Geometry of multidimensional Farey summation algorithm and frieze patterns

TL;DR

This paper introduces a geometric framework for multidimensional Farey summation algorithms, extending classical concepts to higher dimensions and applying them to classify polyhedra and generalize frieze patterns with new algebraic relations.

Contribution

It develops a novel geometric approach to multidimensional Farey summation, introduces Farey polyhedra and sails, and generalizes Conway-Coxeter frieze patterns to higher dimensions.

Findings

Defined Farey polyhedra and their duality properties.

Classified Farey polyhedra using prismatic diagrams.

Established multidimensional frieze patterns satisfying Ptolemy relations.

Abstract

In this paper we develop a new geometric approach to subtractive continued fraction algorithms in high dimensions. We adapt a version of Farey summation to the geometric techniques proposed by F. Klein in 1895. More specifically we introduce Farey polyhedra and their sails that generalise respectively Klein polyhedra and their sails, and show similar duality properties of the Farey sail integer invariants. The construction of Farey sails is based on the multidimensional generalisation of the Farey tessellation provided by a modification of the continued fraction algorithm introduced by R. W. J. Meester. We classify Farey polyhedra in the combinatorial terms of prismatic diagrams. Prismatic diagrams extend boat polygons introduced by S. Morier-Genoud and V. Ovsienko in the two-dimensional case. As one of the applications of the new theory we get a multidimensional version of Conway-Coxeter frieze patterns. We show that multidimensional frieze patterns satisfy generalised Ptolemy relations.