  In this PhD thesis we develop the frame work of triple crossing diagram maps
(TCD maps), which describes constrained configurations of points in projective
spaces and discrete dynamics on these configurations. We are able to capture
the constraints and dynamics of a large list of examples that occur in discrete
differential geometry (DDG), discrete integrable systems and exactly solvable
models. We explain how to apply various geometric operations to TCD maps,
including projections, intersections with hyperplanes and projective
dualization. In fact, we show how many examples in the literature are related
by the aforementioned operations. Moreover, we introduce a hierarchy of cluster
structures on TCD maps, thus answering the open question how objects of DDG
relate to cluster structures. At the same time, the general cluster structure
reproduces cluster structures known for the pentagram map, T-graphs and
t-embeddings. We also explain how the cluster structures behave under geometric
operations. Via the cluster structures, the TCD maps are also related to the
probabilistic dimer model. The spanning tree model and the Ising model can be
obtained as special cases of the dimer model, and we investigate how these
special cases relate to geometry. This also leads to two new incidence theorems
in 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 spaces
of projective flag configurations.
