Circle complexes and the discrete CKP equation

TL;DR

This paper explores the geometric and algebraic structures of fundamental line complexes within various geometries, revealing new integrable systems and circle patterns related to the discrete CKP equation.

Contribution

It introduces a novel discrete integrable system linked to circle geometries and extends classical incidence theorems with new identities and patterns.

Findings

Development of a discrete integrable equation for circle patterns

Connection between line complexes and the discrete CKP equation

New incidence theorems and cross-ratio identities in hypercomplex settings

Abstract

In the spirit of Klein's Erlangen Program, we investigate the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix which constitute associated Pl\"ucker coordinates. Particular emphasis is put on the restriction to Lie circle geometry which is intimately related to the master dCKP equation of discrete integrable systems theory. The geometric interpretation, construction and integrability of fundamental line complexes in M\"obius, Laguerre and hyperbolic geometry are discussed in detail. In the process, we encounter various avatars of classical and novel incidence theorems and associated cross- and multi-ratio identities for particular hypercomplex numbers. This leads to a discrete integrable equation which, in the context of M\"obius geometry, governs novel doubly hexagonal circle patterns.