  The new concept of a system of hex equations is introduced as an
overdetermined system of six five-point face-centered quad equations defined on
six vertices of a hexagon. For a consistent system of hex equations, two
variables on neighbouring vertices of the hexagon can be solved for uniquely in
terms of the other four. A consistent system of hex equations has a
well-defined unique evolution in the hexagonal lattice under suitable initial
value problems defined on a single connected staircase of points.
Multidimensional consistency for systems of hex equations is proposed in terms
of their consistency on certain polytopes which have both hexagonal and
quadrilateral faces, and specific examples are presented for the hexagonal
prism, the elongated dodecahedron, the truncated octahedron, and the
6-6-duoprism. Consistent systems of hex equations on such polytopes may be
constructed from face-centered quad equations which satisfy
consistency-around-a-face-centered-cube, in combination with regular quad
equations that satisfy consistency-around-a-cube.
