Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain. It observed there exist infinitely many types of integral Apollonian packings in which all circles had integer curvatures, with the integral structure being related to the integral nature of the Apollonian group. Here we consider the action of a larger discrete group, the super-Apollonian group, also having an integral structure, whose orbits describe the Descartes quadruples of a geometric object we call a super-packing. The circles in a super-packing never cross each other but are nested to an arbitrary depth. Certain Apollonian packings and super-packings are strongly integral in the sense that the curvatures of all circles are integral and the curvature$\times$centers of all circles are integral. We show that (up to scale) there are exactly 8 different (geometric) strongly integral super-packings, and that each contains a copy of every integral Apollonian circle packing (also up to scale). We show that the super-Apollonian group has finite volume in the group of all automorphisms of the parameter space of Descartes configurations, which is isomorphic to the Lorentz group $O(3, 1)$.Comment: 37 Pages, 11 figures. The second in a series on Apollonian circle packings beginning with math.MG/0010298. Extensively revised in June, 2004. More integral properties are discussed. More revision in July, 2004: interchange sections 7 and 8, revised sections 1 and 2 to match, and added matrix formulations for super-Apollonian group and its Lorentz version. Slight revision in March 10, 200

Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)$\times$(center) is an integer vector. This series of papers explain such properties. A {\em Descartes configuration} is a set of four mutually tangent circles with disjoint interiors. We describe the space of all Descartes configurations using a coordinate system \sM_\DD consisting of those $4 \times 4$ real matrices \bW with \bW^T \bQ_{D} \bW = \bQ_{W} where \bQ_D is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 -{1/2}(x_1 +x_2 +x_3 + x_4)^2$ and \bQ_W of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. There are natural group actions on the parameter space \sM_\DD. We observe that the Descartes configurations in each Apollonian packing form an orbit under a certain finitely generated discrete group, the {\em Apollonian group}. This group consists of $4 \times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups, the dual Apollonian group and the super-Apollonian group, which have nice geometrically interpretations. We show these groups are hyperbolic Coxeter groups.Comment: 42 pages, 11 figures. Extensively revised version on June 14, 2004. Revised Appendix B and a few changes on July, 2004. Slight revision on March 10, 200

Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions

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).Comment: 37 pages. The third in a series on Apollonian circle packings beginning with math.MG/0010298. Revised and extended. Added: Apollonian groups and Apollonian Cluster Ensembles (Section 4),and Presentation for n-dimensional Apollonian Group (Section 5). Slight revision on March 10, 200

Systematic Biases in Panel Surveys Due to Differential Nonresponse.

Panel surveys involve repeated observations on the same sample units of the population. In some of these studies, systematic biases have appeared. It is shown that these systematic biases can also be he result of the characteristics of the response probabilities

Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

Apollonian circle packings arise by repeatedly filling the intersticesbetween four mutually tangent circles with further tangent circles.We observe that there exist Apollonian packings which have strong integralityproperties, in which all circles in the packing have integer curvatures andrational centers such that (curvature) $times$ (center) is an integer vector. This series of papers explain such properties. A Descartes configuration is a set of four mutually tangent circles with disjoint interiors. An Apollonian circle packing can be described in terms of the Descartes configuration it contains. We describe the space of all ordered, oriented Descartes configurations using a coordinate system $M_ D$ consisting of those $4 times 4$ real matrices $W$ with $W^T Q_{D} bW = Q_{W}$ where $Q_D$ is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 - frac{1}{2}(x_1 +x_2 +x_3 + x_4)^2$ and $Q_W$ of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. On the parameter space$M_ D$ the group $mathop{it Aut}(Q_D)$ acts on the left, and $mathop{it Aut}(Q_W)$ acts on the right, giving two different "geometric" actions. Both these groups are isomorphic to the Lorentz group $O(3, 1)$. The right action of $mathop{it Aut}(Q_W)$ (essentially) corresponds to Mobius transformations acting on the underlying Euclidean space $rr^2$ while the left action of $mathop{it Aut}(Q_D)$ is defined only on the parameter space. We observe thatthe Descartes configurations in each Apollonian packing form an orbit of a single Descartes configuration under a certain finitely generated discrete subgroup of $mathop{it Aut}(Q_D)$, which we call the Apollonian group. This group consists of $4 times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups in $mathop{it Aut}(Q_D)$, the dual Apollonian group produced from the Apollonian group by a "duality" conjugation, and the super-Apollonian group which is the group generated by the Apollonian anddual Apollonian groups together. These groups also consist of integer $4 times 4$ matrices. We show these groups are hyperbolic Coxeter groups.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41356/1/454_2005_Article_1196.pd

Minimizing the expected minimum

AbstractThe problem is to find a probability measure, supported in the unit cube, such that the projections onto each of the three coordinate planes are uniform, that minimizes the expectation of the minimum of x1,x2,x3. This is an infinite linear programming problem. Its solution by way of discrete approximation seems to be far out of reach. In this paper I describe a solution