  We initiate an exploration of the conformal bootstrap for $n>4$ point
correlation functions. Here we bootstrap correlation functions of the lightest
scalar gauge invariant operators in planar non-abelian conformal gauge theories
as their locations approach the cusps of a null polygon. For that we consider
consistency of the OPE in the so-called snowflake channel with respect to
cyclicity transformations which leave the null configuration invariant. For
general non-abelian gauge theories this allows us to strongly constrain the OPE
structure constants of up to three large spin $J_j$ operators (and large
polarization quantum number $l_{j}$) to all loop orders. In $ \mathcal{N}=4$ we
fix them completely through the duality to null polygonal Wilson loops and the
recent origin limit of the hexagon explored by Basso, Dixon and Papathanasiou.
