  In [1], two of the present authors along with P. Raman attempted to extend
the Amplituhedron program for scalar field theories [2] to quartic scalar
interactions. In this paper we develop various aspects of this proposal. Using
recent seminal results in Representation theory [3,4], we show that
projectivity of scattering forms and existence of kinematic space associahedron
completely capture planar amplitudes of quartic interaction. We generalise the
results of [1] and show that for any $n$-particle amplitude, the positive
geometry associated to the projective scattering form is a convex realisation
of Stokes polytope which can be naturally embedded inside one of the ABHY
associahedra defined in [2,5]. For a special class of Stokes polytopes with
hyper-cubic topology, we show that they have a canonical convex realisation in
kinematic space as boundaries of kinematic space associahedra.
  We then use these kinematic space geometric constructions to write worldsheet
forms for $\phi^{4}$ theory which are forms of lower rank on the CHY moduli
space. We argue that just as in the case of bi-adjoint $\phi^3$ scalar
amplitudes, scattering equations can be used as diffeomorphisms between certain
$\frac{n-4}{2}$ forms on the worldsheet and $\frac{n-4}{2}$ forms on ABHY
associahedron that generate quartic amplitudes.
