assembled from two different tetrahedron types, which can be distinguished by their orientation with respect to the preferred foliation. As can be seen from Fig. \[fig:foliation\], spacelike edges of a tetrahedron are always contained in a spatial submanifold of integer $t$, whereas timelike edges always connect different spatial slices. The (3,1)-tetrahedron has three spacelike links forming a spacelike triangle, while the (2,2)-tetrahedron contains only two spacelike links. The notation $(i,j)$ indicates that $i$ vertices of the tetrahedron are located on one spatial slice and the remaining $j$ vertices on an adjacent one.

The kinematical set-up of CDT quantum gravity in 3+1 dimensions can be defined in a similar way. The leaves of the preferred foliation are three-dimensional Euclidean triangulations of fixed topology, and neighbouring slices are connected using Minkowskian four-simplices. Path integral configurations are simplicial manifolds assembled from two types of these building blocks, denoted by (4,1) and (3,2), depending on how their vertices are distributed among adjacent spatial slices. By labeling the foliation with increasing integers we again get a time variable for free, with every vertex being assigned a definite discrete time label.

Another ingredient that needs to be specified to make the model complete is an implementation 