  A new link between tetrahedra and the group SU(2) is pointed out: by
associating to each face of a tetrahedron an irreducible unitary SU(2)
representation and by imposing that the faces close, the concept of quantum
tetrahedron is seen to emerge. The Hilbert space of the quantum tetrahedron is
introduced and it is shown that, due to an uncertainty relation, the ``geometry
of the tetrahedron'' exists only in the sense of ``mean geometry''. A
kinematical model of quantum gauge theory is also proposed, which shares the
advantages of the Loop Representation approach in handling in a simple way
gauge- and diff-invariances at a quantum level, but is completely
combinatorial. The concept of quantum tetrahedron finds a natural application
in this model, giving a possible intepretation of SU(2) spin networks in terms
of geometrical objects.
