  In this paper, we study the discrete classical phase space of loop gravity,
which is expressed in terms of the holonomy-flux variables, and show how it is
related to the continuous phase space of general relativity. In particular, we
prove an isomorphism between the loop gravity discrete phase space and the
symplectic reduction of the continuous phase space with respect to a flatness
constraint. This gives for the first time a precise relationship between the
continuum and holonomy-flux variables. Our construction shows that the fluxes
depend on the three-geometry, but also explicitly on the connection, explaining
their non commutativity. It also clearly shows that the flux variables do not
label a unique geometry, but rather a class of gauge-equivalent geometries.
This allows us to resolve the tension between the loop gravity geometrical
interpretation in terms of singular geometry, and the spin foam interpretation
in terms of piecewise flat geometry, since we establish that both geometries
belong to the same equivalence class. This finally gives us a clear
understanding of the relationship between the piecewise flat spin foam
geometries and Regge geometries, which are only piecewise-linear flat: While
Regge geometry corresponds to metrics whose curvature is concentrated around
straight edges, the loop gravity geometry correspond to metrics whose curvature
is concentrated around not necessarily straight edges.
