  In this paper we study a relation between two positive geometries: the
momentum amplituhedron, relevant for tree-level scattering amplitudes in
$\mathcal{N} = 4$ super Yang-Mills theory, and the kinematic associahedron,
encoding tree-level amplitudes in bi-adjoint scalar $\phi^3$ theory. We study
the implications of restricting the latter to four spacetime dimensions and
give a direct link between its canonical form and the canonical form for the
momentum amplituhedron. After removing the little group scaling dependence of
the gauge theory, we find that we can compare the resulting reduced form with
the pull-back of the associahedron form. In particular, the associahedron form
is the sum over all helicity sectors of the reduced momentum amplituhedron
forms. This relation highlights the common singularity structure of the
respective amplitudes; in particular the factorization channels, corresponding
to vanishing planar Mandelstam variables, are the same. Additionally, we also
find a relation between these canonical forms directly on the kinematic space
of the scalar theory when reduced to four spacetime dimensions by Gram
determinant constraints. As a by-product of our work we provide a detailed
analysis of the kinematic spaces relevant for the four-dimensional gauge and
scalar theories, and provide direct links between them.
