  The polarization process of conventional polar codes in binary erasure
channel (BEC) is recast to the Domany-Kinzel cellular automaton model of
directed percolation in a tilted square lattice. Consequently, the former's
scaling exponent, $\mu$, can be analogously expressed as the inverse of the
percolation critical exponent, $\beta$. Relying on the vast percolation theory
literature and the best known numerical estimate for $\beta$, the scaling
exponent can be easily estimated as
$\mu_{\text{num}}^{\text{perc}}\simeq1/0.276486(8)\simeq3.617$, which is only
about $0.25\%$ away from the known exponent computation from coding theory
literature based on numerical approximation, $\mu_{\text{num}}\simeq3.627$.
Remarkably, this numerical result for the critical exponent, $\beta$, can be
analytically approximated (within only $0.028\%$) leading to the closed-form
expression for the scaling exponent
$\mu\simeq2+\varphi=2+1.618\ldots\simeq3.618$, where
$\varphi\triangleq(1+\sqrt{5})/2$ is the ubiquitous golden ratio. As the
ultimate achievable scaling exponent is quadratic, this implies that the
penalty for polar codes in BEC, in terms of the scaling exponent, can be very
well estimated by the golden ratio, $\varphi$, itself.
