A good way to conceptualize what’s going on is not the idea that balls become "spiky" in high dimensions – like the article says, balls are always perfectly symmetrical by definition. But it’s the box becoming spiky, "caltrop-shaped", its vertices reaching farther and farther out from the origin as the square root of dimension, while the centers of its sides remain at exactly +-1. And the 2^N surrounding balls are also getting farther from the origin, while their radius remains 1/2. Now it should be quite easy to imagine how the center ball gets more and more room until it grows out of the spiky box.
Exactly: a corner of a square covers 1/4 of that part of the plane. A corner of a cube covers 1/8, a corner of a hypercube in dimension n covers just 1/(2^n) of the space. But each side/face/hyperface divides the plane/space/n-dim space just in half.
Why did I imagine that this would be about two shapes that are merely topologically n-balls, each having part of their boundary be incident with one of the two hemi(n-1)-spheres of the boundary of an n-ball (and otherwise not intersecting it)? (So like, in 3D, if you took some ball and two lumps of clay of different colors, and smooshed each piece of clay over half of the surface of the ball, with each of the two lumps of clay remaining topologically a 3-ball.)
I don’t know that there would even be anything interesting to say about that.
I am struggling to juggle the balls in my mind. Are there any stepping-stone visual pieces like this to hopefully get me there? Very neat write-up, but I can't wait to share the realized absurdity of the red ball's green box eclipsing in our 3D intersection of the fully diagonalized 10D construct
The hypercube is the strange thing, not the red sphere. Placing the blue spheres tangent to the hypercube is an artificial construct which only “bounds” the red sphere in small dimensions. Our intuition is wrong because we think of the problem the wrong way (“the red sphere must be bounded by the box”, but there is no geometrical argument for that in n dimensions).
> 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.
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