  The standard toolkit of operators to probe quanta of geometry in loop quantum
gravity consists in area and volume operators as well as holonomy operators.
New operators have been defined, in the U(N) framework for intertwiners, which
allow to explore the finer structure of quanta of geometry. However these
operators do not carry information on the global shape of the intertwiners.
Here we introduce dual multipole moments for continuous and discrete surfaces,
defined through the normal vector to the surface, taking special care to
maintain parametrization invariance. These are raised to multipole operators
probing the shape of quantum surfaces. Further focusing on the quadrupole
moment, we show that it appears as the Hessian matrix of the large spin
Gaussian approximation of coherent intertwiners, which is the standard method
for extracting the semi-classical regime of spinfoam transition amplitudes.
This offers an improvement on the usual loop quantum gravity techniques, which
mostly focus on the volume operator, in the perspective of modeling (quantum)
gravitational waves as shape fluctuations waves propagating on spin network
states.
