  We review Lie polynomials as a mathematical framework that underpins the
structure of the so-called double copy relationship between gauge and gravity
theories (and a network of other theories besides). We explain how Lie
polynomials naturally arise in the geometry and cohomology of
$\mathcal{M}_{0,n}$, the moduli space of $n$ points on the Riemann sphere up to
Mobi\"us transformation. We introduce a twistorial correspondence between the
cotangent bundle $T^*_D\mathcal{M}_{0,n}$, the bundle of forms with logarithmic
singularities on the divisor $D$ as the twistor space, and $\mathcal{K}_n$ the
space of momentum invariants of $n$ massless particles subject to momentum
conservation as the analogue of space-time. This gives a natural framework for
Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering
amplitudes of gauge and gravity theories as being the corresponding Penrose
transform. In particular we show that it gives a natural correspondence between
CHY half-integrands and scattering forms, certain $n-3$-forms on
$\mathcal{K}_n$, introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also
give a generalization and more invariant description of the associahedral
$n-3$-planes in $\mathcal{K}_n$ introduced by ABHY.}
