  We propose a new diagrammatic formulation of the all-loop scattering
amplitudes/Wilson loops in planar N=4 SYM, dubbed the "momentum-twistor
diagrams". These are on-shell-diagrams obtained by gluing trivalent black and
white vertices defined in momentum twistor space, which, in the reduced diagram
case, are known to be related to diagrams in the original twistor space. The
new diagrams are manifestly Yangian invariant, and they naturally represent
factorization and forward-limit contributions in the all-loop BCFW recursion
relations in momentum twistor space, in a fashion that is completely different
from those in momentum space. We show how to construct and evaluate
momentum-twistor diagrams, and how to use them to obtain tree-level amplitudes
and loop-level integrands; in particular for the latter we identify an isolated
bubble-structure for each loop variable, arising from a forward limit, or
entangled removal of particles. From a given diagram one can directly read off
the C, D matrices via a generalized "boundary measurement"; this in turn
determines a cell in the amplituhedron associated with the amplitude, and our
diagrammatic representations of the amplitude can provide triangulations of the
amplituhedron with generally very intricate geometries. To demonstrate the
computational power of the formalism, we give explicit results for general
two-loop integrands, and the cells of the complete amplituhedron for two-loop
MHV amplitudes.
