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Comment by spenczar5 | Hacker News Reader

Substep 6 is not obvious to me. It is not clear that the point where the inner sphere and the space-filling spheres intersect must be along the line from the center of the cube to the center of each sub-cube.

To put it another way, it's not obvious to me that the point of contact between the inner pink circle and the outer black circle is along the green line.

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Both spheres are symmetric around this line, so a point on intersection anywhere but on the line would give you a circle of intersection (and so a disk, by convexity) ...

How do you know they are both symmetric around this line? The rest of your argument makes sense to me, yeah.

The spheres are symmetric around the line because the line goes through the center points of the spheres.

I think this is a property spheres. It seems to me that any two spheres that are touching have a straight line from one center to the other center exactly through the point of contact. Try thinking of just two spheres and adding more in step-by-step.

Then the result follows because all the spheres are defined as centered on the cube/sub-cubes respectively.

The inner sphere is not defined as centered on the cube; it is defined as touching all the other spheres.

That said, there is a symmetry argument that if it were centered anywhere else, something is wrong. But that only works if there is only one unique sphere that touches all the other spheres, which is also not obvious to me in higher dimensions.

You can go ahead and define it as centered on the cube. That still demonstrates the strange nature of high-dimensional spheres even if there wasn't a unique solution for touching all the other spheres.

Aha! Right, this is pretty convincing to me. Thanks!

Not obvious for higher dimensions or even D=2? Surely you agree the center of both circles (regardless of dimension) occur along the line from the center of the cube to one of the corners. Therefore, if you just radially grow these spheres they must touch for the first time along this line. To be clear, this is nothing special about the cube, if you draw a line between the center of any two circles, the first time they touch will be somewhere along this line. In this case, the inner circle is obviously along the diagonal and it doesn't take much to see that the outer circles are as well by their construction. Therefore, the diagonal is the line that connects the centers.

I don't agree that the center of both circles needs to lie along that line. The space-filling ones do by definition, but I don't see why the center one has to be on that line. It seems like it must in D=2, of course, but I couldn't prove that it must for D=9, or even that it is unique.

The line goes from the center of the cube to the corner. The central sphere is concentric with the cube. Therefore the center of the central sphere is on the line from the center to the corner. In fact, it the center of that sphere is on any line from the center of the cube to anywhere.

Ah, ok, I will try to appeal to symmetry while giving geometric intuition. I think the best way to visualize this (and explains a little what exactly is going on) consider cutting a plane through the hypercube through diametric corners. For instance, (2,2,2,...,2) and (-2,-2,-2,...,-2). This will reduce down to a 2-dimensional problem but things are no longer square. The vertical edges of the quadrants are 2 but the horizontal lengths are 2sqrt(D-1). To see this, calculate it for 3-dimensions as you can still draw it out. By construction, the circle inside the quadrant kisses the top and bottom and is centered horizontally. That is, there is a gap between the edges of the circle horizontally and the quadrant. Again, to visualize, picture how the plane is cutting in 3-dimensions. Now, since you seem to accept that the circle is on the origin in two dimensions, the circle is on the origin in this plane. Repeat this procedure for all diametric corners and, by symmetry, the inner sphere is always centered at the origin. For completeness, because the inner sphere is centered at the origin, the radius must grow along the diagonal. So, we can use our squished quadrant again to calculate the radius, r, of the inner sphere. It is going to be sqrt(2^2 + (2sqrt(D-1)^2)-2 = 2r, where the subtraction is the outer sphere. Some arithmetic gets you to r = sqrt(D)-1 which is of course what they got in the post.

Hopefully this was the correct amount of words and equations that you can reconstruct this on paper. I think doing it this way shows you that the outer spheres get increasingly smaller relative to their containing "quadrant". For me, it also elucidates that the final drawing in the post is very misleading because the hyperspheres don't spike like that. Rather, they look more like 4-leaf clovers and that is what allows it to escape the cube in the additional "horizontal space" of the aforementioned quadrant.

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