  We derive the nonlinear sigma model as a peculiar dimensional reduction of
Yang-Mills theory. In this framework, pions are reformulated as
higher-dimensional gluons arranged in a kinematic configuration that only
probes cubic interactions. This procedure yields a purely cubic action for the
nonlinear sigma model which exhibits a symmetry enforcing color-kinematics
duality. Remarkably, the associated kinematic algebra originates directly from
the Poincare algebra in higher dimensions. Applying the same construction to
gravity yields a new quartic action for Born-Infeld theory and, applied once
more, a cubic action for the special Galileon theory. Since the nonlinear sigma
model and special Galileon are subtly encoded in the cubic sectors of
Yang-Mills theory and gravity, respectively, their double copy relationship is
automatic.
