  This Letter presents a study of the flavor-changing neutral current radiative
$B^{\pm} \to K^{\pm}\pi^{\mp}\pi^{\pm}\gamma$ decays performed using data
collected in proton-proton collisions with the LHCb detector at $7$ and
$8\,$TeV center-of-mass energies. In this sample, corresponding to an
integrated luminosity of $3\,\text{fb}^{-1}$, nearly $14\,000$ signal events
are reconstructed and selected, containing all possible intermediate resonances
with a $K^{\pm}\pi^{\mp}\pi^{\pm}$ final state in the $[1.1, 1.9]\,$GeV/$c^{2}$
mass range. The distribution of the angle of the photon direction with respect
to the plane defined by the final-state hadrons in their rest frame is studied
in intervals of $K^{\pm}\pi^{\mp}\pi^{\pm}$ mass and the asymmetry between the
number of signal events found on each side of the plane is obtained. The first
direct observation of the photon polarization in the $b \to s\gamma$ transition
is reported with a significance of $5.2\,\sigma$.
