In this article we will present pure three dimensional analytic solutions for the
Navier-Stokes and the continuity equations in Cartesian coordinates. The key idea
is the three-dimensional generalization of the well-known self-similar Ansatz of Barenblatt.
A geometrical interpretation of the Ansatz is given also. The results are the Kummer
functions or strongly related. Our final formula is compared with other results obtained
from group theoretical approaches.