into a sum over triangulations, typically subject to some manifold conditions, which ensure that the triangulation looks like a two-dimensional space everywhere.

By contrast, the flat triangles used in CDT quantum gravity have Lorentzian signature, something that cannot be achieved for equilateral triangles. The standard choice of an elementary CDT building block in 1+1 dimensions is one whose base has squared length $\ell_s^2>0$ (and therefore is spacelike) and whose remaining two edges both have squared length $\ell_t^2<0$ (and therefore are timelike). From the point of view of triangulating Lorentzian spacetimes, one could in principle have chosen $\ell_t^2$ to be space- or lightlike, but then CDT’s prescription of a Wick rotation – to be described below – would no longer be applicable.

Fig. \[fig:dt2d\] illustrates the fundamental building blocks of two-dimensional DT (left) and CDT (right), and how they are put together to obtain piecewise flat manifolds of Euclidean and Lorentzian signature. The two-dimensional graphs correctly represent the neighbourhood relations of adjacent triangles, but not the actual length assignments, since it is impossible to flatten out a curved surface and preserve the edge lengths at the same time. Note that in the Lorentzian case the spacelike edges form a sequence of 