Consider the volume of a cap. Take a plane that is 90% of the distance from the centre to the edge, and look at what percentage of the volume is "outside" that plane. When the dimension is high, that volume is negligible.
And when the dimension is really high you can get quite close to the centre, and still the volume you cut off is very small. In our 3D world the closest thing that has this property is a spike. You can cut off quite close to the centre, and the volume excised is small.
The sense in which a high-dimensional ball is not a spikey thing is in the symmetry, and the smoothness.
So when you want to develop an intuition for a high dimensional ball you need to think of it as simultaneously symmetrical, smooth, and spikey.
Then think of another five impossible things, and you can have breakfast.
However, if I have the n-sphere x_1^2 + x_2^2 + ... + x_n^2 = 1, knowing the value of x_1 gives me very little information about all the other coordinates. And humans' interaction with the geometry is reality is somewhat limited to manipulating one coordinate at a time, i.e., our intuition is for built on things like moving our body linearly through space, not dilating the volume or surface area of our bodies.
This to me feels similar in many ways to how a corner in a high dimensional n-cube, although 90 degrees, no matter how you measure it, seems extremely spiky. As the shape does not increase in width, but the corners extend arbitrarily far away from the center. A property reserved for spiky things in 3D.
"So instead of considering n-balls to be spiky, it’s the space around them that outgrows them."
You should have seen how few replies read the last article I posted. https://news.ycombinator.com/item?id=40525629 The majority of the comments, including all the top ones, expressed insights as original, that were pretty thoroughly analyzed in the article. Just read my mildly frustrated replies.
I have thoughts about this, but I can find no way to contact you to open a conversation. You might want to think about adding something to your HN profile.
Meanwhile ... from this and other sources, I feel your pain.
[0]: https://en.m.wikipedia.org/wiki/Curse_of_dimensionality
I don’t know that there would even be anything interesting to say about that.
That article doesn't have the nice animations, but it is from 14 years ago ...
https://news.ycombinator.com/item?id=12998899
https://news.ycombinator.com/item?id=3995615
And from October 29, 2010:
It great to see these interesting math facts continue to be discussed and presented in new ways.
2D: 3 mutually touching 2-spheres (circles)
3D: 4 mutually touching 3-spheres (or spheres)
...
This variation of the problem doesn't rely on an artificial construct of a hypercube, I wonder if this yields a similarly unintuitive result.
The course was also the basis for his book _The Art of Doing Science and Engineering_ (1997). At first it takes some getting used to as you have the feeling it may be outdated, but it's about teaching a style of thinking. It's great.
Two of the lectures were spent on building intuition for very high dimensionality (this one), and another on neural networks, because he thought there was a big chance they were going to be important. In the early 90s, not bad.
We don't have special words for the voluminous versions of other 3D shapes, so why do spheres need one?
I completely agree. N-ball is maybe more mathematically precise. It might rule out some strange edge case I couldn't think of. I chose it mostly for stylistic reasons. n-ball is shorter than n-sphere, and ball is a more playful term.
I do however understand the need to define the two object classes, and see little issue with giving them distinct names.
Topologists don't need them because they already have ball and sphere.
In analysis, I can imagine them calling full hyperrectangles "brick", and empty hyperrectangles "box", but both words start with "b", so there is no shorthand for them on paper. I^n and ∂I^n are just fine.
And for 3D shapes have Torus, and, well, ‘solid-torus’.
“…At what dimension would the red ball extend outside the box?”
If anyone has o1-preview it’d be interesting to hear how well it does on this.
> There is a geometric thought experiment that is often used to demonstrate the counterintuitive shape of high-dimensional phenomena. We start with a 4×4 square. There are four blue circles, with a radius of one, packed into the box. One in each corner. At the center of the box is a red circle. The red circle is as large as it can be, without overlapping the blue circles. When extending the construct to 3D, many things happen. All the circles are now spheres, the red sphere is larger while the blue spheres aren’t, and there are eight spheres while there were only four circles.
> There are more than one way to extend the construct into higher dimensions, so to make it more rigorous, we will define it like so: An n-dimensional version of the construct consists of an n-cube with a side length of 4. On the midpoint between each vertex and the center of the n-cube, there is an n-ball with a radius of one. In the center of the n-cube there is the largest n-ball that does not intersect any other n-ball.
> At what dimension would the red ball extend outside the box?
Response: "[...] Conclusion: The red ball extends outside the cube when n≥10n≥10."
It calculated it with a step-by-step explanation. This is the first time I'm actually pretty stunned. It analysed the problem, created an outline. Pretty crazy.
Reboot on TV this year.