Thinking outside the 10-dimensional box
Why Hacker News recommends it
HN recommends it as the go-to intuitive visualization of high-dimensional geometry, relevant to machine learning, optimization and sphere packing.
What it is
3Blue1Brown visualizes high-dimensional spheres to explain a surprising puzzle about how a sphere fits in a box as dimensions grow. It builds geometric intuition for counterintuitive properties of high-dimensional space.
Recommended by 6 HN commenters across 6 different months, 2017 to 2025.
What Hacker News said
Strong rec sillysaurus3 · 2017-10 · on “What Is a Neural Network?”
A trick for visualizing higher dimensions: [this video] Only 383k subscribers. Hmm. Remember to subscribe, comment, and smash that like button. Real Engineering is another great channel. And of course Veritasium and Numberphile.
Strong rec jedimastert · 2018-07 · on “Random Points on a Sphere”
Don't forget 3Blue1Brown's incredible video about visualizing higher dimensions[1]. He's getting to xkcd levels of "Of course there's a relevant video" [1]: [this video]
Strong rec kibwen · 2025-07 · on “New sphere-packing record stems from an unexpected source”
There's a great 3blue1brown video that demonstrates this unintuitive result in a relatively intuitive way: [this video]
Recommends gilbetron · 2018-03 · on “Counterintuitive Properties of High Dimensional Space”
Great article! I've been thinking about this on and off recently, as I wonder if we might be having intuitive issues when it comes to gradient descent optimization. 3 Blue,1 Brown had a video recently that kicked off my head scratching and is a great complement to your article: [this video]
Recommends jerf · 2022-01 · on “Geometry from Another Universe”
As you increase the dimensionality of a sphere, an ever-increasing proportion of the sphere is within epsilon of the surface. This 3Blue1Brown video addresses enough related stuff that the rest should become reasonably comprehensible: [this video] This particular result is tossed in as a side note around 23:25, so it's not addressed directly, but it'll help.
Recommends NeoTar · 2024-05 · on “Strangely Curved Shapes Break 50-Year-Old Geometry Conjecture”
Three Blue, One Brown has a decent video which perhaps helps get a handle on how one can use higher-dimensional spaces, without needing to wrap you head around trying to envision a four-dimensional cube, or similar. [this video]
Cites darkmighty · 2017-11 · on “What is backpropagation and what is it doing? [video]”
I would reiterate that simply because there are problems with naive visualization, we shouldn't discredit visual thinking. There are several key elements to effective visual thinking. The primary importance is to keep it grounded in proofs and theorems, so you know exactly what are your limitations. Often you can use a geometric argument on top of a few theorems and you get a very strong result intuitively, and then use this intuition with a tiny amount of algebra to prove it (which might take you forever to arrive from a purely algebraic perspective). Another key is that there are several ways of visualizing things. You can almost always transform a problem into an equivalent one that is ea
Cites wruza · 2018-08 · on “A Visual, Intuitive Guide to Imaginary Numbers (2007)”
Yeah, I see now. “times i” is discrete 90 degree rotation itself, not just ‘i’, nor ‘b’. Thanks everyone for making that clear. This though shows that explanations via analogies or non-strict wording may confuse one rather than enlighten. I’m not good at math, but once understood to not search analogies or geometry in things. Instead it is better to “shut up and calculate”. Not sure if imagining something is required to manage it. It’s only our brain’s faulty quirk. [this video]