  This paper considers the planar figure of a combinatorial polytope or
tessellation identified by the Coxeter symbol $k_{i,j}$ , inscribed in a conic,
satisfying the geometric constraint that each octahedral cell has a centre.
This realisation exists, and is movable, on account of some constraints being
satisfied as a consequence of the others. A close connection to the birational
group found originally by Coble in the different context of invariants for sets
of points in projective space, allows to specify precisely a determining subset
of vertices that may be freely chosen. This gives a unified geometric view of
certain integrable discrete systems in one, two and three dimensions. Making
contact with previous geometric accounts in the case of three dimensions, it is
shown how the figure also manifests as a configuration of circles generalising
the Clifford lattices, and how it can be applied to construct the spatial
point-line configurations called the Desargues maps.
