Circle packings in different geometries (Q916095)

This paper contains a discussion of how the three classical geometries of constant curvature (the sphere, the plane and the unit disc) control the possible circle packings in that geometry. A circle packing has degree k if each circle in the packing is tangent to exactly k other circles, and the main result is that degree k circle packings exist in the sphere if and only if \(k\leq 5\), in the plane if and only if \(k=6\), and in the disc if and only if \(k\geq 7\). In each case, the packing is unique up to an automorphism of the geometry.