Vsauce
6 HN recommenders · recommended in 10 months, 2015–2024
24:14 · 49M views · Aug 1, 2015
Vsauce's roughly 24 minute explainer on the Banach-Tarski paradox, building from infinity and set theory to the decomposition of a sphere into pieces that form two spheres.
Recommended as an accurate, entertaining on-ramp to set theory and infinity, worth pausing and rewatching.
12 comments from Hacker News
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 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]
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.