The photography method: solving pentagon equation

In the present paper, we consider two applications of the pentagon equation. The first deals with actions of flips on edges of triangulations labelled by rational functions in some variables. The second can be formulated as a system of linear equations with variables corresponding to triangles of a triangulation. The general method says that if there is some general {\em data} (say, edge lengths or areas) associated with {\em states} (say, triangulations) and a general {\em data transformation rule} (say, how lengths or areas are changed under flips) then after returning to the initial state we recover the initial data.
Comment: 9 pages, 4 figures, comments are welcome!