Now let me show the relation between the decomposition and trajectories of triangles.
Given subset A of standard lattice L3. If Bd(Cone A, Roof* A) is empty, then …
In particular, In(Cone A, Roof* A) consists of all the closed trajectories, and In(Cone A, Roof* A) consists of all the open trajectories.
In the left example, there is no “Boundary” triangle and “In” triangles form the only closed trajectory of length six.
In the middle example, there are two “Boundary” triangles (colored yellow) and “In” triangles are part of a open trajectory.
In the right example, there is no “Boundary” triangle and “In” triangles form the only closed trajectory of length ten.