  We explore single and multi-loop conformal integrals, such as the ones
appearing in dual conformal theories in flat space. Using Mellin amplitudes, a
large class of higher loop integrals can be written as simple
integro-differential operators on star integrals: one-loop $n$-gon integrals in
$n$ dimensions. These are known to be given by volumes of hyperbolic simplices.
We explicitly compute the five-dimensional pentagon integral in full generality
using Schl\"afli's formula. Then, as a first step to understanding higher
loops, we use spline technology to construct explicitly the $6d$ hexagon and
$8d$ octagon integrals in two-dimensional kinematics. The fully massive hexagon
and octagon integrals are then related to the double box and triple box
integrals respectively. We comment on the classes of functions needed to
express these integrals in general kinematics, involving elliptic functions and
beyond.
