  In this paper, we consider the holograph principle emergent from
noncommutative geometry, based on the spectral action principle. We show that
under some appropriate conditions, the gravity theory on a manifold with
boundary could be equivalent to a gauge theory $SU(N)$ on the boundary. Then an
expression for $N$ with the geometrical quantities of the manifold is given.
Based on this result, we find that the volume of the manifold and the boundary
have some discrete structure. Applying the result to the black hole, we get
that the radium of the Schwarzschild black hole is quantized. We also find an
explanation why the extremal RN-black hole has zero temperature but with finite
entropy.
