Exactly: a corner of a square covers 1/4 of that part of the plane. A corner of a cube covers 1/8, a corner of a hypercube in dimension n covers just 1/(2^n) of

Exactly: a corner of a square covers 1/4 of that part of the plane. A corner of a cube covers 1/8, a corner of a hypercube in dimension n covers just 1/(2^n) of the space. But each side/face/hyperface divides the plane/space/n-dim space just in half.