  The modeling of fast flavor evolution of neutrinos in dense environments has
been traditionally carried out by relying on a two flavor approximation for
simplicity. In addition, vacuum mixing has been deemed negligible. For the
first time, we highlight that the fast flavor evolution in three flavors is
intrinsically different from the one obtained in the two flavor approximation.
This is due to the exponential growth of flavor mixing in the $e$--$\mu$ and
$e$--$\tau$ sectors generated by the vacuum term in the Hamiltonian. As a
result, substantially larger flavor mixing is found in three flavors. Our
findings highlight that the two flavor approximation is not justified for fast
pairwise conversion, even if the angular distributions of non-electron type
neutrinos are initially identical.
