  We describe various kaleidoscopic and self-similar aspects of the integral
Apollonian gaskets - fractals consisting of close packing of circles with
integer curvatures. Self-similar recursive structure of the whole gasket is
shown to be encoded in transformations that forms the modular group $SL(2,Z)$.
The asymptotic scalings of curvatures of the circles are given by a special set
of quadratic irrationals with continued fraction $[n+1: \overline{1,n}]$ - that
is a set of irrationals with period-2 continued fraction consisting of $1$ and
another integer $n$. Belonging to the class $n=2$, there exists a nested set of
self-similar kaleidoscopic patterns that exhibit three-fold symmetry.
Furthermore, the even $n$ hierarchy is found to mimic the recursive structure
of the tree that generates all Pythagorean triplets
