  We propose a novel inverse-design method that enables brute-force discovery
of photonic crystal (PhC) structures with complex spectral degeneracies. As a
proof of principle, we demonstrate PhCs exhibiting third-order Dirac points
formed by the \emph{accidental} degeneracy of modes of monopolar, dipolar, and
quadrupolar nature. We show that under suitable conditions, these modes can
coalesce and form a third-order exceptional point (EP3), leading to diverging
Petermann factors. We show that the spontaneous emission (SE) rate of emitters
at such EP3s, related to the local density of states, can be enhanced by a
factor of 8 in purely lossy (passive) structures, with larger enhancements
$\sim \sqrt{n^3}$ possible at exceptional points of higher order $n$ or in
materials with gain.
