Integrable lattices of hyperplanes related to billiards within confocal quadrics

TL;DR

This paper introduces a new integrable discrete system derived from ellipsoidal billiards, defined on a honeycomb lattice of hypercubes and cross polytopes, revealing complex dynamics in multiple dimensions.

Contribution

It presents a novel discrete system linked to billiard dynamics within confocal quadrics, expanding the understanding of integrable lattice models in higher dimensions.

Findings

The system is integrable and related to double reflection nets.

In 2D, the lattice is regular with dual space dynamics.

In 3D, the lattice comprises tetrahedra and cuboctahedra.

Abstract

We introduce a new discrete system that arises from ellipsoidal billiards and is closely related to the double reflection nets. The system is defined on the lattice of a uniform honeycomb consisting of rectified hypercubes and cross polytopes. In the $2$-dimensional case, the lattice is regular and it incorporates dynamics both in the original space and its dual. In the $3$-dimensional case, the lattice consists of tetrahedra and cuboctahedra.