  Noncommutative geometry has been slowly emerging as a new paradigm of
geometry which starts from quantum mechanics. One of its key features is that
the new geometry is spectral in agreement with the physical way of measuring
distances. In this paper we present a detailed introduction with an overview on
the study of the quantum nature of space-time using the tools of noncommutative
geometry. In particular we examine the suitability of using the spectral action
as action functional for the theory. To demonstrate how the spectral action
encodes the dynamics of gravity we examine the accuracy of the approximation of
the spectral action by its asymptotic expansion in the case of the round three
sphere. We find that the two terms corresponding to the cosmological constant
and the scalar curvature term already give the full result with remarkable
accuracy. This is then applied to the physically relevant case of the product
of the three sphere by a circle where we show that the spectral action in this
case is also given, for any test function, by the sum of two terms up to an
astronomically small correction, and in particular all higher order terms
vanish. This result is confirmed by evaluating the spectral action using the
heat kernel expansion where we check that the higher order terms a4 and a6 both
vanish due to remarkable cancelations. We also show that the Higgs potential
appears as an exact perturbation when the test function used is a smooth cutoff
function.
