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.