$\pi/3$ at the vertex (Fig. \[fig:reggecurvature\], right). This also coincides with the rotation angle undergone by a two-dimensional vector parallel-transported around the vertex, and is therefore an [*intrinsic*]{} property of the two-dimensional disc. The principle of encoding curvature through deficit angles (located at subsimplices of dimension $d-2$ in a $d$-dimensional triangulation) works in any dimension and for any metric signature.

Let us review the difference between the (Euclidean) DT and the (Lorentzian) CDT path integral, again for simplicity in dimension two. Recall that the geometric properties of a flat triangle (or, in higher dimensions, a flat simplex) are completely determined by the lengths of its edges.[^5], including the [*signature*]{} of the flat metric in the building block’s interior. In the Euclidean DT approach in two dimensions one works with a single type of building block, an equilateral triangle of Euclidean signature, all of whose edges have some fixed spacelike length, the same length for all triangles in the triangulation. As we have seen above, the number of such triangles around each interior vertex of the triangulation characterizes the local curvature at that vertex. In this way the formal integral over curved geometries in the continuum path integral (\[eq:pigrav\]) is turned 