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: [VIDEO]

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": [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]

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] [VIDEO] [2] [LINK]

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 " [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.

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 [VIDEO] that explains the Banach–Tarski Paradox with some technical details, but not too many.

[VIDEO] nicely visualizes it. It's an infinite number of infinitely thin filaments from the center of the sphere going outward in every direction.

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: [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]: [VIDEO]