We study the circles of the Apollonian band circle super-packing, which are also the images of the real projective line by all linear fractional transformations of the group PLS (2 ,Z [ i ]) . This super-packing takes place on the torus rather than on the plane. We focus first on the connection between these circles and lattices in the ring of Gaussian integers. Then we explore the group structure of the set of such circles with given curvature. The end of this paper is more geometrically-minded and describes first a test to determine whether a Gaussian circle is Apollonian or not, then a process which yields quickly all Gaussian circles with curvature smaller than a given integer. We develop algorithms for each of the computational problems we encounter.
MSC Classification: 11E25 , 11G99 , 20K01 , 52C26