These are the basic ideas of the 2-dim'l case.
First, divide ...
Shown in the left figure is the division on 3-cube. Since 3-cubes are to be projected onto a hyperplane, it is enough to consider the "upper" facets. And these three upper facets are divided into 6 triangles a[x1x2], ....
Next, pile ...
Then, as shown in the middle figure, we obtain "peaks and valleys" of 3-cubes.
Note that the division of facets makes up a division of the surface of the Ps&Vs.
Finally, project ...
Then, we obtain a "flow" of ... as shown in the middle figure.
As you see, the division of the surface of the Ps&Vs specifies trajectories of triangles on the hyperplane. For example, the blue triangles form a closed trajectory of length 10.
Shown in the right figure is the projection of triangles. "Slant tiles" are triangles in R3 and "flat tiles" are triangles in R2. And it is a projection from R3 to R2.
In the next slide, we use the projection pi to contsruct DDG of triangle tiles.