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

## TL;DR

This paper explores the geometric and integrable properties of Coxeter-symboled polytopes and tessellations inscribed in conics, revealing connections to Coble's birational group, circle configurations, and discrete systems.

## Contribution

It provides a unified geometric framework linking Coxeter polytopes, Coble's birational group, and integrable discrete systems across multiple dimensions.

## Key findings

- Existence of movable Coxeter inscribed figures satisfying geometric constraints.
- Identification of a subset of vertices determining the entire configuration.
- Connection to circle configurations and Desargues maps in projective geometry.

## Abstract

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.