  We uncover an unexpected connection between the physics of loop integrals and
the mathematics of spline functions. One loop integrands are Laplace transforms
of splines. This clarifies the geometry of the associated loop integrals, since
a $n$-node spline has support on an $n$-vertex polyhedral cone. One-loop
integrals are integrals of splines on a hyperbolic slice of the cone, yielding
polytopes in $AdS$ space. Splines thus give a geometrical counterpart to the
rational function identities at the level of the integrand. Spline technology
also allows for a clear, simple, algebraic decomposition of higher point loop
integrals in lower dimensional kinematics in terms of lower point integrals -
e.g. an hexagon integral in 2d kinematics can be written as a sum of scalar
boxes. Higher loops can also be understood directly in terms of splines - they
map onto spline convolutions, leading to an intriguing representation in terms
of hyperbolic simplices integrated over other hyperbolic simplices. We finish
with speculations on the interpretation of one-loop integrals as partition
functions, inspired by the use of splines in counting points in polytopes.
