  Ferromagnetic ground states have often been overlooked in comparison to
seemingly more interesting antiferromagnetic ground states. However, both the
physical and mathematical structure of ferromagnetic ground states are
particularly rich. We show that the highly degenerate and highly entangled
ground states of the ferromagnetic spin-1 biquadratic model are scale
invariant, originating from spontaneous symmetry breaking from ${\rm SU}(3)$ to
${\rm U}(1)\times {\rm U}(1)$ with two type-B Goldstone modes. The ground state
degeneracies are characterized as the Fibonacci-Lucas sequences -- an ancient
mathematical gem, under open and periodic boundary conditions, with the
residual entropy being non-zero. This implies that the ground state
degeneracies for this model are asymptotically the golden spiral. In addition,
sequences of degenerate ground states generated from highest and generalized
highest weight states are constructed to establish that the entanglement
entropy scales logarithmically with the block size in the thermodynamic limit,
with the prefactor being half the number of type-B Goldstone modes. The latter
in turn is identified to be the fractal dimension.
