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In this PhD thesis we develop the frame work of triple crossing diagram maps(TCD maps), which describes constrained configurations of points in projectivespaces and discrete dynamics on these configurations. We are able to capturethe constraints and dynamics of a large list of examples that occur in discretedifferential geometry (DDG), discrete integrable systems and exactly solvablemodels. We explain how to apply various geometric operations to TCD maps,including projections, intersections with hyperplanes and projectivedualization. In fact, we show how many examples in the literature are relatedby the aforementioned operations. Moreover, we introduce a hierarchy of clusterstructures on TCD maps, thus answering the open question how objects of DDGrelate to cluster structures. At the same time, the general cluster structurereproduces cluster structures known for the pentagram map, T-graphs andt-embeddings. We also explain how the cluster structures behave under geometricoperations. Via the cluster structures, the TCD maps are also related to theprobabilistic dimer model. The spanning tree model and the Ising model can beobtained as special cases of the dimer model, and we investigate how thesespecial cases relate to geometry. This also leads to two new incidence theoremsin relation to quadrics and null-polarities in $\mathbb C \mathrm P^3$.Finally, we also show how TCD maps relate to the Fock-Goncharov moduli spacesof projective flag configurations.

Discrete Differential Geometry and Cluster Algebras via TCD maps