  We present new triangulations of the $m=4$ amplituhedron relevant for
scattering amplitudes in planar $\mathcal{N}=4$ super-Yang-Mills, obtained
directly from the combinatorial definition of the geometry. Using the "sign
flip" characterization of the amplituhedron, we reproduce the canonical forms
for the all-multiplicity next-to-maximally helicity violating (NMHV) and
next-to-next-to-maximally helicity violating ($\text{N}^2$MHV) tree-level as
well as the NMHV one-loop cases, without using any input from traditional
amplitudes methods. Our results provide strong evidence for the equivalence of
the original definition of the amplituhedron \cite{Arkani-Hamed:2013jha} and
the topological one \cite{Arkani-Hamed:2017vfh}, and suggest a new path forward
for computing higher loop amplitudes geometrically. In particular, we realize
the NMHV one-loop amplituhedron as the intersection of two amplituhedra of
lower dimensionality, which is reflected in the novel structure of the
corresponding canonical form.
