Apollonian circle packings: Geometry and group theory III. Higher dimensions Academic Article uri icon

abstract

  • This paper gives n-dimensional analogues of the Apollonian circle packings in Parts I and II. Those papers considered circle packings described in terms of their Descartes configurations, which are sets of four mutually touching circles. They studied packings that had integrality properties in terms of the curvatures and centers of the circles. Here we consider collections of n-dimensional Descartes configurations, which consist of n+2 mutually touching spheres. We work in the space MDn of all n-dimensional oriented Descartes configurations parametrized in a coordinate system, augmented curvature-center coordinates, as those (n+2) (n+2) real matrices W with WTQD,n W = QW,n where QD,n = x T2 +....+ xn+22 - (1/n)(x 1 +...+ xn+2 )2 is the n-dimensional Descartes quadratic form, QW,n = -8x1x2 + 2x 32 + ... + 2xn+22, and Q D,n and QW,n are their corresponding symmetric matrices. On the parameter space MDn of augmented curvature-center matrices, the group it Aut(QD,n) acts on the left and Aut(Q W,n) acts on the right. Both these groups are isomorphic to the (n+2)-dimensional Lorentz group O(n+1,1), and give two different "geometric" actions. The right action of Aut(QW,n) (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space n while the left action of it Aut(Q D,n) is defined only on the parameter space MQDn We introduce n-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups in it Aut(QD,n), with the following integrality properties: the dual Apollonian group consists of integral matrices in all dimensions, while the other two consist of rational matrices, with denominators having prime divisors drawn from a finite set S depending on the dimension. We show that the Apollonian group and the dual Apollonian group are finitely presented, and are Coxeter groups. We define an Apollonian cluster ensemble to be any orbit under the Apollonian group, with similar notions for the other two groups. We determine in which dimensions there exist rational Apollonian cluster ensembles (all curvatures are rational) and strongly rational Apollonian sphere ensembles (all augmented curvature-center coordinates are rational). 2005 Springer Science+Business Media, Inc.

published proceedings

  • DISCRETE & COMPUTATIONAL GEOMETRY

author list (cited authors)

  • Graham, R. L., Lagarias, J. C., Mallows, C. L., Wilks, A. R., & Yan, C. H.

citation count

  • 22

complete list of authors

  • Graham, RL||Lagarias, JC||Mallows, CL||Wilks, AR||Yan, CH

publication date

  • January 2006