drawing a future-pointing arrow onto the edge. Conversely, these arrow assignments fix the orientation of the triangle uniquely. To follow the “flow of time" in a triangulation it is convenient to also associate a future-pointing arrow with each spacelike edge, which is drawn perpendicular to the edge, see Fig. \[fig:flowoftime2d\].

Choosing a consistent orientation for all building blocks in standard CDT in this way is completely straightforward: each triangle sits in a strip between discrete times $t$ and $t+1$, which fixes its orientation uniquely. Independent of the spatial boundary conditions, there are no closed timelike curves (unless we impose periodic boundary conditions [*in time*]{}, which trivially makes any timelike curve closed, a situation we are not considering here).

By contrast, the situation in nonfoliated CDT is slightly more involved. Given a time-oriented triangle, the orientation of a neighbouring triangle that shares an edge with the first one is uniquely determined by consistency. It is easy to see that when vertex causality is violated (like in the example of Fig. \[fig:causality2d\], left), inductively assigning orientations in this way will fail – i.e. lead to contradictions – even for a local vertex neighbourhood. If vertex causality is satisfied, one can show that 