Suppose you have four equal circles in a box that they just fit into, and you're interested in the largest circle you can fit in the middle.....
That means that in 4 dimensions the sphere in the middle will be of radius (square root of 4) - 1 , which is 1. The central sphere is the same size as the spheres around it.
That's odd, but it gets even more interesting.
In 9 dimensions the central sphere is of size (square root of 9) - 1 which is 2. Remember, that's the radius of the central sphere, so the diameter is 4. That's the size of the containing box. The central sphere actually touches the sides of the containing box.
But wait - it gets better.
In 10 dimensions the central sphere is of size (square root of 10) - 1 which is about 2.162. The diameter is about 4.325. It pokes out the sides (and top and bottom, etc) of the "containing" box.
In fact it's not just the central sphere that gets more spikey, the surrounding spheres are also getting spikey. Each corner sphere's volume is getting smaller (as a proportion of the enclosing cube) as the dimensions go up. So it's not just just the sphere pokes through, it's also that there's more space for it in the first place.
You can also think of the corners of the cube being spikey, and the spheres are therefore packed away into the corners, leaving loads of space.
Somehow we have to see the central sphere as "poking out between" the surrounding spheres. It's almost as if a sphere in high dimensions isn't smooth, and round. It's almost as if it's somehow "spikey."
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