5 papers
Beyond the Descartes circle theorem
Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks
The Descartes circle theorem states that if four circles are mutually tangent with disjoint intersion, then their curvatures (or "bends) b_j = 1/r_j satisfy the relation (b_1 + b_2…
Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions
R. L. Graham, J. C. Lagarias, C. L. Mallows +2
This paper gives -dimensional analogues of the Apollonian circle packings in parts I and II. We work in the space $\sM_{\dd}^n$ of all -dimensional oriented Descartes configu…
Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings
R. L. Graham, J. C. Lagarias, C. L. Mallows +2
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 o…
Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group
R. L. Graham, J. C. Lagarias, C. L. Mallows +2
Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian pa…
Apollonian Circle Packings: Number Theory
R. L. Graham, J. C. Lagarias, C. L. Mallows +2
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a pack…