An interesting article. And a little frustrating. The upper bounds are hard. How do we know that you cannot fit 25 balls around a ball in 4 dimensional space? That is the tricky side of the proof, best left to professional mathematicians. How do we know that you can fit 24 balls around a ball in 4 dimensional space? Well, the article could give locations for the centers of the balls and readers could use Pythagoras' theorem to check the distances.

In two dimensions, the center ball is at the origin, (0,0) and the six surrounding balls are at (1,0), (1/2, √3/2), (-1/2, √3/2), (1/2, -√3/2), (-1/2, √3/2). Since it is past my bed time, I'm not going to check that I've got that right.

The article says how to do it in 3D dimensions: use an icosahedron. That doesn't sound right, an icosahedron has twenty faces. Perhaps they mean dodecahedron, the one with twelve faces? No, they were right with icosahedron, because it has twelve corners, top, bottom and two rings of five. I could come up with the coordinates in three or four hours :-)

But four dimensions? They give a drawing of a four cell in the article. But it is a two dimensional drawing of a four dimensional shape, so just the four dimensional shape projected to create an incomprehensible mess of lines.

Wait, I think https://en.wikipedia.org/wiki/24-cell gives the game away

8 vertices obtained by permuting the integer coordinates:

(±1, 0, 0, 0)

and 16 vertices with half-integer coordinates of the form:

(±1/2, ±1/2, ±1/2, ±1/2)

I'll have to think about that tomorrow.