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This book provides a number of combinatorial tools that allow a systematic study of richer building blocks and of the spaces they generate in the continuum. In any dimension D, we can discretize Euclidean gravity in the absence of matter over random discrete spaces obtained by gluing families of polytopes together in all possible ways. These spaces are then classified according to their curvature. In D=2, it results in a theory of random discrete spheres, which converge in the continuum limit towards the Brownian sphere, a random fractal space interpreted as a quantum random space-time. In this limit, the continuous Liouville theory of D=2 quantum gravity is recovered. Previous results in higher dimension regarded triangulations, converging towards Aldou's continuum random tree, or gluings of simple building blocks of small sizes, for which multi-trace matrix model results are recovered. In this book, the author develops a bijection with stacked two-dimensional discrete surfaces, and details how it can be used to classify discrete spaces according to their curvature and topology.
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