During many years the problem of the Apollonius tiling has interested a great number of scientists. Many well known mathematicians and physicists have worked on this for several hundred years. These include W. L. Bragg , F. Soddy , H.S.M. Coxeter , R. Descartes , P. Beecroft , and more recently P. G. De Gennes , whose works have a great importance in Soft Matter Physics. René Descartes, in a letter of November 1643 to Princess Elisabeth of Bohemia, developed a formula relating the radii of four mutually tangent circles (Descartes’s theorem), which is: ddeeff+ddeexx+ddffxx+eeffxx=2deffxx+2deeffx+2deefxx+2ddeffx+2ddefxx+2ddeefx where d, e, f are the radii of the three externally tangent circles and where x is the radius of the fourth circle. These notations have been simplified, demonstrated by Coxeter  and generalized to the case of internal tangent circles in order to obtain:(1)1R4=1R1+1R2+1R3±21R1R2+1R1R3+1R2R3 where R1, R2, R3 denote the radii of the three initial circles and R4 the radii of the fourth circles (see Figure 1a).