The photography method: how to solve equations without writing them

We shall talk about a universal method which allows one to guess solutions of different equations which can be written differently.
Very often it leads to solutions of equations coming from physics and applicable to topology.
The simplest but structurally very important is the proof (without calculations) that the Ptolemy transformation satisfies the pentagon identity (in other words, being applied five times it gives the identity map).
Then we will tell how to "draw equations" and solve them by using "proper sense" arguments (соображения здравого смысла).
Usual logical considerations will lead to various generalisations (for example, tropical ones). The photography method is related to various branches of mathematics: cluster algebras, braids, Conway-Coxeter friezes, Stasheff polytopes, associators ets.
We shall describe some directions of further research and list the papers where this was initiated and colleagues and students working on them.