  This work analyses the quantum state of twin photons and twin beams generated
by parametric-down conversion in a hexagonally poled photonic
crystal,characterized by the simultaneous presence of two nonlinear processes
sustained by two vectors of the reciprocal lattice. In those special points of
the fluorescence spectrum where the two processes coexist,we show that a
tripartite entangled state is realized, equivalent to a single parametric
process followed by a beam-splitter. By proper angle tuning a peculiar
resonance condition is reached, with a transition to a 4-mode entanglement,
dominated by the Golden Ratio of the segment $\phi= (1+\sqrt 5)/2$. A maximal
coherence between the two nonlinear processes is here estabilished, as the
overall process is shown to be equivalent to two independent parametric
processes followed by a beam-splitter. We offer an interpretation of the
occurrence of the golden ratio in this system based on an analogy between the
evolution of the light modes and the Fibonacci sequence.
