The radii of an Apollonian circle packing are computed from the initial curvatures e.g. (-10, 18, 23, 27) solving Descartes equation $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ and using the four matrices to generate more solutions \[ \left[\begin{array}{cccc} -1 & 2 & 2 & 2 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \end{array}\right] \hspace{0.25 in} \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 2 & -1 & 2 & 2 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \end{array}\right] \hspace{0.25in} \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0\\\\ 2 & 2 & -1 & 2 \\\\ 0 & 0 & 0 & 1 \end{array}\right] \hspace{0.25in} \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0\\\\ 0 & 0 & 1 & 0 \\\\ 2 & 2 & 2&-1 \end{array}\right] \]

How to compute the centers of circles in the Apollonian circle packing? The formulas probably simplify if you use complex numbers.

Also in what sense it is the circle packing the limit set of a Kleinian group?

The radii of an Apollonian circle packing are computed from the initial curvatures e.g. (-10, 18, 23, 27) solving Descartes equation $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ and using the four matrices to generate more solutions $$ \left[\begin{array}{cccc} -1 & 2 & 2 & 2 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \end{array}\right] \hspace{0.25 in} \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 2 & -1 & 2 & 2 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \end{array}\right] \hspace{0.25in} \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0\\\\ 2 & 2 & -1 & 2 \\\\ 0 & 0 & 0 & 1 \end{array}\right] \hspace{0.25in} \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0\\\\ 0 & 0 & 1 & 0 \\\\ 2 & 2 & 2&-1 \end{array}\right] $$ How to compute the centers of circles in the Apollonian circle packing? The formulas probably simplify if you use complex numbers.

Also in what sense it is the circle packing the limit set of a Kleinian group?

The radii of an Apollonian circle packing are computed from the initial curvatures e.g. (-10, 18, 23, 27) solving Descartes equation $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ and using the four matrices to generate more solutions \[ \left[\begin{array}{cccc} -1 & 2 & 2 & 2 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \end{array}\right] \hspace{0.25 in} \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 2 & -1 & 2 & 2 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \end{array}\right] \hspace{0.25in} \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0\\\\ 2 & 2 & -1 & 2 \\\\ 0 & 0 & 0 & 1 \end{array}\right] \hspace{0.25in} \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0\\\\ 0 & 0 & 1 & 0 \\\\ 2 & 2 & 2&-1 \end{array}\right] \]

How to compute the centers of circles in the Apollonian circle packing? The formulas probably simplify if you use complex numbers.

Also in what sense it is the circle packing the limit set of a Kleinian group?

The radii of an Apollonian circle packing are computed from the initial curvatures e.g. (-10, 18, 23, 27) solving Descartes equation $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ and using the four matrices to generate more solutions \[ \left[\begin{array}{cccc} -1 & 2 & 2 & 2 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \end{array}\right] \hspace{0.25 in} \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 2 & -1 & 2 & 2 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \end{array}\right] \hspace{0.25in} \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0\\\\ 2 & 2 & -1 & 2 \\\\ 0 & 0 & 0 & 1 \end{array}\right] \hspace{0.25in} \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0\\\\ 0 & 0 & 1 & 0 \\\\ 2 & 2 & 2&-1 \end{array}\right] \]

How to compute the centers of circles in the Apollonian circle packing? The formulas probably simplify if you use complex numbers.

Also in what sense it is the circle packing the limit set of a Kleinian group?