  We create a passive wave splitter, created purely by geometry, to engineer
three-way beam splitting in electromagnetism in transverse electric
polarisation. We do so by considering arrangements of Indium Phosphide
dielectric pillars in air, in particular we place several inclusions within a
cell that is then extended periodically upon a square lattice. Hexagonal
lattice structures more commonly used in topological valleytronics but, as we
discuss, three-way splitting is only possible using a square, or rectangular,
lattice. To achieve splitting and transport around a sharp bend we use
accidental, and not symmetry-induced, Dirac cones. Within each cell pillars are
either arranged around a triangle or square; we demonstrate the mechanism of
splitting and why it does not occur for one of the cases. The theory is
developed and full scattering simulations demonstrate the effectiveness of the
proposed designs.
