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This paper considers the planar figure of a combinatorial polytope ortessellation 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 beingsatisfied as a consequence of the others. A close connection to the birationalgroup found originally by Coble in the different context of invariants for setsof points in projective space, allows to specify precisely a determining subsetof vertices that may be freely chosen. This gives a unified geometric view ofcertain integrable discrete systems in one, two and three dimensions. Makingcontact with previous geometric accounts in the case of three dimensions, it isshown how the figure also manifests as a configuration of circles generalisingthe Clifford lattices, and how it can be applied to construct the spatialpoint-line configurations called the Desargues maps.

Coble's group and the integrability of the Gosset-Elte polytopes and tessellations