Correct ternary periodicity of parallelohedra volumes

Abstract

If we divide a cube by three mutually perpendicular central planes into eight small cubes, we obtain a periodic operation that we can continue infinitely in both directions. In this process, the volume of the cube decreases (or increases) by a factor of eight, that is, two cubed. But this period is very simply divided into three smaller periods with a decrease (or increase) in volume by a factor of two. 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.

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