  The dual picture of quantum geometry provided by a spin network state is
discussed. From this perspective, we introduce a new operator in Loop Quantum
Gravity - the length operator. We describe its quantum geometrical meaning and
derive some of its properties. In particular we show that the operator has a
discrete spectrum and is diagonalized by appropriate superpositions of spin
network states. A series of eigenstates and eigenvalues is presented and an
explicit check of its semiclassical properties is discussed.
