  According to the holographic picture of 't Hooft and Susskind, the black hole
entropy, $S_{\rm BH}$, is carried by the chaotic microscopic degrees of
freedom, that live in the near horizon geometry and have a Hilbert space of
states of finite dimension, $d=\exp(S_{\rm BH})$. In previous work we have
proposed that the near horizon geometry, when the microscopic degrees of
freedom can be resolved, can be described by the discrete, finite, random
geometry, AdS$_2[\mathbb{Z}_N]$, where $N$ is proportional to $S_{\rm BH}$.
What had remained as an open problem was how the smooth AdS2 geometry can be
recovered, in the limit when N goes to infinity. In this contribution we
present the salient points of the solution to this problem, which involves
embedding AdS$_2[\mathbb{Z}_N]$ in a family of finite geometries,
AdS$_2^M[\mathbb{Z}_N]$, where $M$ is another integer, within 2+1 Minkowski
spacetime. In this construction $N$ and $M$ can be considered IR and UV
cutoffs. The continuum limit, corresponding to the smooth AdS$_2$ geometry, is
obtained by taking $N$ and $M$ to infinity in a correlated way, using
properties of the Fibonacci and $k$-Fibonacci sequences. This method can be
directly applied to higher-dimensional AdS spacetimes, also.
