  Motivated by relativistic materials, we develop a numerical scheme to support
existing or state new conjectures in the spectral optimisation of eigenvalues
of the Dirac operator, subject to infinite-mass boundary conditions. We study
the optimality of the regular polygon (respectively, disk) among all polygons
of a given number of sides (respectively, arbitrary sets), subject to area or
perimeter constraints. We consider the three lowest positive eigenvalues and
their ratios. Roughly, we find results analogous to known or expected for the
Dirichlet Laplacian, except for the third eigenvalue which does not need to be
minimised by the regular polygon (respectively, the disk) for all masses. In
addition to the numerical results, a new, mass-dependent upper bound to the
lowest eigenvalue in rectangles is proved and its extension to arbitrary
quadrilaterals is conjectured.
