Catalan Solids Derived From 3D-Root Systems and Quaternions

TL;DR

This paper explores the construction of Catalan solids as duals of Archimedean solids using quaternion representations of Coxeter-Dynkin diagrams, simplifying calculations without computer assistance.

Contribution

It introduces a quaternion-based method to derive vertices and faces of Catalan solids from 3D-root systems, avoiding complex computational tools.

Findings

Vertices obtained from Weyl group orbits on fundamental weights.

Faces represented by orbits from specific weights.

Quaternion representations simplify calculations.

Abstract

Catalan Solids are the duals of the Archimedean solids, vertices of which can be obtained from the Coxeter-Dynkin diagrams A3, B3 and H3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A3), W(B3) and W(H3) acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result as the orbits derived from fundamental weights. The Platonic solids are dual to each others however duals of the Archimedean solids are the Catalan solids whose vertices can be written as the union of the orbits, up to some scale factors, obtained by applying the above Weyl groups on the fundamental highest weights (100), (010), (001) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011) and (111) which correspond to the vertices of the Archimedean solids. Representations of the Weyl groups W(A3), W(B3) and W(H3) by the quaternions simplify the calculations with no reference to the computer calculations.