Before moving to the 3-dim'l case, let me show another view point of the "division of facets."
That is, "division along diagonal" ...
Recall that facets of n-cube ...
Shown on the left is the standard lattce generated by x1, x2, and x3. So far, we have used the standard lattice only. And "peaks and valleys" are a cone of the standard lattice generated by its peaks.
On the other hand, the conjugate lattice shown on the right is the lattice geneated by y1, y2, and y3, which are defined as shown below. Here we used the monomial notation and y1, y2, and y3 correspond to vector (0,1,1), (1,0,1), and (1,1,0) respectively.
Then, as shown in the middle figure, we could use a cone of the conjugate lattice to specify an ("affine") trajectory. In this case, the closed trajectory of blue tiles is specified by the gray cone.
Moreover, one could consider "algebra of closed affine trajectories." But it is not today's subject.