We study Apollonian circle packings using the properties of a certain rank 4 indefinite Kac-Moody root system
Φ
\Phi
. We introduce the generating function
Z
(
s
)
Z(\mathbf {s})
of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for
Φ
\Phi
. By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of
Φ
\Phi
, with automorphic Weyl denominators, we express
Z
(
s
)
Z(\mathbf {s})
in terms of Jacobi theta functions and the Siegel modular form
Δ
5
\Delta _5
. We also show that the domain of convergence of
Z
(
s
)
Z(\mathbf {s})
is the Tits cone of
Φ
\Phi
, and discover that this domain inherits the intricate geometric structure of Apollonian packings.