  Open problems in fluid dynamics, such as the existence of finite-time
singularities (blowup), explanation of intermittency in developed turbulence,
etc., are related to multi-scale structure and symmetries of underlying
equations of motion. Significantly simplified equations of motion, called
toy-models, are traditionally employed in the analysis of such complex systems.
In such models, equations are modified preserving just a part of the structure
believed to be important. Here we propose a different approach for constructing
simplified models, in which instead of simplifying equations one introduces a
simplified configuration space: velocity fields are defined on
multi-dimensional logarithmic lattices with proper algebraic operations and
calculus. Then, the equations of motion retain their exact original form and,
therefore, naturally maintain most scaling properties, symmetries and
invariants of the original systems. Classification of such models reveals a
fascinating relation with renowned mathematical constants such as the golden
mean and the plastic number. Using both rigorous and numerical analysis, we
describe various properties of solutions in these models, from the basic
concepts of existence and uniqueness to the blowup development and turbulent
dynamics. In particular, we observe strong robustness of the chaotic blowup
scenario in the three-dimensional incompressible Euler equations, as well as
the Fourier mode statistics of developed turbulence resembling the full
three-dimensional Navier-Stokes system.
