  We develop a new perspective on the discretization of the phase space
structure of gravity in 2+1 dimensions as a piecewise-flat geometry in 2
spatial dimensions. Starting from a subdivision of the continuum geometric and
phase space structure into elementary cells, we obtain the loop gravity phase
space coupled to a collection of effective particles carrying mass and spin,
which measure the curvature and torsion of the geometry. We show that the new
degrees of freedom associated to the particle-like elements can be understood
as edge modes, which appear in the decomposition of the continuum theory into
subsystems and do not cancel out in the gluing of cells along codimension 2
defects. These new particle-like edge modes are gravitationally dressed in an
explicit way. This provides a detailed explanation of the relations and
differences between the loop gravity phase space and the one deduced from the
continuum theory.
