  We propose a framework for understanding the fragmentation criterion for
self-gravitating discs which, in contrast to studies that emphasise the
`gravoturbulent' nature of such discs, instead focuses on the properties of
their quasi-regular spiral structures. Within this framework there are two
evolutionary paths to fragmentation: i) collapse on the free-fall time, which
requires that the ratio of cooling time to dynamical time ($\beta$) $< 3$ and
ii) quasistatic collapse on the cooling time at a rate that is sufficiently
fast that fragments are compact enough to withstand disruption when they
encounter spiral features in the disc.
  We perform 2D grid simulations which demonstrate numerically converged
fragmentation at $\beta < 3$ (in good agreement with Paardekooper et al. (2011)
and others) and argue that this is a consequence of the fact that such
simulations smooth the gravitational force on the scale $H$, the scale height
of the disc. Such simulations thus only allow fragmentation via route i) above
since they suppress the quasistatic contraction of fragments on scales $< H$;
the inability of fragments to contract to significantly smaller scales then
renders them susceptible to disruption at the next spiral arm encounter.
  On the other hand, 3D simulations indeed show fragmentation at higher $\beta$
via route ii). We derive an analytic prediction of fragmentation by route ii)
when $\beta \lesssim 12$, based on the requirement that fragments must contract
sufficiently to withstand disruption by spiral arms. We also discuss the
necessary numerical requirements on both grid based and SPH codes if they are
to model fragmentation via route ii).
