  We explore the interplay of fractal geometry and quantum entanglement by
analyzing the von Neumann entropy (known as entanglement entropy) and the
entanglement contour in the scaling limit. Focusing on free-fermion quantum
models known for their simplicity and effectiveness in studying highly
entangled quantum systems, we uncover intriguing findings. For gapless ground
states exhibiting a finite density of states at the chemical potential, we
reveal a super-area law characterized by the presence of a logarithmic
divergence in the entanglement entropy. This extends the well-established
super-area law observed on translationally invariant Euclidean lattices where
the Gioev-Klich-Widom conjecture regarding the asymptotic behavior of Toeplitz
matrices holds significant influence. Furthermore, we observe the emergence of
a self-similar and universal pattern termed an ``entanglement fractal'' in the
entanglement contour data as we approach the scaling limit. Remarkably, this
pattern bears resemblance to intricate Chinese paper-cutting designs. We
provide general rules to artificially generate this fractal, offering insights
into the universal scaling of entanglement entropy. Building upon the insights
gained from the entanglement fractal, we explicitly elucidate the origin of the
logarithmic divergence on fractals where translation symmetry is broken and the
Widom conjecture is inapplicable. For gapped ground states, we observe that the
entanglement entropy adheres to a generalized area law, with its dependence on
the Hausdorff dimension of the boundary between complementary subsystems.
