  Energy Correlators measure the energy deposited in multiple detectors as a
function of the angles between the detectors. In this paper, we analytically
compute the three particle correlator in the collinear limit in QCD for quark
and gluon jets, and also in $\mathcal{N}=4$ super Yang-Mills theory. We find an
intriguing duality between the integrals for the energy correlators and
infrared finite Feynman parameter integrals, which maps the angles of the
correlators to dual momentum variables. In $\mathcal{N}=4$, we use this duality
to express our result as a rational sum of simple Feynman integrals (triangles
and boxes). In QCD our result is expressed as a sum of the same transcendental
functions, but with more complicated rational functions of cross ratio
variables as coefficients. Our results represent the first analytic calculation
of a three-prong jet substructure observable of phenomenological relevance for
the LHC, revealing unexplored simplicity in the energy flow of QCD jets. They
also provide valuable data for improving the understanding of the light-ray
operator product expansion.
