  In a remarkable recent work [arXiv : 1711.09102] by Arkani-Hamed et al, the
amplituhedron program was extended to the realm of non-supersymmetric
scattering amplitudes. In particular it was shown that for tree-level planar
diagrams in massless $\phi^{3}$ theory (and its close cousin, bi-adjoint
$\phi^{3}$ theory) a polytope known as the associahedron sits inside the
kinematic space and is the amplituhedron for the theory. Precisely as in the
case of amplituhedron, it was shown that scattering amplitude is nothing but
residue of the canonical form associated to the associahedron. Combinatorial
and geometric properties of associahedron naturally encode properties like
locality and unitarity of (tree level) scattering amplitudes. In this paper we
attempt to extend this program to planar amplitudes in massless $\phi^{4}$
theory. We show that tree-level planar amplitudes in this theory can be
obtained from geometry of objects known as the Stokes polytope which sits
naturally inside the kinematic space. As in the case of associahedron we show
that residues of the canonical form on these Stokes polytopes can be used to
compute scattering amplitudes for quartic interactions. However unlike
associahedron, Stokes polytope of a given dimension is not unique and as we
show, one must sum over all of them to obtain the complete scattering
amplitude. Not all Stokes polytopes contribute equally and we argue that the
corresponding weights depend on purely combinatorial properties of the Stokes
polytopes. As in the case of $\phi^{3}$ theory, we show how factorization of
Stokes polytope implies unitarity and locality of the amplitudes.
