  A surprising connection has recently been made between the amplitudes for
Tr($\Phi^3$) theory and the non-linear sigma model (NLSM). A simple shift of
kinematic variables naturally suggested by the associahedron/stringy
representation of Tr$(\Phi^3$) theory yields pion amplitudes at all loops. In
this note we provide an elementary motivation and proof for this link going in
the opposite direction, starting from the non-linear sigma model and
discovering its formulation as a sum over triangulations of surfaces with
simple numerator factors. This uses an ancient connection between "circles" and
"triangles", interpreting the equation $y = \sqrt{1 - x^2}$ both as
parametrizing points on a circle as well as generating the number of
triangulations of polygons. A further simplification of the numerator factors
exposes them as arising from the kinematically shifted Tr($\Phi^3$) theory, and
gives rise to novel tropical representations of NLSM amplitudes. The connection
to Tr$(\Phi^3)$ theory defines a natural notion of "surface-soft limit"
intrinsic to curves on surfaces. Remarkably, with this definition, the soft
limit of pion amplitudes vanishes directly at the level of the integrand, via
obvious pairwise cancellations. We also give simple, explicit expressions for
the multi-soft factors for tree and loop-level integrands in the limit as any
number of pions are taken "surface-soft".
