Correct triple periodicity of volumes of parallelohedrons

Abstract

If we divide a cube into eight cubes by three mutually perpendicular central planes, we get a periodic operation that we can continue endlessly in both directions. In this case, the volume of the cube decreases (or increases) eight times, that is, two in the cube. But this period is very simply divided into three smaller periods with a decrease (or increase) in volume by half. In this periodicity, a special role is played by A) the centers of the faces of the original cube and B) the centers of the cubes of the lower period.