Tessellations

TL;DR

This paper explores n-dimensional tessellations and polytopes, focusing on their geometric structures, symmetries, and algorithms for their construction and classification.

Contribution

It provides a comprehensive framework for constructing and classifying regular, quasi-regular, and uniform tessellations and polytopes in multiple dimensions, including symmetry group analysis.

Findings

Classification of Coxeter groups and root systems

Algorithms for constructing tessellations and polytopes

Extension to quasi-regular and uniform generalizations

Abstract

This work presents the tessellations and polytopes from the perspective of both n-dimensional geometry and abstract symmetry groups. It starts with a brief introduction to the terminology and a short motivation. In the first part, it engages in the construction of all regular tessellations and polytopes of n dimensions and extends this to the study of their quasi-regular and uniform generalizations. In the second part, the symmetries of polytopes and tessellations are considered and the Coxeter groups and their associated root systems are introduced and classified. In the last part, the algorithms developed for this work are described and their results discussed.