Consider that the distance between opposite corners of a square is √2 meters, but for a cube it's √3 meters. In general, for an n-dimensional hypercube of diame

Consider that the distance between opposite corners of a square is √2 meters, but for a cube it's √3 meters. In general, for an n-dimensional hypercube of diameter 1m, the exactly-opposite corners are √n meters apart. Meaning, in a million-dimensional meter-diameter hypercube, exactly-opposite corners will be a kilometer apart. There's a lot more diagonal wiggle room than you think.