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We introduce twisted triple crossing diagram maps, collections of points inprojective space associated to bipartite graphs on the cylinder, and use themto provide geometric realizations of the cluster integrable systems ofGoncharov and Kenyon constructed from toric dimer models. Using this notion, weprovide geometric proofs that the pentagram map and the cross-ratio dynamicsintegrable systems are cluster integrable systems. We show that in appropriatecoordinates, cross-ratio dynamics is described by geometric R-matrices, whichsolves the open question of finding a cluster algebra structure describingcross-ratio dynamics.

Integrable dynamics in projective geometry via dimers and triple crossing diagram maps on the cylinder