This Letter presents a study of the flavorchanging neutral current radiative $B^{\pm} \to K^{\pm}\pi^{\mp}\pi^{\pm}\gamma$ decays performed using data collected in protonproton collisions with the LHCb detector at $7$ and $8\,$TeV centerofmass 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 finalstate 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 $5.2\,\sigma$.
Mass distribution of the selected $ B ^+ \!\rightarrow K ^+ \pi ^ \pi ^+ \gamma $ candidates. The blue solid curve shows the fit results as the sum of the following components: signal (red solid), combinatorial background (green dotted), missing pion background (black dashed) and other partially reconstructed backgrounds (purple dashdotted). 
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Backgroundsubtracted $ K ^+\pi ^\pi ^+$ mass distribution of the $ B ^+ \!\rightarrow K ^+ \pi ^ \pi ^+ \gamma $ signal. The four intervals of interest, separated by dashed lines, are shown. 
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Distributions of $\cos\hat{{\theta}}$ for $ B ^+ \!\rightarrow K ^+ \pi ^ \pi ^+ \gamma $ signal in four intervals of $ K ^+\pi ^\pi ^+$ mass. The solid blue (dashed red) curves are the result of fits allowing all (only even) Legendre components up to the fourth power. 
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Legendre coefficients obtained from fits to the normalized backgroundsubtracted $\cos\hat{{\theta}}$ distribution in the four $ K ^+\pi ^\pi ^+$ mass intervals of interest. The updown asymmetries are obtained from Eq. 4. The quoted uncertainties contain statistical and systematic contributions. The $ K ^+\pi ^\pi ^+$ mass ranges are indicated in $ {\mathrm{\,GeV\!/}c^2}$ and all the parameters are expressed in units of $10^{2}$. The covariance matrices are given in the supplementary material. 
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Legendre coefficients obtained from fits to the normalized backgroundsubtracted $\cos\hat{{\theta}}$ distribution, using the alternative normal $\vec{p}_{\pi ^ }\times\vec{p}_{\pi ^+ }$ for defining the direction of the photon, in the four $ K ^+\pi ^\pi ^+$ mass intervals of interest. The updown asymmetries are obtained from Eq. 4. The quoted uncertainties contain statistical and systematic contributions. The $ K ^+\pi ^\pi ^+$ mass ranges are indicated in $ {\mathrm{\,GeV\!/}c^2}$ and all the parameters are expressed in units of $10^{2}$. The covariance matrices are given in the supplementary material. 
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Covariance matrices (in units of $10^{3}$) for the fitted values of $c_1$, $c_2$, $c_3$ and $c_4$ of Table 1, for the four $ K ^+\pi ^\pi ^+ $ mass intervals. 
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Covariance matrices (in units of $10^{3}$) for the fitted values of $c^{\prime}_1$, $c^{\prime}_2$, $c^{\prime}_3$ and $c^{\prime}_4$ of Table 2, for the four $ K ^+\pi ^\pi ^+ $ mass intervals. 
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This ZIP file contains supplementary material for the publication LHCbPAPER2014001. The files are: LHCbPAPER2014001supplementary.pdf : An overview of the extra figures *.pdf, *.png, *.eps *.C : The figures in various formats 
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Created on 16 February 2019.Citation count from INSPIRE on 21 February 2019.