  We consider nine geometric systems: Miquel dynamics, P-nets, integrable
cross-ratio maps, discrete holomorphic functions, orthogonal circle patterns,
polygon recutting, circle intersection dynamics, (corrugated) pentagram maps
and the short diagonal hyperplane map. Using a unified framework, for each
system we prove an explicit expression for the solution as a function of the
initial data; more precisely, we show that the solution is equal to the ratio
of two partition functions of an oriented dimer model on an Aztec diamond whose
face weights are constructed from the initial data. Then, we study the Devron
property [Gli15], which states the following: if the system starts from initial
data that is singular for the backwards dynamics, this singularity is expected
to reoccur after a finite number of steps of the forwards dynamics. Again,
using a unified framework, we prove this Devron property for all of the above
geometric systems, for different kinds of singular initial data. In doing so,
we obtain new singularity results and also known ones [Gli15, Yao14]. Our
general method consists in proving that these nine geometric systems are all
related to the Schwarzian octahedron recurrence (dSKP equation), and then to
rely on the companion paper [AdTM22], where we study this recurrence in
general, prove explicit expressions and singularity results.
