  Predictive theory to geometrically engineer devices and materials in
continuum systems to have desired topological-like effects is developed here by
bridging the gap between quantum and continuum mechanical descriptions. A
platonic crystal, a bosonic-like system in the language of quantum mechanics,
is shown to exhibit topological valley modes despite the system having no
direct physical connection to quantum effects. We emphasise a predictive,
first-principle, approach, the strength of which is demonstrated by the ability
to design well-defined broadband edge states, resistant to backscatter, using
geometric differences; the mechanism underlying energy transfer around gentle
and sharp corners is described. Using perturbation methods and group theory,
several distinct cases of symmetry-induced Dirac cones which when gapped yield
non-trivial band-gaps are identified and classified. The propagative behavior
of the edge states around gentle or sharp bends depends strongly upon the
symmetry class of the bulk media and we illustrate this via numerical
simulations.
