch of them, therefore I don't even need to list any examples here to support that. So it seems that YouTube is a pretty good place for long-form in-depth exploration of what

The logic embedded in Supertasks and the Banach-Tarski Paradox has always seemed strange. It's not flawed, but it's so counterintuitive that it can seem like evidence that something in the logic of infinities must be wrong. link ▶ this video

I actually think it's very possible to have an "intuitive understanding" of Banach Tarski (I would say I have one, but perhaps we disagree on what is mean by such an understanding). My "intuitive understanding" of this comes via an intuitive understanding of a paradoxical decomposition of the free group and its Cayley graph, which is flashed briefly in the video here[0] but sadly not discussed at length. [0]: ▶ this video