Mathematics – Metric Geometry
Scientific paper
2000-10-31
Discrete and Computational Geometry, 35: 37-72(2006)
Mathematics
Metric Geometry
37 pages. The third in a series on Apollonian circle packings beginning with math.MG/0010298. Revised and extended. Added: Apo
Scientific paper
This paper gives $n$-dimensional analogues of the Apollonian circle packings in parts I and II. We work in the space $\sM_{\dd}^n$ of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system, ACC-coordinates, as those $(n+2) \times (n+2)$ real matrices $\bW$ with $\bW^T \bQ_{D,n} \bW = \bQ_{W,n}$ where $Q_{D,n} = x_1^2 +... + x_{n+2}^2 - \frac{1}{n}(x_1 +... + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + ... + 2x_{n+2}^2$, and $\bQ_{D,n}$ and $\bQ_{W,n}$ are their corresponding symmetric matrices. There are natural actions on the parameter space $\sM_{\dd}^n$. We introduce $n$-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups 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 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 one can find rational Apollonian cluster ensembles (all curvatures rational) and strongly rational Apollonian sphere ensembles (all ACC-coordinates rational).
Graham Ronald L.
Lagarias Jeffrey C.
Mallows Colin L.
Wilks Allan R.
Yan Chi-Hung
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